Definite Integrals of Special Functions

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Table of Integrals, Series, and Products. http://dx.doi.org/10.1016/B978-0-12-384933-5.00006-0 c 2015 Elsevier Inc. All rights reserved. Copyright

6–7 Deﬁnite Integrals of Special Functions 6.1 Elliptic Integrals and Functions Notation: k =

√ 1 − k 2 (cf. 8.1).

6.11 Forms containing F (x, k)

6.111 6.112

π/2

0

π/2

1. 0

π/2

2. 0

π/2

3. 0

6.113

π/2

1. 0

π/2

2. 0

F (x, k) cot x dx =

π 1 K(k ) + ln k K(k) 4 2

BI (350)(1)

√ (1 + k) k π 1 sin x cos x K(k) ln + K(k ) dx = F (x, k) 4k 2 16k 1 + k sin2 x F (x, k)

2 1 sin x cos x π √ − K(k) ln K(k ) dx = 4k 1 − k sin2 x (1 − k) k 16k

F (x, k)

1 sin x cos x dx = − 2 ln k K(k) 2k 1 − k 2 sin2 x

F (x, k ) F (x, k)

2 1 sin x cos x dx √ K(k ) ln = 4(1 − k) (1 + k) k cos2 x + k sin2 x

dx sin x cos x · 1 − k 2 sin2 t sin2 x 1 − k 2 sin2 x =−

BI (350)(6)

BI (350)(7)

BI (350)(2)a, BY(802.12)a

BI (350)(5)

π 1 K(k) arctan (k F (t, k) tan t) − k 2 sin t cos t 2 BI (350)(12)

6.114

6.115

dx 1 K(k) K 1 − tan2 u cot2 v = 2 cos u sin v u sin2 x − sin2 u sin2 v − sin2 x 2 k = 1 − cot2 u · cot2 v BI (351)(9) √ 1 (1 + k) k π x dx 1 K(k) ln + K(k ) F (arcsin x, k) = 2 1 + kx 4k 2 16k 0 BI (466)(1) (cf. 6.112 2) v

F (x, k)

637

638

Elliptic Integrals and Functions

6.116*

This and similar formulas can be obtained from formulas 6.111–6.113 by means of the substitution x = arcsin t. π/2 sin x cos x 6.116* dx F (x, k) 2 0 (1 + k 2 sinh μ sin2 x) 1 − k 2 sin2 x

−1 π K (k) arctanh(k tanh μ) − F (φ, k ) = 2 k sinh μ cosh μ 2 k = 1 − k 2 , φ = arcsin(tanh μ), 0 < tanh μ < 1, 0 < k < 1 KM (4.48) 6.117*

sin x cos x dx (1 − k 2 cosh ν sin2 x) 1 − k 2 sin2 x

1 tanh ν π = 2 K (k)arctanh F (φ, k − ) k sinh ν cosh ν k 2

tanh ν 2 , 0 < tanh ν < k < 1 k = 1 − k , φ = arcsin k F (x, k)

0

6.118*

π/2

KM (4.49)

π/2

sin x cos x dx (1 − ψ sin2 x) 1 − k 2 sin2 x

1 tan ψ π = 2 K (k)arctan F (β, k) − k sin ψ cos ψ k 2 F (x, k)

0

2

k2

cos2

k = 1 − k2 ,

β = arctan

tan ψ k

,

0<ψ<

π , 2

0
KM (4.50)

6.12 Forms containing E (x, k)

1 sin x cos x 2 dx = 1 + k K(k) − (2 + ln k ) E(k) BI (350)(4) 2k 2 1 − k 2 sin2 x 0 π/2 1 dx = {E(k) K(k) − ln k } 6.122 E (x, k) BI (350)(10), BY (630.02) 2 2 2 0 1 − k sin x π/2 dx sin x cos x · 6.123 E (x, k) 2 sin2 t sin2 x 1 − k 0 1 − k 2 sin2 x

π π 1 E(k) arctan (k tan t) − E (t, k) + cot t 1 − 1 − k 2 sin2 t =− 2 k sin t cos t 2 2 BI (350)(13) v 2 dx tan u 1 6.12412 E (x, k) 1− 2 = 2 cos u sin v E(k) K tan2 v 2 2 2 u sin x − sin u sin v − sin x ⎛ ⎞ k 2 sin v sin2 2u ⎠ K⎝ 1− + 2 cos u sin2 2v 2 BI (351)(10) k = 1 − cot2 u cot2 v 6.121

π/2

E (x, k)

6.142

Complete elliptic integrals

6.125∗

π/2

sin x cos x dx (1 − ν sin2 x) 1 − k 2 sin2 x 1 tanh ν π π tanh ν = 2 E (k) arctanh − [F (φ, k − ) − E(φ, k )] sinh ν cosh ν 2 2 k k

tanh ν k = 1 − k 2 , φ = arcsin , 0 < tanh ν < k < 1 KM (4.45) k

E(x, k) 0

639

k 2 cosh2

π/2

sin x cos x dx 0 (1 − ψ sin2 x) 1 − k 2 sin2 x

π tan ψ tan ψ π 1 E (k) arctan − E(β, k) + = 2 1 − 1 − k 2 cos k sin ψ cos ψ k 2 2 1 − k 2 cos2 ψ

π tan ψ , 0 < k < 1 , 0 < ψ < KM (4.46) k = 1 − k 2 , β = arctan k 2 π/2 sin x cos x ∗ dx 6.127 E(x, k) 2 2 0 (1 +k sinh μ sin2 x) 1 − k 2 sin2 x −1 π = 2 E (k) arctanh(k tanh μ) − F (φ, k ) − E(φ, k ) + tanh μ 1 + k 2 sinh2 μ k sinh μ cosh μ 2 π − coth μ 1 − 1 + k 2 sinh2 μ 2 k = 1 − k 2 , φ = arcsin (tanh μ) , 0 < tanh μ < 1, 0 < k < 1 KM (4.47) 6.126

∗

E(x, k)

k 2 cos2

6.13 Integration of elliptic integrals with respect to the modulus

1

x 1 − cos x = tan sin x 2 0 1 sin2 x + 1 − cos x 6.132 E (x, k)k dk = 3 sin x 0 1 2 1 + r sin x x − r2 Π x, r2 , 0 6.133 Π x, r , k k dk = tan − r ln 2 1 − r sin x 0 6.131

F (x, k)k dk =

BY (616.03) BY (616.04) BY (616.05)

6.14–6.15 Complete elliptic integrals 6.141

1

1. 0

1

2. 0

6.142

1. 0

1

K(k) dk = 2G

FI II 755

π2 4

BY (615.03)

K(k ) dk =

K(k) −

π dk = π ln 2 − 2G 2 k

BY (615.05)

640

2.∗

6.1437 6.144 6.145 6.146 6.147 6.148

Elliptic Integrals and Functions

π dk π = −1 2 2 k 2 0 √

1 2 1 4 1 dk 2 = Γ K(k) = K k 2 16π 4 0 1 2 π dk = K(k) 1 + k 8 0 1 dk 1 4 2 = 24 (ln 2) − π 2 K(k ) − ln k k 12 0 1 1 2 n 2 n k K(k) dk = (n − 1) k n−2 K(k) dk + 1 0 0 1 1 n k n K(k ) dk = (n − 1) k n−2 E(k) dk 1

1

0

1

2. 0

1

2. 0

4.

E(k ) dk =

π2 8

0

1 +G 2

1

1.

E(k) dk =

E(k) dk = 1 1+k

0

1

E(k) −

BY (615.08) BY (615.09) BY (615.13) BY (615.12)

[n > 1]

0

1

3.

3.

K(k) −

0

1.

6.149

6.143

π π dk = π ln 2 − 2G + 1 − 2 k 2

(E(k ) − 1)

dk = 2 ln 2 − 1 k

E(k) dk = 1 1 +k 0

1 dx 4 π E(x) π 2 2 K(x) − √ √ x ln = − a − x + 3 4 4 e 1 − x2 0 x

(see 6.152)

BY (615.11)

BY (615.02)

BY (615.04)

BY (615.06)

BY (615.07)

6.157

Complete elliptic integrals

⎡ 6.151

0

6.152 6.15312 ∗

1

2. 6.156∗

x K (ax) π √ arcsin(a) dx = 2 2a 1−x 0 1

π x E (ax) √ arcsin(a) + a 1 − a2 dx = 4a 1 − x2 0

2 a ≤1

0

1

1

1

2. 0

1

3. 0

1

4. 0

1

5.

3.

0

[K (k) − E (k)]2

dk 1 = k 2

[K (k) − E (k)]2

dk 1 π2 − = 3 k 8 2

[(2 − k 2 ) K (k) − 2 E (k)]2

dk 1 π2 − = 5 k 16 2

[(2 − k 2 ) K (k) − 2 E (k)]2

dk 1 π2 − = k7 32 6

[(2 − k 2 ) K (k) − 2 E (k)]2

π2 2 − k2 dk = 5 k k 8

(see 6.147)

4. c

1

FI II 489

2 a ≤1

2 1 1 B a + 1, [Re a > −1] 2 2 0 2 1 1 a+1 B a + 1, xa RG (0, x, 1) dx = [Re a > −1] 4a + 6 2 0 2 1 1 3 B a + 1, xa (1 − x)RD (0, x, 1) dx = 4a + 6 2 0

BY (615.14)

2 a <1

2 p <1

0

2.

[n > 1]

E (p sin x) π sin x dx = 2 2 1 − p sin x 2 1 − p2

1.

BY (615.10)

π/2

1.

6.157∗

⎥ ⎥ π2 √ ⎥ ⎥ 2 ⎦ K2 2

2 a ≤1

6.15412

1.

√ ⎢ 2 dk 1⎢ 2 4K + E(k) = ⎢ k 8⎢ 2 ⎣

K(k) − E(k) π √ 1 − 1 − a2 dk = 2a k a2 − k 2

a

0

6.155∗

⎤

1 1 (n + 2) k n E(k ) dk = (n + 1) k n K(k ) dk 0 0 1 π x E (ax) √ dx = √ 2 2 2 2 1 − a2 0 (1 − a x ) 1 − x

1.

641

xa RF (0, x, 1) dx =

DLMF (19.28.1)

DLMF (19.28.2)

[Re a > −1]

DLMF (19.28.3)

[a, b, c > 0]

DLMF (19.28.5)

∞

RD (a, b, x) dx = 6RF (a, b, c)

642

Elliptic Integrals and Functions

5. 0

1

RD (a, b, x2 c + (1 − x2 )d) dx = RJ (a, b, c, d)

∞

6. 0

∞

7. 0

RJ (a, b, c, x2 ) dx =

3π RF (ab, ac, bc) 2

6 RJ (ax, b, c, xd) dx = √ RC (d, a)RF (0, b, c) d

6.161

[a, b, c, d > 0]

DLMF (19.28.6)

[a, b, c > 0]

DLMF (19.28.7)

[a, b, c, d > 0]

DLMF (19.28.8)

6.16 The theta function 6.161

∞

1. 0

∞

2. 0

∞

3. 0

s xs−1 ϑ2 0 | ix2 dx = 2s 1 − 2−s π − 2 Γ 12 s ζ(s) s xs−1 ϑ3 0 | ix2 − 1 dx = π − 2 Γ 12 s ζ(s)

[Re s > 2]

ET I 339(20)

[Re s > 2]

ET I 339(21)

1 xs−1 1 − ϑ4 0 | ix2 dx = 1 − 21−s π − 2 s Γ 12 s ζ(s) [Re s > 2]

∞

4. 0

ET I 339(22)

1 xs−1 ϑ4 0 | ix2 + ϑ2 0 | ix2 − ϑ3 0 | ix2 dx = − (2s − 1) 21−s − 1 π − 2 s Γ 12 s ζ(s) ET I 339(24)

6.162 1.11

∞

0

∞

2. 0

3.11

∞

0

4.11

0

∞

e−ax ϑ4

e−ax ϑ1

e−ax ϑ2

e−ax ϑ3

bπ 2l

! ! iπx √ √ l ! dx = √ cosh b a cosech l a ! l2 a

[Re a > 0, ! √ √ bπ !! iπx l dx = − √ sinh b a sech l a ! 2 2l l a

|b| ≤ l]

ET I 224(1)a

[Re a > 0, ! √ √ (l + b)π !! iπx l dx = − √ sinh b a sech l a ! l2 2l a

|b| ≤ l]

ET I 224(2)a

[Re a > 0, ! √ √ (l + b)π !! iπx l dx = √ cosh b a cosech l a ! 2 2l l a

|b| ≤ l]

ET I 224(3)a

[Re a > 0,

|b| ≤ l]

ET I 224(4)a

6.165

6.16310 1.12

Generalized elliptic integrals

∞

0

2.10

√ √ √ 1 √ √ e−(a−μ)x ϑ3 (π μx |iπx ) dx = √ tanh a + μ + tanh a − μ 2 a [Re a > 0]

∞

0

643

ϑ3 (iπkx | iπx) e−(k

2

+l2 )x

dx =

ET I 224(7)a

sinh 2l l (cosh 2l − cos 2k)

∞

1 ϑ4 0 | ie2x + ϑ2 0 | ie2x − ϑ3 0 | ie2x e 2 x cos(ax) dx 0

1 1 1 1 1 +ia 22 − 1 1 − 2 2 −ia π − 4 − 2 ia Γ 14 + 12 ia ζ 12 + ia = 2 [a > 0] ET I 61(11) ∞ 1 6.165 e 2 x ϑ3 0 | ie2x − 1 cos(ax) dx 0 2 1 1 1 + a2 + 14 π − 2 ia− 4 Γ 12 ia + 14 ζ ia + 12 = 2 1 + 4a ET I 61(12) [a > 0] 6.16411

6.1710 Generalized elliptic integrals 1.

Set Ωj (k) ≡

π

0

−(j+ 12 ) dφ, 1 − k 2 cos φ

j! (4m + 2j)! π αm (j) = m (64) (2j)! (2m + j)!

1 m!

2 ,

π λ= 2

(2j + 1)k 2 , 1 − k2

then

⎡

π 1 1 2 −j ⎣ −1 (2j + 1) 1 + 1 − k αm (j)k 4m = Ωj (k) = erf λ + (2j + 1)k 2 2 2k 2 m=0 $

#

1 13 2 2 1 2 −λ2 −2 λe − (2j + 1) 16 + 2 + 4 1+ λ × erf λ − √ π 3 12 k k ⎤ $

#

2 2 4 2 λe−λ + . . .⎦ 1 + λ2 + λ4 × erf λ − √ π 3 15

∞ "

while for large λ lim Ωj (k) =

j→∞

−j π k2 1 − k2 (2j + 1) # $ # $ 1 4 13 1 1 + . . . × 1 + (2j + 1)−1 1 + 2 − (2j + 1)−2 1 + + 2 2k 3 16k 2 16k 4

644

The Exponential Integral Function and Functions Generated by It

2.

6.211

Set Rμ (k, α, δ) =

π

cos2α−1 (θ/2) sin2δ−2α−1 (θ/2) dθ

, μ+ 1 [1 − k 2 cos θ] 2 0 < k < 1, Re δ > Re α > 0, Re μ > −1/2, (−1)ν 2ν μ + 12 ν Γ(α) Γ (δ − α + ν) , Mν (μ, α, δ) = ν! Γ(δ + ν) with (λ)ν = Γ(λ + ν)/ Γ(λ), and ν 2 μ + 12 ν Γ(α + ν) Γ (δ − α) , Wν (μ, α, δ) = ν! Γ(δ + ν) 0

then: • for small k; ∞ −(μ+ 12 ) " 2 ν Rμ (k, α, δ) = 1 − k 2 k / 1 − k2 Mν (μ, α, δ) ν=0

∞ −(μ+ 12 ) " 2 ν = 1 + k2 k / 1 + k2 W ν (μ, α, δ), ν=0

• for k 2 close to 1; Rμ (k, α, δ) 2 α−δ δ−α−μ− 12 = Γ(δ − α) Γ μ + α − δ + 12 Γ μ + 12 2k 1 − k2 μ+ 12 × Γ δ − α − μ − 12 Γ(α) Γ δ − μ − 12 2k 2 Re μ + α − δ + 12 not an integer 1 = 2μ+ 2 k 2μ+1 Γ μ + 12 Γ(1 − α) ×

∞ " α−δ+μ−n+ 1 2 Γ (δ − α + n) Γ(1 − α + n) Γ α − δ + μ − n + 12 n! 2k 2 / 1 − k 2 n=0 α − δ + μ + 12 = m, with m a non-negative integer

6.2–6.3 The Exponential Integral Function and Functions Generated by It 6.21 The logarithm integral

6.211 6.212

1

0

1. 0

1

li(x) dx = − ln 2

1 x dx = 0 li x

BI (79)(5)

BI (255)(1)

6.214

The logarithm integral

1

2. 0

1 li(x)xp−1 dx = − ln(p + 1) p

1

3.

li(x) 0

∞

4.

li(x) 1

6.213 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

dx 1 = − ln(q − 1) q+1 x q

[p > −1]

BI (255)(2)

[q < 1]

BI (255)(3)

[q > 1]

BI (255)(4)

1 π

1 sin (a ln x) dx = [a > 0] a ln a − 2 x 1+a 2 0 ∞

1 π 1 sin (a ln x) dx = − + a ln a [a > 0] li x 1 + a2 2 1 1 1 π

1 cos (a ln x) dx = − a [a > 0] ln a + li x 1 + a2 2 0 ∞ 1 π

1 cos (a ln x) dx = a [a > 0] ln a − li x 1 + a2 2 1 1 ln 1 + a2 dx = [a > 0] li(x) sin (a ln x) BI(479)(1), x 2a 0 1 arctan a dx =− li(x) cos (a ln x) x a 0 1 π

dx 1 [a > 0] a ln a + li(x) sin (a ln x) 2 = 2 x 1+a 2 0 ∞

dx 1 π − a ln a [a > 0] li(x) sin (a ln x) 2 = x 1 + a2 2 1 1 π

dx 1 a [a > 0] ln a − li(x) cos (a ln x) 2 = x 1 + a2 2 0 ∞ π

dx 1 ln a + li(x) cos (a ln x) 2 = − a [a > 0] x 1 + a2 2 1 $ # 1 a a 1 2 2 ln (1 + p) − p arctan li(x) sin (a ln x) xp−1 dx = 2 + a a + p2 2 1+p 0

1

li

1

12. 0

6.214

dx 1 = ln(1 − q) xq+1 q

645

1. 0

1

li(x) cos (a ln x) xp−1 dx = −

a2

1 + p2

# a arctan

1 (− ln x)p−1 dx = −π cot pπ · Γ(p) li x

[p > 0] p a + ln (1 + p)2 + a2 1+p 2

$

BI (475)(1) BI (475)(9)

BI (475)(2) BI (475)(10)

ET I 98(20)a

BI (479)(2)

BI (479)(3) BI (479)(13)

BI (479)(4) BI (479)(14)

BI (477)(1)

[p > 0]

BI (477)(2)

[0 < p < 1]

BI (340)(1)

646

The Exponential Integral Function and Functions Generated by It

∞

2. 1

6.215

1

1. 0

xp−1 dx = −2 li(x) √ − ln x

1

2.

li(x) 0

6.216 1.12

1 π (ln x)p−1 dx = − Γ(p) li x sin pπ

1

xp+1

dx √ = −2 − ln x

1

2.

π √ arcsinh p = −2 p

[0 0]

BI (444)(3)

[1 > p > 0]

BI (444)(4)

li(x)[− ln x]p−1

dx 1 = − Γ(p) x p

[0 < p < 1]

BI (444)(1)

li(x)[− ln x]p−1

dx π Γ(p) =− x2 sin pπ

[0 0, q > 0] FI II 653, NT 53(3) ∞ Γ(μ) Ei(−βx)xμ−1 dx = − μ [Re β ≥ 0, Re μ > 0] μβ 0 Ei(αx) dx = p Ei(αp) +

NT 55(7), ET I 325(10)

6.224

∞

1.

Ei(−βx)e 0

−μx

μ 1 dx = − ln 1 + μ β

[Re(β + μ) ≥ 0,

= −1/β

μ > 0]

[μ = 0] FI II 652, NT 48(8)

∞

2. 0

Ei(ax)e−μx dx = −

μ

1 ln −1 μ a

[a > 0,

Re μ > 0,

μ > a]

ET I 178(23)a, BI (283)(3)

6.225

∞

1. 0

2. 0

∞

2 Ei −x2 e−μx dx = −

2 Ei −x2 epx dx = −

π √ arcsinh μ = − μ

π √ ln μ+ 1+μ μ [Re μ > 0]

π √ arcsin p p

[1 > p > 0]

BI (283)(5), ET I 178(25)a NT 59(9)a

6.233

6.226 1. 2. 3. 4.

6.227 1. 2.

The exponential integral function

647

1 2 √ e−μx dx = − K 0 ( μ) Ei − [Re μ > 0] 4x μ 0 ∞ 2 a 2 √ e−μx dx = − K 0 (a μ) Ei [a > 0, Re μ > 0] 4x μ 0 ∞ π √ 1 −μx2 Ei (− μ) Ei − 2 e dx = [Re μ > 0] 4x μ 0 ∞ 2 1 π √ 1 √ √ √ [cos μ ci μ − sin μ si μ] Ei − 2 e−μx + 4x2 dx = 4x μ 0

∞

∞

1 1 − 2 ln(1 + μ) μ(μ + 1) μ 0 ∞ −ax ax Ei(ax) e Ei(−ax) e − dx = 0 x−b x+b 0 = π 2 e−ab Ei(−x)e−μx x dx =

MI 34 MI 34 MI 34

[Re μ > 0]

MI 34

[Re μ > 0]

MI 34

[a > 0,

b < 0]

[a > 0,

b > 0] ET II 253(1)a

6.228 1.

∞

0

Ei(−x)ex xν−1 dx = −

∞

2.

Ei(−βx)e 0

6.229 6.231

1. 2. 6.233

x

[0 < Re ν < 1]

μ Γ(ν) dx = − 2 F 1 1, ν; ν + 1; ν(β + μ)ν β+μ [|arg β| < π, Re(β + μ) > 0,

ET II 308(13)

Re ν > 0]

√ 1 1 dx √ √ √ √ 2 Ei − 2 exp −μx + 2 = 2 π (cos μ si μ − sin μ ci μ) 2 4x 4x x 0 [Re μ > 0] ∞ −x −μx 1 [a < 1, Re μ > 0] Ei(−a) − Ei −e e dx = γ(μ, a) μ − ln a

6.232

−μx ν−1

π Γ(ν) sin νπ

∞

b2 ∞ ln 1 + 2 a Ei(−ax) sin bx dx = − 2b 0 ∞ b 1 Ei(−ax) cos bx dx = − arctan b a 0

1. 0

ET II 308(14)

∞

Ei(−x)e−μx sin βx dx = −

1 β 2 + μ2

#

MI 34 MI 34

[a > 0,

b > 0]

BI (473)(1)a

[a > 0,

b > 0]

BI (473)(2)a

β β ln (1 + μ)2 + β 2 − μ arctan 2 1+μ [Re μ > |Im β|]

$

BI (473)(7)a

648

The Exponential Integral Function and Functions Generated by It

∞

2.

Ei(−x)e

−μx

0

1 cos βx dx = − 2 β + μ2

#

6.234

μ β ln (1 + μ)2 + β 2 + β arctan 2 1+μ

$

[Re μ > |Im β|] 6.234

∞

0

BI (473)(8)a

Ei(−x) ln x dx = C + 1

NT 56(10)

6.24–6.26 The sine integral and cosine integral functions 6.241

∞

1. 0

π 2p

[p ≥ q]

BI II 653, NT 54(8)

ci(px) ci(qx) dx =

π 2p

[p ≥ q]

FI II 653, NT 54(7)

∞

2. 0

si(px) si(qx) dx =

∞

3. 0

2 2

2 p − q2 p+q 1 1 ln ln si(px) ci(qx) dx = + 4q p−q 4p q4 1 = ln 2 q

[p = q] [p = q] FI II 653, NT 54(10, 12)

6.242 6.243 1. 2. 6.244 1.8

0

ci(ax) 1 2 2 dx = − [si(aβ)] + [ci(aβ)] β+x 2

∞

si(px)

∞

si(px) 0

6.245

0

1. 0

ET II 224(1)

[a > 0,

b > 0]

ET II 253(3)

∞

[a > 0]

ET II 253(2)

[p > 0,

q > 0]

BI (255)(6)

x dx π = − ci(pq) q 2 − x2 2

[p > 0,

q > 0]

BI (255)(6)

ci(px)

q2

dx π = Ei(−pq) 2 +x 2q

[p > 0,

q > 0]

BI (255)(7)

ci(px)

q2

dx π si(pq) = 2 −x 2q

[p > 0,

q > 0]

BI (255)(8)

[a > 0,

0 < Re μ < 1]

∞

2. 6.246

|arg β| < π]

x dx π = Ei(−pq) 2 +x 2

q2

∞

1. 0

[a > 0,

∞

si (a|x|) sign x dx = π ci (a|b|) −∞ x − b ∞ ci (a|x|) dx = −π sign b · si (a|b|) −∞ x − b

0

2.8

∞

si(ax)xμ−1 dx = −

Γ(μ) μπ sin μ μa 2

NT 56(9), ET I 325(12)a

6.252

The sine integral and cosine integral functions

∞

2. 0

ci(ax)xμ−1 dx = −

Γ(μ) μπ cos μaμ 2

[a > 0,

649

0 < Re μ < 1] NT 56(8), ET I 325(13)a

6.247

∞

μ 1 arctan μ β 0 ∞ 1 μ2 −μx ci(βx)e dx = − ln 1 + 2 μ β 0

1.

2. 6.248

si(βx)e−μx dx = −

[Re μ > 0]

NT 49(12), ET I 177(18)

[Re μ > 0]

NT 49(11), ET I 178(19)a

1 π Φ −1 [Re μ > 0] 1. si(x)e x dx = √ 4μ 2 μ 0

∞ 2 1 1 π Ei − [Re μ > 0] 2. ci(x)e−μx dx = 4 μ 4μ 0 % 2 2 2 & ∞ 2 π −μx 1 1 μ2 π μ e S − − si x + 6.249 dx = + C 2 μ 4 2 4 2 0 8

6.251 1. 2. 6.252

∞

−μx2

1 2 √ e−μx dx = kei (2 μ) x μ 0 ∞ 1 2 √ e−μx dx = − ker (2 μ) ci x μ 0

MI 34

[Re μ > 0]

ME 26

[Re μ > 0]

MI 34

[Re μ > 0]

MI 34

∞

si

MI 34

∞

1. 0

2 p > q2 2 p = q2 2 p < q2

π 2p π =− 4p

sin px si(qx) dx = −

=0

FI II 652, NT 50(8)

2.6

∞

0

cos px si(qx) dx = − =

1 ln 4p

1 q

p+q p−q

2

p = 0,

p2 = q 2

[p = 0] FI II 652, NT 50(10)

3. 0

∞

sin px ci(qx) dx = − =0

1 ln 4p

p2 −1 q2

2

p = 0,

p2 = q 2

[p = 0] FI II 652, NT 50(9)

650

The Exponential Integral Function and Functions Generated by It

∞

2 p > q2 2 p = q2 2 p < q2

π 2p π =− 4p

cos px ci(qx) dx = −

4. 0

6.253

=0

FI II 654, NT 50(7)

6.253

∞

0

m+1

π rm + r si(ax) sin bx dx = − 2 1 − 2r cos x + r2 4b(1 − r) (1 −mr ) m+1 π 2 + 2r − r − r =− 4b(1 − r) (1 − r2 ) πrm+1 =− 2b(1 − r2) − r) (1 m+1 π 1+r−r =− 2b(1 − r) (1 − r2 )

[b = a − m] [b = a + m] [a − m − 1 < b < a − m] [a + m < b < a + m + 1] ET I 97(10)

6.254

∞

1. 0

1 1 1 dx = L2 − L2 − ci(x) sin x x 2 2 2 2

2.12

6.255

z

log(1 − t) dt and this in turn can t 0 be expressed as L2 (z) = Φ(z, 2, 1) in terms of the Lerch function deﬁned in 9.550, with z real. ∞ dx π a π cos bx = ln if a > b > 0 si(ax) + 2 x 2 b 0 =0 if a2 ≤ b2 ET I 41(11)

where L2 (x) is the Euler dilogarithm deﬁned as L2 (z) = −

∞

1.

[cos ax ci (a|x|) + sin (a|x|) si (a|x|)] −∞

dx = −π [sign b cos ab si (a|b|) − sin ab ci (a|b|)] x−b [a > 0]

∞

2. −∞

[sin ax ci (a|x|) − sign x cos ax si (a|x|)]

dx = −π [sin (a|b|) si (a|b|) + cos ab ci (a|b|)] x−b [a > 0]

6.256

∞

1. 0

2.

∗

0

3.∗

0

∞

∞

2 π si (x) + ci2 (x) cos ax dx = ln(1 + a) a [si(x) cos x − ci(x) sin x]2 dx = si2 (x) cos(ax) dx =

[a > 0]

π 2

π log(1 + a) 2a

ET II 253(4)

[0 ≤ a ≤ 2]

ET II 253(5)

6.259

4.

∗

6.257 6.258

The sine integral and cosine integral functions

∞

π log(1 + a) 2a 0 ∞

√

π a sin bx dx = − J 0 2 ab si x 2b 0

ci2 (x) cos(ax) dx =

∞

si(ax) +

1. 0

651

[0 ≤ a ≤ 2] [b > 0]

ET I 42(18)

dx π sin bx 2 2 x + c2 ( π ' −bc e = [Ei(bc) − Ei(−ac)] + ebc [Ei(−ac) − Ei(−bc)] 4c π = e−bc [Ei(ac) − Ei(−ac)] 4c

[0 0]

[0 < a ≤ b,

c > 0] BI (460)(1)

∞

si(ax) +

2. 0

x dx π cos bx 2 2 x + c2 ( π ' −bc e =− [Ei(bc) − Ei(−ac)] + ebc [Ei(−bc) − Ei(−ac)] 4 π = e−bc [Ei(−ac) − Ei(ac)] 4

[0 0]

[0 < a ≤ b,

c > 0]

BI (460)(2, 5)

6.259

∞

1.

si(ax) sin bx 0

x2

∞

ci(ax) sin bx 0

ci(ax) cos bx 0

4.∗

0

c > 0]

[0 < a ≤ b,

c > 0]

x dx π = − sinh(bc) Ei(−ac) x2 + c2 2 π π = − sinh(bc) Ei(−bc) + e−bc [Ei(−bc) + Ei(bc) 2 4 − Ei(−ac) − Ei(ac)]

[0 0]

[0 < a ≤ b,

c > 0]

BI (460)(3)a, ET I 97(15)a ∞

3.

[0 < b ≤ a,

ET I 96(8)

2.

dx π Ei(−ac) sinh(bc) = 2 +c 2c π −cb = e [Ei(−bc) + Ei(bc) − Ei(−ac) − Ei(ac)] 4c π + Ei(−bc) sinh(bc) 2c

dx + c2 π = cosh bc Ei(−ac) 2c ( π ' −bc e [Ei(ac) + Ei(−ac) − Ei(bc)] + ebc Ei(−bc) = 4c

x2

[0 0]

[0 < a ≤ b,

c > 0]

BI (460)(4), ET I 41(15) ∞

[ci(x) sin x − Si(x) cos x] sin x

2 x dx 1 Ei(a)e−a − Ei(−a)ea = a2 + x2 8 [a real]

652

5.∗

The Exponential Integral Function and Functions Generated by It

∞

0

[ci(x) sin x − Si(x) cos x]

2

6.261

2 x dx π π 3 e−|a| sinh(a) − Ei(a)e−a − Ei(−a)ea = 2 2 a +x 8a 8|a| [a real]

6.261

∞

1.

si(bx) cos (ax) e

−px

0

1 2bp a p2 + (a + b)2 dx = − + p arctan 2 ln 2 (a2 + p2 ) 2 p2 + (a − b)2 b − a2 − p 2 [a > 0,

∞

2.

si(βx) cos (ax) e

−μx

0

∞

ci(bx) sin (ax) e

1.

−μx

0

∞

2. 0

3.

dx = −

2. 3. 6.264

Re μ > |Im β|]

ET I 40(9)

) * 2 2 μ + b2 − a2 + 4a2 μ2 2aμ 1 a μ arctan 2 dx = − ln 2 (a2 + μ2 ) μ + b 2 − a2 2 b4 [a > 0, b > 0, Re μ > 0] ⎡

−1 ⎣ p ln ci(bx) cos (ax) e−px dx = 2 2 2 (a + p ) 2

b 2 + p 2 − a2

2

+ 4a2 p2

b4

ET I 98(16)a

⎤ 2ap ⎦ + a arctan 2 b + p 2 − a2

[a > 0, b > 0, Re p > 0] (μ − ai)2 (μ + ai)2 ∞ ln 1 + − ln 1 + β2 β2 − ci(βx) cos (ax) e−μx dx = 4(μ + ai) 4(μ − ai) 0

π − μ ln μ [ci(x) cos x + si(x) sin x] e−μx dx = 2 1 + μ2 0 π ∞ − μ + ln μ −μx [si(x) cos x − ci(x) sin x] e dx = 2 1 + μ2 0 ∞ ln 1 + μ2 [sin x − x ci(x)] e−μx dx = 2μ2 0

1.

ET I 40(8)

μ − ai μ + ai arctan β β − 2(μ + ai) 2(μ − ai)

[a > 0, 6.263

p > 0]

arctan

[a > 0, 6.262

b > 0,

∞

0

2.

si(x) ln x dx = C + 1

∞

ci(x) ln x dx = 0

ET I 41(17)

−

∞

1.

Re μ > |Im β|]

ET I 41(16)

π 2

[Re μ > 0]

ME 26a, ET I 178(21)a

[Re μ > 0]

ME 26a, ET I 178(20)a

[Re μ > 0]

ME 26

NT 46(10) NT 56(11)

6.282

The probability integral

653

6.27 The hyperbolic sine integral and hyperbolic cosine integral functions 6.271 1.

∞

μ+1 1 1 ln = arccoth μ [Re μ > 1] 2μ μ − 1 μ 0 ∞ 1 ln μ2 − 1 2.11 chi(x)e−μx dx = − [Re μ > 1] 2μ 0

∞ 2 1 1 π Ei [p > 0] 6.27211 chi(x)e−px dx = 4 p 4p 0 6.273 ∞ ln μ [Re μ > 0] [cosh x shi(x) − sinh x chi(x)] e−μx dx = 2 1.11 μ −1 0 ∞ μ ln μ 2.11 [cosh x chi(x) + sinh x shi(x)] e−μx dx = [Re μ > 2] 1 − μ2 0

∞ 1 1 π 4μ 1 −μx2 11 e Ei − 6.274 [cosh x shi(x) − sinh x chi(x)] e dx = 4 μ 4μ 0 [Re μ > 0] 2 ∞ ln μ − 1 6.275 [x chi(x) − sinh x] e−μx dx = − [Re μ > 1] 2μ2

0 ∞ 1 1 π 1 −μx2 Ei − 6.276 [cosh x chi(x) + sinh x shi(x)] e x dx = exp 8 μ3 4μ 4μ 0 [Re μ > 0] 6.277 ∞ ln μ4 − 1 −μx [Re μ > 1] [chi(x) + ci(x)] e dx = − 1. 2μ 0 ∞ μ2 + 1 1 ln 2 [Re μ > 1] 2. [chi(x) − ci(x)] e−μx dx = 2μ μ − 1 0 shi(x)e−μx dx =

MI 34 MI 34

MI 35

MI 35 MI 35

MI 35 MI 35

MI 35

MI 34 MI 35

6.28–6.31 The probability integral 6.281 1.

6

∞

0

2q−1

[1 − Φ(px)] x

Γ q + 12 dx = √ 2 πqp2q

[Re q > 0,

Re p > 0] NT 56(12), ET II 306(1)a

2.6

1−α

∞ 2b b 2α b α 1 − Φ at ± α dt = √ K 1+α (2ab) ± K 1−α (2ab) e±2ab 2α 2α t π a 0

[a > 0, 6.282

1. 0

∞

Φ(qt)e−pt dt =

2 p p 1 1−Φ exp p 2q 4q 2

b > 0,

Re p > 0,

α = 0]

|arg q| <

π 4

MO 175, EH II 148(11)

654

2.

12

The Exponential Integral Function and Functions Generated by It

∞

0

6.283 1. 2.

(μ + 1)2 1 1 1 μ+1 −μx+ 14 −Φ e 1−Φ Φ x+ dx = exp 2 2 μ 4 2

√ √ α 1 √ −1 eβx 1 − Φ αx dx = β α−β 0 ∞ √ √ q 1 √ Φ qt e−pt dt = p p +q 0

ME 27

∞

[Re α > 0,

Re β < Re α]

[Re p > 0,

Re(q + p) > 0]

ET II 307(5)

EH II 148(12)

∞ √ q 1 √ e−px dx = e−q p 1−Φ 2 x p 0

6.284

6.283

Re p > 0,

|arg q| <

π 4

EF 147(235), EH II 148(13)

6.285 1. 0

∞

∞

2.

2

[1 − Φ(x)] e−μ Φ(iat)e−a

2 2

x2

t −st

dx =

dt =

0

arctan μ √ πμ

−1 √ exp 2ai π

6.286

[Re μ > 0]

s2 4a2

∞

1. 0

∞

2. 0

[1 − Φ(βx)] e

) 1−Φ

√

μ2 x2 ν−1

2x 2

x

Γ dx =

ν +1 2 √ πνβ ν

s2 Ei − 2 4a Re s > 0,

MI 37

|arg a| <

π 4

EH II 148(14)a

2F 1

μ2 ν ν +1 ν , ; + 1; 2 2 2 2 β Re β 2 > Re μ2 ,

Re ν > 0

ET II 306(2)

* e

x2 2

ν

xν−1 dx = 2 2 −1 sec

νπ ν

Γ 2 2 [0 < Re ν < 1]

6.287 1.

∞

2

Φ(βx)e−μx x dx =

0

∞

2. 0

2

Re μ > − Re β 2 ,

β 2μ μ + β 2

[1 − Φ(βx)] e−μx x dx =

1 2μ

β 1− μ + β2

ET I 325(9)

Re μ > 0

ME 27a, ET I 176(4)

Re μ > − Re β 2 ,

Re μ > 0

NT 49(14), ET I 177(9)

3.12

0

∞

1 1 r2 r A B − α arctan + β arctan Q(rA) Q(rB) dr = exp − σ2 2σ 2 4 2π αB βA

1 1 1 = − α arctan π α

4 ∞ 1 σ 2 A2 x 1 −t2 /2 Q(x) = √ 1 − erf √ , α= e dt = , 2 1 + σ 2 A2 2π x 2

β=

B = A B=A

σ2 B 2 1 + σ2 B 2

BEA

6.295

4.

∗

The probability integral

0

A2 + 2p A2 3 2ν−2 ABΓ(ν) F1 1, ν, 1; ν + ; 2 , r e Q(Ar)Q(Br)dr = πc(1 + 2ν) 2 A + B 2 + 2p A2 + B 2

* B 2 + 2p B2 3 , +F1 1, ν, 1; ν + ; 2 LEI c = (A2 + B 2 )(A2 + B 2 + 2p)ν 2 A + B 2 + 2p A2 + B 2

∞

∞

6.288 6.289

655

2ν−1 −pr 2

2

Φ(iax)e−μx x dx =

0

∞

1.

Φ(βx)e(β

2

−μ2 )x2

ai 2μ μ − a2

x dx =

0

β 2μ (μ2 − β 2 )

a > 0,

Re μ > Re a2

Re μ2 > Re β 2 ,

MI 37a

|arg μ| <

π 4

ET I 176(5)

∞

2. 0

[1 − Φ(βx)] e(β

2

−μ2 )x2

x dx =

1 2μ(μ + β)

Re μ2 > Re β 2 ,

√ ∞ √

2 b−a √ [Re μ > −a > 0, Φ b − ax e−(a+μ)x x dx = 2(μ + a) μ + b 0

∞ 2 μ 2 i μ2 1 + eμ /4 Ei − Φ(ix)e−(μx+x ) x dx = √ π μ a 4 0 [Re μ > # $ 0] ∞ 2 2 arctan μ 1 1 [1 − Φ(x)] e−μ x x2 dx = √ − 2 2 2 π μ3 μ (μ + 1) 0 |arg μ| < π4 √ ∞ 2 dx μ+1+1 1 = ln √ = arccoth μ + 1 Φ(x)e−μx x 2 μ+1−1 0 [Re μ > 0]

|arg μ| <

π 4

ET I 177(10)

3. 6.29112 6.292

6.293 6.294

∞

1. 0

∞

2. 0

6.295 1.

2.

2 2 β 1 e−μ x x dx = 2 exp(−2βμ) 1−Φ x 2μ

2 2 dx 1 e−μ x = − Ei(−2μ) 1−Φ x x

|arg β| <

π 4,

b > a]

|arg μ| <

MI 37

MI 37

MI 37a π 4

ET I 177(11)

|arg μ| <

π 4

1 1 1 2 2 exp −μ x + 2 dx = √ [sin 2μ ci(2μ) − cos 2μ si(2μ)] 1−Φ x x πμ 0 |arg μ| < π4

∞ 1 π 1 1 exp −μ2 x2 + 2 x dx = [H1 (2μ) − Y 1 (2μ)] − 2 1−Φ x x 2μ μ 0 |arg μ| < π4

ME 27

MI 37

∞

MI 37

MI 37

656

The Exponential Integral Function and Functions Generated by It

∞

3. 0

6.296 6.297

π 1 dx 1 2 2 exp −μ x + 2 = [H0 (2μ) − Y 0 (2μ)] 1−Φ x x x 2 |arg μ| < π4

∞

2

0

∞

0

∞

2. 0

2

1−Φ

a √ 2x

−

a2 2 ax · e− 2x2 π

&

2 2 1 −aμ√2 e−μ x x dx = e 2μ4 |arg μ| < π4 , a > 0

MI 37

MI 38a

2 2 β √ 1 exp [−2 (βγ + β μ)] e(γ −μ)x x dx = √ √ 1 − Φ γx + x 2 μ μ+γ [Re β > 0, Re μ > 0]

2 b + 2ax e−bμ exp − μ2 − a2 x2 + ab x dx = 1−Φ 2x 2μ(μ + a)

∞ #

3. 0

6.299

x +a

1.

6.298

%

6.296

1−Φ

ET I 177(12)a

MI 38 [a > 0, b > 0, Re μ > 0] $

2 b − 2ax2 b + 2ax2 1 e−ab + 1 − Φ eab e−μx x dx = exp −b a2 + μ 2x 2x μ

MI 38 [a > 0, b > 0, Re μ > 0] $

∞# 2 2 b − 2ax2 b + 2ax2 1 √ − eab Φ e−(μ−a )x x dx = 2 cosh ab − e−ab Φ exp (−b μ) 2 2x 2x μ−a 0 [a > 0, b > 0, Re μ > 0] MI 38 ∞ 1 2 2 1 2 exp 2 a K ν a cosh(2νt) exp (a cosh t) [1 − Φ (a cosh t)] dt = 2 cos(νπ) 0 Re a > 0, − 12 < Re ν < 12

b2 1 1 − e− 4a2 [a > 0, b > 0] [1 − Φ(ax)] sin bx dx = b √ √ 0 ∞ 2 + a 2b 2b 1 b + a a √ + 2 arctan Φ(ax) sin bx2 dx = √ ln 2 b − a2 4 2πb b + a − a 2b 0 [a > 0, b > 0]

6.311 6.312

∞

6.313

∞

1. 0

2. 0

∞

ET I 96(4)

ET I 96(3)

⎛ α ⎞ 12 − 12 1 √ 1 α2 + β 2 2 − α sin(βx) 1 − Φ αx dx = − ⎝ 2 2 2 ⎠ β α +β

[Re α > |Im β|]

ET II 308(10)

⎛

√ cos(βx) 1 − Φ αx dx = ⎝

ET II 307(6)

⎞ 12

α − 12 1 2 2 2 2 ⎠ α + β + α α2 + β 2 [Re α > |Im β|]

ET II 307(7)

6.321

6.314

Fresnel integrals

∞

1. 0

a 1 1 dx = b−1 exp −(2ab) 2 cos (2ab) 2 sin(bx) 1 − Φ x

[Re a > 0, b > 0]

∞ 1 1 a dx = −b−1 exp −(2ab) 2 sin (2ab) 2 cos(bx) 1 − Φ x 0 [Re a > 0,

6.315

ET II 307(8)

2.

657

∞

ν−1

x

1. 0

∞

2. 0

2F 2

Re ν > −1] β2 ν ν+1 1 ν , ; , + 1; − 2 2 2 2 2 4α [Re α > 0,

[a > 0,

2 2 1 p p dx = Ei − 2 − Ei [Φ(ax) − Φ(bx)] cos px x 2 4b 4a2

0

∞

5. 0

6.316

Γ 2 + 12 ν √ πναν

∞

4.

xν−1 cos(βx) [1 − Φ(αx)] dx =

1

[Re α > 0, Re ν > 0]

2 1 b2 1 b [1 − Φ(ax)] cos bx · x dx = 2 exp − 2 − 2 1 − exp − 2 2a 4a b 4a

0

√ 1 1 x− 2 Φ a x sin bx dx = √ 2 2πb

∞

1

e2x

ET II 307(9)

Γ 1 + 12 ν β 3 ν +3 β2 ν +1 ν , + 1; , ; − sin(βx) [1 − Φ(αx)] dx = √ F 2 2 2 2 2 2 4α2 π(ν + 1)αν+1

∞

3.

b > 0]

2

0

1−Φ

x √ 2

%

sin bx dx =

ET II 307(4)

b > 0]

ET I 40(5)

[a > 0, b > 0, p > 0] * ) √ *& √ a 2b b + a 2b + a2 √ + 2 arctan ln b − a2 b − a 2b + a2

)

π b2 e 2 1−Φ 2

[a > 0, b √ 2 [b > 0]

b > 0]

6.3176 6.318

∞

2

ET I 40(6)

ET I 96(3)

ET I 96(5)

√ i π − b22 e 4a e−a x Φ(iax) sin bx dx = [b > 0] a 2 0 ∞

2 2 2 1 − e−p − √ (1 − Φ(p)) [1 − Φ(x)] si(2px) dx = πp π 0 [p > 0]

ET II 307(3)

2

ET I 96(2)

NT 61(13)a

6.32 Fresnel integrals 6.321 1. 0

∞

1 − S (px) x2q−1 dx = 2

√ 2q + 1 π 2 Γ q + 12 sin 4 √ 2q 4 πqp

0 < Re q < 32 ,

p>0

NT 56(14)a

658

The Exponential Integral Function and Functions Generated by It

∞

2. 0

6.322 1. 2. 6.323

2. 6.324 1. 2. 6.325

√

2q + 1 π 2 Γ q + 12 cos 4 √ 4 πqp2q

# p p2 1 p2 cos −C + sin S (t)e 4 2 2 4 0 # ∞

p2 1 p p2 1 cos −S − sin C (t)e−pt dt = p 4 2 2 4 0

∞

−pt

1 dt = p

0 < Re q < 32 ,

p>0

p $ 1 −S 2 2 $ p

1 −C 2 2

∞

√

S t e−pt dx =

∞

∞

1. 0

∞

2. 0

NT 56(13)a

MO 173a MO 172a

1 + sin p2 − cos p2 1 − S (x) sin 2px dx = 2 4p 0 ∞ 1 − sin p2 − cos p2 1 − C (x) sin 2px dx = 2 4p 0

12 p2 + 1 − p 2p p2 + 1 0

12 ∞

2+1+p p √ C t e−pt dt = 2p p2 + 1 0

1.

1 − C (px) x2q−1 dx = 2

6.322

EF 122(58)a

EF 122(58)a

√ π −5 2 2 S (x) sin b x dx = b =0 2 2

√ π −5 2 2 C (x) cos b2 x2 dx = b =0

[p > 0]

NT 61(12)a

[p > 0]

NT 61(11)a

0 < b2 < 1 2 b >1 ET I 98(21)a

0 < b2 < 1 2 b >1 ET I 42(22)

6.326

∞

1. 0

2. 0

∞

π 1/2 1 1 + sin p2 − cos p2 − S (x) si(2px) dx = [S (p) + C (p) − 1] − 2 8 4p [p > 0] π 1/2 1 1 − sin p2 − cos p2 − C (x) si(2px) dx = [S (p) − C (p)] − 2 8 4p [p > 0]

NT 61(15)a

NT 61(14)a

6.414

The gamma function

659

6.4 The Gamma Function and Functions Generated by It 6.41 The gamma function 6.411

6.412

12

∞

−∞

12

Γ(α + x) Γ(β − x) dx

i∞

−i∞

= −iπ21−α−β Γ(α + β)

[Re(α + β) < 1,

Im α > 0,

Im β > 0] ET II 297(3)

= iπ21−α−β Γ(α + β)

[Re(α + β) < 1,

Im α < 0,

Im β < 0] ET II 297(2)

=0

[Re(α + β) < 1,

(Im α) (Im β) < 0]

Γ(α + s) Γ(β + s) Γ(γ − s) Γ(δ − s) ds = 4πi

ET II 297(1)

Γ(α + γ) Γ(α + δ) Γ(β + γ) Γ(β + δ) Γ(α + β + γ + δ) [Re α, Re β, Re γ, Re δ > 0] ET II 302(32)

6.413

∞

1. 0

2.

6.414 1.

3.

5.

∞

Γ(α + x) dx = 0 −∞ Γ(β + x)

π Γ(a) Γ a + 12 Γ(b) Γ b + 12 Γ(a + b) 2 Γ a + b + 12 [a > 0,

b > 0]

ET II 302(27)

0 < a 1] Γ(α + β − 1) −∞ Γ(α + x) Γ(β − x) ∞ Γ(γ + x) Γ(δ + x) dx = 0 −∞ Γ(α + x) Γ(β + x) [Re(α + β − γ − δ) > 1, Im γ, Im δ > 0]

4.

√

√ ! ∞! 1 1 ! Γ(a + ix) !2 ! dx = π Γ(a) Γ a + 2 Γ b − a − 2 ! ! Γ(b + ix) ! 2 Γ(b) Γ b − 12 Γ(b − a) 0

2.

2

|Γ(a + ix) Γ(b + ix)| dx =

ET II 297(5)

ET II 299(18)

∞

Γ(γ + x) Γ(δ + x) ±2π 2 i Γ(α + β − γ − δ − 1) dx = sin[π(γ − δ)] Γ(α − γ) Γ(α − δ) Γ(β − γ) Γ(β − δ) −∞ Γ(α + x) Γ(β + x)

[Re(α + β − γ − δ) > 1, Im γ < 0, Im δ < 0. In the numerator, we take the plus sign if Im γ > Im δ and the minus sign if Im γ < Im δ.] ET II 300(19) 1 ∞ π exp ± 2 π(δ − γ)i Γ(α − β − γ + x + 1) dx = Γ(α + x) Γ(β − x) Γ(γ + x) Γ(β + γ − 1) Γ 12 (α + β) Γ 12 (γ − δ + 1) −∞ [Re(β + γ) > 1, δ = α − β − γ + 1, Im δ = 0. The sign is plus in the argument if the exponential for Im δ > 0 and minus for Im δ < 0]. ET II 300(20)

660

The Gamma Function and Functions Generated by It

6.

6.415

∞

dx Γ(α + β + γ + δ − 3) = Γ(α + x) Γ(β − x) Γ(γ + x) Γ(δ − x) Γ(α + β − 1) Γ(β + γ − 1) Γ(γ + δ − 1) Γ(δ + α − 1) −∞ [Re(α + β + γ + δ) > 3]

6.415

−∞

1. −∞

R(x) dx Γ(α + x) Γ(β − x) Γ(γ + x) Γ(δ − x) =

1 Γ(α + β + γ + δ − 3) R(t) dt Γ(α + β − 1) Γ(β + γ − 1) Γ(γ + δ − 1) Γ(δ + α − 1) 0 [Re(α + β + γ + δ) > 3, R(x + 1) = R(x)] ET II 301(24)

1

R(t) cos 12 π(2t + α − β) dt R(x) dx = 0 γ+δ α+β Γ(α + x) Γ(β − x) Γ(γ + x) Γ(δ − x) −∞ Γ Γ(α + δ − 1) Γ 2 2 [α + δ = β + γ, Re(α + β + γ + δ) > 2, R(x + 1) = −R(x)]

2.

ET II 300(21)

∞

ET II 301(25)

6.42 Combinations of the gamma function, the exponential, and powers 6.421

∞

1. −∞

Γ(α + x) Γ(β − x) exp [2(πn + θ)xi] dx = 2πi Γ(α + β)(2 cos θ)−α−β exp[(β − α)iθ]

) Re(α + β) < 1,

2.

3.

4.

− π2 < θ <

π 2,

× [ηn (β) exp(2nπβi) − ηn (−α) exp(−2nπαi)] * % 0 if 12 − n Im ξ > 0 n an integer, ηn (ξ) = sign 12 − n if 12 − n Im ξ < 0 ET II 298(7)

∞

eπicx dx =0 −∞ Γ(α + x) Γ(β − x) Γ(γ + kx) Γ(δ − kx) [Re(α + β + γ + δ) > 2, c and k are real,

∞

∞

|c| > |k| + 1]

ET II 301(26)

Γ(α + x) exp[(2πn + π − 2θ)xi] dx −∞ Γ(β + x) (2 cos θ)β−α−1 = 2πi sign n + 12 exp[−(2πn + π − θ)αi + θi(β − 1)] Γ(β − α) Re(β − α) > 0, − π2 < θ < π2 , n is an integer, n + 12 Im α < 0 ET II 298(8) Γ(α + x) exp[(2πn + π − 2θ)xi] dx = 0 −∞ Γ(β + x) Re(β − α) > 0, − π2 < θ < π2 ,

n is an integer,

n+

1 2

Im α > 0

ET II 297(6)

6.422

6.422

The gamma function, the exponential, and powers

i∞

1. −i∞

Γ(s − k − λ) Γ λ + μ − s + 12 Γ λ − μ − s + 12 z s ds

γ+i∞

2. γ−i∞

γ+i∞

3.

z − k − μ Γ 12 − k + μ z λ e 2 W k,μ (z) Re λ > |Re μ| − 12 , |arg z| < 32 π ET II 302(29)

= 2πi Γ Re(k + λ) < 0,

Γ(−s) Γ(β + s)ts ds = 2πi Γ(β)(1 + t)−β

∞i

4.

Γ −∞i

Γ(s) Γ −i∞

6.

t−p 2

i∞

5.

3

c+i∞

c−i∞

2

[0 > γ > Re(1 − β),

1

|arg t| < π] EH I 256, BU 75

√ t−p−2 1 2 Γ(−t) 2 z t dt = 2πie 4 z Γ(−p) D p (z) |arg z| < 34 π, p is not a positive integer

2ν +

1 4

− s Γ 12 ν −

1 4

−s

z2 2

−c+i∞

−c−i∞

−c+i∞

−c−i∞

WH

s ds 1 1 1 3 2 = 2πi2 4 − 2 ν z − 2 e 4 z Γ 12 ν + 14 Γ 12 ν − 14 D ν (z) |arg z| < 34 π, ν = 12 , − 21 , − 32 , . . . EH II 120

[x > 0, − Re ν < c < 1]

8.

EH I 256(5)

1 −s 1 −1 Γ 2 ν + 12 s Γ 1 + 12 ν − 12 s ds = 4πi J ν (x) 2x

7.

1

Γ(α + s) Γ(−s) Γ(1 − c − s)xs ds = 2πi Γ(α) Γ(α − c + 1)Φ(α, c; x) − Re α < γ < min (0, 1 − Re c) , − 32 π < arg x < 32 π

γ−i∞

661

ν+2s 1 Γ(−ν − s) Γ(−s) − 21 iz ds = −2π 2 e 2 iνπ H (1) ν (z) |arg(−iz)| < Γ(−ν − s) Γ(−s)

1 ν+2s 1 ds = 2π 2 e− 2 iνπ H (2) ν (z) 2 iz |arg(iz)| <

π 2,

π 2,

EH II 21(34)

0 < Re ν < c

EH II 83(34)

0 < Re ν < c

EH II 83(35)

1 ν+2s i∞ x ds = 2πi J ν (x) Γ(−s) 2 [x > 0, Re ν > 0] EH II 83(36) Γ(ν + s + 1) −i∞ i∞ 5 Γ(−s) Γ(−2ν − s) Γ ν + s + 12 (−2iz)s ds = −π 2 e−i(z−νπ) sec(νπ)(2z)−ν H (1) ν (z) −i∞ 3 |arg(−iz)| < 2 π, 2ν = ±1, ±3 . . .

9. 10.

EH II 83(37)

662

The Gamma Function and Functions Generated by It

i∞

i∞

6.422

5 Γ(−s) Γ(−2ν − s) Γ ν + s + 12 (2iz)s ds = π 2 ei(z−νπ) sec(νπ)(2z)−ν H (2) ν (z) −i∞ |arg(iz)| < 32 π, 2ν = ±1, ±3 . . .

11.

EH II 84(38)

12.

Γ(s) Γ −i∞

1 2

3 3 1 − s − ν Γ 12 − s + ν (2z)s ds = 2 2 π 2 iz 2 ez sec(νπ) K ν (z) |arg z| < 32 π, 2ν = ±1, ±3, . . .

EH II 84(39)

13.

− 12 +i∞

− 12 −i∞

Γ(−s) 2s x ds = 4π s Γ(1 + s)

∞

2x

J 0 (t) dt t

[x > 0]

MO 41

i∞

Γ(α + s) Γ(β + s) Γ(−s) Γ(α) Γ(β) (−z)s ds = 2πi F (α, β; γ; z) Γ(γ + s) Γ(γ) −i∞

14.

[For arg(−z) < π, the path of integration must separate the poles of the integrand at the points s = 0, 1, 2, 3, . . . from the poles s = −α − n and s = −β − n (for n = 0, 1, 2, . . . )].

δ+i∞

15. δ−i∞

Γ(α + s) Γ(−s) 2πi Γ(α) (−z)s ds = 1 F 1 (α; γ; z) Γ(γ + s) Γ(γ) − π2 < arg(−z) < π2 , 0 > δ > − Re α, γ = 0, 1, 2, . . .

) *2 1

1 Γ 12 − s 1 z s ds = 2πiz 2 2π −1 K 0 4z 4 − Y 0 4z 4 Γ(s) −i∞

16.

i∞

[z > 0] 1 Γ λ+μ−s+ Γ λ−μ−s+ 2 s z z ds = 2πiz λ e− 2 W k,μ (z) Γ(λ − k − s + 1) −i∞ Re λ > |Re μ| − 12 ,

17.

i∞

ET II 303(33)

1 2

i∞

1 2

1 2

|arg z| <

m + 12

i∞

−i∞

j=1 q + j=m+1

Γ (bj − s)

n +

Γ (1 − aj + s)

j=1

Γ (1 − bj + s)

p + j=n+1

Γ (aj − s)

π 2

ET II 302(30)

Γ k+μ+ Γ(k − λ + s) Γ λ + μ − s + z z λ e− 2 M k,μ (z) z s ds = 2πi 1 Γ(2μ + 1) Γ μ−λ+s+ 2 −i∞ Re(k − λ) > 0, Re(λ + μ) > − 21 , |arg z| <

18.

19.

EH I 62(15), EH I 256(4)

π 2

ET II 302(31)

! ! a1 , . . . , ap ! z z s ds = 2πi G mn pq ! b1 , . . . , bq

|arg z| < m + n − 12 p − 12 q π; k = 1, . . . , n; Re bj > 0, j = 1, . . . , m

p + q < 2(m + n); Re ak < 1,

ET II 303(34)

6.433

6.423

Gamma functions and trigonometric functions

∞

1.

e−αx

0

∞

663

dx = ν e−α Γ(1 + x)

MI 39, EH III 222(16)

dx = eβα ν e−α , β MI 39, EH III 222(16) Γ(x + β + 1) 0 ∞ xm dx = μ e−α , m Γ(m + 1) 3. e−αx [Re m > −1] MI 39, EH III 222(17) Γ(x + 1) 0 ∞ xm dx = enα μ e−α , m, n Γ(m + 1) 4. e−αx MI 39, EH III 222(17) Γ(x + n + 1) 0

α+β−2 θ 1 ∞ 2 cos R(x) exp[(2πn + θ)xi] dx 1 2 = exp θ(β − α)i 6.424 R(t) exp(2πnti) dt Γ(α + x) Γ(β − x) Γ(α + β − 1) 2 −∞ 0 [Re(α + β) > 1, −π < θ < π, n is an integer, R(x + 1) = R(x)] ET II 299(16) 2.

e−αx

6.43 Combinations of the gamma function and trigonometric functions 6.431 1.12

r p+q−2 r(q − p) 2 cos sin sin rx dx 2 2 = Γ(p + q − 1) −∞ Γ(p + x) Γ(q − x)

∞

=0

[|r| > π] [r is real;

2.

∞

cos rx dx = Γ(p + x) Γ(q − x) −∞

2 cos

r p+q−2 r(q − p) cos 2 2 Γ(p + q − 1)

=0

Re(p + q) > 1]

[|r| < π]

Re(p + q) > 1]

sin(mπx) dx =0 sin(πx) Γ(α + x) Γ(β − x) −∞

[m is an even integer]

2α+β−2 Γ(α + β − 1)

[m is an odd integer] [Re(α + β) > 1]

1.

MO 10a, ET II 299(13, 14)

∞

= 6.433

MO 10a, ET II 298(9, 10)

[|r| > π] [r is real;

6.432

[|r| < π]

∞

ET II 298(11, 12)

sin π2 (β − α)

γ+δ α+β Γ Γ(α + δ − 1) 2Γ 2 2 [α + δ = β + γ, Re(α + β + γ + δ) > 2]

sin πx dx = −∞ Γ(α + x) Γ(β − x) Γ(γ + x) Γ(δ − x)

ET II 300(22)

664

The Gamma Function and Functions Generated by It

6.441

cos π2 (β − α)

γ+δ α+β Γ Γ(α + δ − 1) 2Γ 2 2 [α + δ = β + γ, Re(α + β + γ + δ) > 2]

∞

cos πx dx = −∞ Γ(α + x) Γ(β − x) Γ(γ + x) Γ(δ − x)

2.

ET II 301(23)

6.44 The logarithm of the gamma function∗ 6.441

p+1

1. p

√ ln Γ(x) dx = ln 2π + p ln p − p

1

2.

1

ln Γ(x) dx = 0

0

0

√ ln Γ(1 − x) dx = ln 2π

√ ln Γ(x + q) dx = ln 2π + q ln q − q

1

3.

FI II 784

FI II 783

[q ≥ 0]

NH 89(17), ET II 304(40)

√ z(z + 1) + z ln Γ(z + 1) − ln G(z + 1), ln Γ(x + 1) dx = z ln 2π − 2 ∞ # $

z z k z(z + 1) C z 2 + z2 2 − where G(z + 1) = (2π) exp − 1+ exp −z + 2 2 k 2k

z

4. 0

WH

k=1

n

5.

ln Γ(a + x) dx = 0

6.∗

k=0

1

0

6.442

n−1 "

[a ≥ 0, n = 1, 2, . . .] ET II 304(41)

√ √ √ C 4 π 1 ζ (2) ζ (2) ln2 Γ(x)dx = + + + C ln 2π + ln2 2π − C + 2 ln 2π 12 48 3 3 π2 2π 2

1

0

√ 1 (a + k) ln(a + k) − na + n ln 2π − n(n − 1) 2

2

2

exp(2πnxi) ln Γ(a + x) dx = (2πni)−1 [ln a − exp(−2πnai) Ei(2πnai)] [a > 0,

6.443

n = ±1, ±2, . . .]

ET II 304(38)

1

1 NH 203(5), ET II 304(42) [ln(2πn) + C] 2πn 0

1 π 1 1 1 1 ln + 2 1 + + ···+ + ln Γ(x) sin(2n + 1)πx dx = (2n + 1)π 2 3 2n − 1 2n + 1 0 ln Γ(x) sin 2πnx dx =

1. 2.

ET II 305(43)

1

3.

ln Γ(x) cos 2πnx dx = 0

4.

8

0

1

1 4n

2 ln Γ(x) cos(2n + 1)πx dx = 2 π

NH 203(6), ET II 305(44)

)

∞

" ln k 1 (C + ln 2π) + 2 2 (2n + 1) 4k 2 − (2n + 1)2

* NH 203(6)

k=2

∗ Here, we are violating our usual order of presentation of the formulas in order to make it easier to examine the integrals involving the gamma function

6.456

5.12

The incomplete gamma function

1

0

6.12

0

sin(2πnx) ln Γ(a + x) dx = −(2πn)−1 [ln a − cos(2πna) ci(2πna) − sin(2πna) si(2πna)] [a > 0,

1

665

n = 1, 2, . . .]

ET II 304(36)

cos(2πnx) ln Γ(a + x) dx = −(2πn)−1 [− sin(2πna) ci(2πna) + cos(2πna) si(2πna)] [a > 0,

n = 1, 2, . . .]

ET II 304(37)

6.45 The incomplete gamma function 6.451 1. 2. 6.452

∞

1 Γ(β)(1 + α)−β α 0 ∞ 1 1 e−αx Γ(β, x) dx = Γ(β) 1 − α (α + 1)β 0

∞

1. 0

2. 6.453 6.454

∞

e−αx γ(β, x) dx =

[β > 0]

MI 39

[β > 0]

MI 39

2 1 x2 e−μx γ ν, 2 dx = 2−ν−1 Γ(2ν)e(aμ) D −2ν (2aμ) 8a μ |arg a| <

π 4,

Re ν > − 12 ,

√ 2 2 a dx = √ e(aμ) K 14 a2 μ2 μ 3 4

1 x2 , 2 |arg a| < π4 , Re μ > 0 4 8a 0 ∞ a

1 1 √ dx = 2a 2 ν μ 2 ν−1 K ν (2 μa) e−μx Γ ν, |arg a| < π2 , Re μ > 0 x

2 0 ∞ √ α α −βx − 12 ν ν − 12 ν−1 D −ν √ e γ ν, α x dx = 2 α β Γ(ν) exp 8β 2β 0 [Re β > 0, Re ν > 0] e−μx γ

Re μ > 0

ET I 179(36) ET I 179(35)

ET I 179(32)

ET II 309(19), MI 39a

6.455

∞

1.

xμ−1 e−βx Γ(ν, αx) dx =

0

∞

2. 0

6.456 1. 2.

xμ−1 e−βx γ(ν, αx) dx =

β αν Γ(μ + ν) 1, μ + ν; μ + 1; 2F 1 μ(α + β)μ+ν α+β [Re(α + β) > 0, Re μ > 0, Re(μ + ν) > 0]

ET II 309(16)

α αν Γ(μ + ν) 1, μ + ν; ν + 1; F 2 1 ν(α + β)μ+ν α+β [Re(α + β) > 0, Re β > 0, Re(μ + ν) > 0]

ET II 308(15)

√ √ γ (2ν, α) 1 1 dx = π e−αx (4x)ν− 2 γ ν, 1 4x αν+ 2 0

√ √ ∞ π Γ (2ν, α) 1 −αx ν− 12 dx = e (4x) Γ ν, 1 4x αν+ 2 0

∞

MI 39a MI 39a

666

6.457

The Gamma Function and Functions Generated by It

6.457

√ √ γ (2ν + 1, α) 1 √ γ ν + 1, dx = π 1. e MI 39 1 x 4x αν+ 2 0

√ ∞ √ Γ (2ν + 1, α) (4x)ν 1 dx = π 2. e−αx √ Γ ν + 1, MI 39 1 4x x αν+ 2 0

2 ∞ 1 b b D 2ν−2 √ 6.458 x1−2ν exp αx2 sin(bx) Γ ν, αx2 dx = π 2 2−ν αν−1 Γ 32 − ν exp 8α 2α 0 |arg α| < 3π , 0 < Re ν < 1 2

∞

ν −αx (4x)

6.46–6.47 The function ψ(x)

6.461

ψ(t) dt = ln Γ(x) 1

ψ(α + x) dx = ln α 0 ∞ x−α [C + ψ(1 + x)] = −π cosec(πα) ζ(α) 0 1 e2πnxi ψ(α + x) dx = e−2πnαi Ei(2πnαi)

6.463 6.464

1.8

x

1

6.462

6.465

ET II 309(18)

[α > 0]

ET II 305(1)

[1 < Re α < 2]

ET II 305(6)

[α > 0;

0

1

0

k=2

1

2. 0

1 ψ(x) sin(2πnx) dx = − π 2

6.466

∞

0

[n = 1, 2, . . .]

1

0

2.12

NH 204 ET II 305(3)

−1 [ψ(α + ix) − ψ(α − ix)] sin xy dx = iπe−αy 1 − e−y [α > 0,

1.12

ET II 305(2)

) * ∞ " ln k 2 C + ln 2π + 2 ψ(x) sin πx dx = − π 4k 2 − 1 (see 6.443 4)

6.467

n = ±i, ±2, . . .]

y > 0]

ET I 96(1)

sin(2πnx) ψ(α + x) dx = − sin(2πnα) ci(2πnα) + cos(2πnα) si(2πnα) [α ≥ 0;

n = 1, 2, . . .]

ET II 305(4)

1

cos(2πnx) ψ(α + x) dx = sin(2πnα) si(2πnα) + cos(2πnα) ci(2πnα) 0

[α > 0; 6.468 6.469

1

0

1. 0

1

1 ψ(x) sin2 πx dx = − [C + ln(2π)] 2 ψ(x) sin πx cos πx dx = −

π 4

n = 1, 2, . . .]

ET II 305(5) NH 204

NH 204

6.511

2.8

Bessel functions

1

n 1 − n2 1 n−1 = ln 2 n+1

ψ(x) sin πx sin(nπx) dx = 0

667

[n is even] [n > 1 is odd] NH 204(8)a

6.471 ∞ x−α [ln x − ψ(1 + x)] dx = π cosec(πα) ζ(α) [0 < Re α < 1] 1. 0 ∞ 2. x−α [ln(1 + x) − ψ(1 + x)] dx = π cosec(πα) ζ(α) − (α − 1)−1

ET II 306(7)

0

[0 < Re α < 1] ∞

[ψ(x + 1) − ln x] cos(2πxy) dx =

3. 0

6.472 1.

∞

0

ET II 306(8)

1 [ψ(y + 1) − ln y] 2

ET II 306(12)

x−α (1 + x)−1 − ψ (1 + x) dx = −πα cosec(πα) ζ(1 + α) − α−1 [|Re α| < 1]

∞

2. 0

x−α x−1 − ψ (1 + x) dx = −πα cosec(πα) ζ(1 + α) [−2 < Re α < 0]

6.473

ET II 306(9)

∞

x−α ψ (n) (1 + x) dx = (−1)n−1

0

π Γ(α + n) ζ(α + n) Γ(α) sin πα [n = 1, 2, . . . ;

ET II 306(10)

0 < Re α < 1] ET II 306(11)

6.5–6.7 Bessel Functions 6.51 Bessel functions 6.511 ∞ 1 J ν (bx) dx = 1. b 0 ∞ νπ

1 2. Y ν (bx) dx = − tan b 2 0

[Re ν > −1,

b > 0]

[|Re ν| < 1,

b > 0]

ET II 22(3)

WA 432(7), ET II 96(1)

a

3. 0

a

4. 0

5.

0

a

J ν (x) dx = 2

∞ "

J ν+2k+1 (a)

[Re ν > −1]

ET II 333(1)

k=0

J 12 (t) dt = 2 S

√ a

J − 12 (t) dt = 2 C

√ a

WA 599(4) WA 599(3)

668

Bessel Functions

a

6. 0

a

7. 0

J 0 (x) dx = a J 0 (a) +

∞

a

∞

9. a

b

10. a

11. 0

a

I ν (x) dx = 2

[a > 0]

ET II 18(1)

[a > 0]

ET II 7(3)

[a > 0]

ET II 18(2)

[Y ν+2n+1 (b) − Y ν+2n+1 (a)]

ET II 339(46)

∞ "

(−1)n I ν+2n+1 (a)

[Re ν > −1]

ET II 364(1)

n=0 ∞

∞

13. 0

1.11

ET II 7(2)

πa [J 0 (a) H1 (a) − J 1 (a) H0 (a)] 2

J 1 (x) dx = J 0 (a) ∞ "

[a > 0]

n=0

0

6.512

J 0 (x) dx = 1 − a J 0 (a) +

Y ν (x) dx = 2

12.

πa [J 1 (a) H0 (a) − J 0 (a) H1 (a)] 2

J 1 (x) dx = 1 − J 0 (a)

8.

6.512

K 0 (ax) =

π 2a

[a > 0]

K 20 (ax) =

π2 4a

[a > 0]

μ+ν +1

∞ Γ b2 μ+ν+1 ν −μ+1 2 F

, ; ν + 1; 2 J μ (ax) J ν (bx) dx = bν a−ν−1 μ−ν +1 2 2 a 0 Γ(ν + 1) Γ 2 [a > 0, b > 0, Re(μ + ν) > −1, b < a.

For b > a, the positions of μ and ν should be reversed.]

2.7

∞

0

3.8

0

β2 β ν−n−1 Γ(ν) F ν, −n; ν − n; 2 J ν+n (αt) J ν−n−1 (βt) dt = ν−n α n! Γ(ν − n) α n 1 = (−1) 2α =0 [Re(ν) > 0]

∞

β ν−1 αν 1 = 2β

J ν (αx) J ν−1 (βx) dx =

=0

ET II 48(6)

[0 < β < α] [0 < β = α] [0 < α < β] MO 50

[β < α] [β = α] [β > α] [Re ν > 0]

WA 444(8), KU (40)a

6.513

Bessel functions

∞

4. 0

ν −ν−1

J ν+2n+1 (ax) J ν (bx) dx = b a

P (ν,0) n

669

2b2 1− 2 a

=0

[Re ν > −1 − n,

0 −1 − n,

0 < a < b] ET II 47(5)

∞

5. 0

Re ν > − 12 ,

1 2a

J ν+n (ax) Y ν−n (ax) dx = (−1)n+1

a > 0,

n = 0, 1, 2, . . . ET II 347(57)

∞ b2 b−1 ln 1 − 2 J 1 (bx) Y 0 (ax) dx = − π a 0 a ∞ " 2 J ν (x) J ν+1 (x) dx = [J ν+n+1 (a)]

6. 7.

0

8.

9

9. 10. 6.513

ET II 21(31)

[Re ν > −1]

ET II 338(37)

n=0 ∞

1 δ(b − a) a 0

∞ b2 1 ln 1 + 2 K 0 (ax) J 1 (bx) = 2b a 0

∞ b2 1 K 0 (ax) I 1 (bx) = − ln 1 − 2 2b a 0 k J n (ka) J n (kb) dk =

[n = 0, 1, . . .]

JAC 110

[a > 0,

b > 0]

[a > 0,

b > 0]

1 + ν + 2μ ∞ 2

[J μ (ax)]2 J ν (bx) dx = a2μ b−2μ−1 1 + ν − 2μ 2 0 [Γ(μ + 1)] Γ 2 ⎡ ⎛

1.

[0 −1,

∞

2. 0

b−1 Γ [J μ (ax)] K ν (bx) dx = 2 2

2μ + ν + 1 2

3. 0

∞

⎞⎤2 4a2 1 − 2 ⎟⎥ b ⎟⎥ ⎠⎦ 2

0 < 2a < b]

ET II 52(33)

*2 ) 2μ − ν + 1 4a2 −μ Γ P 1 ν− 1 1+ 2 2 2 2 b [2 Re μ > |Re ν| − 1, Re b > 2|Im a|]

ET II 138(18)

ν + 2μ + 1 eμπi Γ 4a2 4a2 2 −μ −μ P 1 ν− 1

I μ (ax) K μ (ax) J ν (bx) dx = 1 + 2 Q 1 ν− 1 1+ 2 2 2 2 2 ν − 2μ + 1 b b bΓ 2 [Re a > 0, b > 0, Re ν > −1, Re(ν + 2μ) > −1] ET II 65(20)

670

Bessel Functions

∞

4. 0

νπ

π sec P μ1 ν− 1 J μ (ax) J −μ (ax) K ν (bx) dx = 2 2 2b 2

∞

5. 0

z

6. 0

8.

0

z

z

WA 415(4)

∞

∞

∞

0

Jν

Jν

a

x a

x

[b > 2a > 0] [2a > b > 0]

√

J ν (bx) dx = b−1 J 2ν 2 ab

Y ν (bx) dx = b−1

a > 0,

√ 2 √ Y 2ν 2 ab + K 2ν 2ab π a > 0,

b > 0,

Re ν > − 12

ET II 57(9)

b > 0,

− 12 < Re ν <

3 2

Jν

a

x

1

K ν (bx) dx = b−1 e 2 i(ν+1)π K 2ν

√ √ 1 1 1 2e 4 iπ ab + b−1 e− 2 i(ν+1)π K 2ν 2e− 4 πi ab a > 0, Re b > 0, |Re ν| < 52

ET II 141(31)

∞

4. 0

0

1 b

2 b = arcsin πb 2a

J 20 (ax) J 1 (bx) =

ET II 110(12)

∞

3.

5.

WA 414(2)

[−1 < Re μ < 2]

0

Re ν > −1]

J μ (x) J 1−μ (z − x) dx = J 0 (z) − cos(z)

2.

[Re μ > −1,

WA 415(4)

0

(−1)k J μ+ν+2k+1 (z)

[−1 < Re μ < 1]

1.

∞ "

ET II 66(28)

J μ (x) J −μ (z − x) dx = sin z

0

ET II 138(21)

1 + ν + 2μ ) *2 e2μπi Γ 4a2 2 2 −μ

Q 1 ν− 1 [K μ (ax)] J ν (bx) dx = 1+ 2 2 2 1 + ν − 2μ b bΓ 2 Re a > 0, b > 0, Re 12 ν ± μ > − 21

J μ (x) J ν (z − x) dx = 2

9.

6.514

4a2 4a2 −μ 1 + 2 P 1 ν− 1 1+ 2 2 2 b b [|Re ν| < 1, Re b > 2|Im a|]

k=0

7. 0

6.514

∞

Yν

Yν

a

x a

x

J ν (bx) dx = −

2b−1 π

√

K 2ν 2 ab −

√

Y ν (bx) dx = −b−1 J 2ν 2 ab

π 2

√ Y 2ν 2 ab a > 0, b > 0, a > 0,

b > 0,

|Re ν| < |Re ν| <

1 2

1 2

ET II 62(37)a

ET II 110(14)

6.516

Bessel functions

∞

6. 0

0

0

∞

1. 0

∞

2. 0

1 √

√

1 1 1 K ν (bx) dx = −b−1 e 2 νπi K 2ν 2e 4 πi ab − b−1 e− 2 νπi K 2ν 2e− 4 πi ab x a > 0, Re b > 0, |Re ν| < 52

Kν

√

√ 3νπ 3νπ Y ν (bx) dx = −2b−1 sin ker2ν 2 ab + cos kei2ν 2 ab x 2 2 Re a > 0, b > 0, |Re ν| < 12

a

ET II 113(28)

∞

8. 6.515

a

ET II 143(37)

∞

7.

Yν

671

∞

3. 0

Kν

Jμ

a

x a

x

√

K ν (bx) dx = πb−1 K 2ν 2 ab

Yμ

a

x

[Re a > 0,

Re b > 0]

√

√

K 0 (bx) dx = −2b−1 J 2μ 2 ab K 2μ 2 ab

[a > 0, Re b > 0] a 2 1 √

√

1 Kμ K 0 (bx) dx = 2πb−1 K 2μ 2e 4 πi ab K 2μ 2e− 4 πi ab x

H (1) μ

2

a x

H (2) μ

ET II 146(54)

2

a x

ET II 143(42)

ET II 147(59) [Re a > 0, Re b > 0]

√ √

J 0 (bx) dx = 16π −2 b−1 cos μπ K 2μ 2eπi/4 a b K 2μ 2e−πi/4 a b |arg a| < π4 , b > 0, |Re μ| < 14

ET II 17(36)

6.516

∞

1. 0

∞

2. 0

3. 0

4.12

5.

∞

√ J 2ν a x J ν (bx) dx = b−1 J ν

a2 4b

√ J 2ν a x Y ν (bx) dx = −b−1 Hν

√ π J 2ν a x K ν (bx) dx = b−1 I ν 2

a > 0,

2

a 4b

a 4b

Re ν > − 12

ET II 58(16)

2

b > 0,

a > 0,

− Lν

2

a 4b

b > 0,

Re ν > − 12

ET II 111(18)

Re b > 0, Re ν > − 21 ET II 144(45)

2

2

2 ∞ √ a a a sec(πν) 2 cos(πν)Yν − Y−ν + H−ν Y 2ν a x J ν (bx) dx = 2b 4b 4b 4b 0 [a > 0, b > 0] MC ∞ √ Y 2ν a x Y ν (bx) dx 0

2

2

2 b−1 a a a sec(νπ) J −ν + cosec(νπ) H−ν − 2 cot(2νπ) Hν = 2 4b 4b 4b a > 0, b > 0, |Re ν| < 12 ET II 111(19)

672

6.

7.

Bessel Functions

⎡

2

2 −1 √ a a πb ⎣ − cot(2νπ) Lν Y 2ν a x K ν (bx) dx = cosec(2νπ) L−ν 2 4b 4b 0 ⎤

2

2 sec(νπ) a a ⎦ − Kν − tan(νπ) I ν 4b π 4b ET II 144(46) Re b > 0, |Re ν| < 12

2

2 ∞ √ a a 1 − Y −ν K 2ν a x J ν (bx) dx = πb−1 sec(νπ) H−ν 4 4b 4b 0 Re a > 0, b > 0, Re ν > − 21

∞

∞

√ K 2ν a x Y ν (bx) dx

2

2

2 1 −1 a a a − cosec(νπ) H−ν + 2 cosec(2νπ) Hν = − πb sec(νπ) J −ν 4 4b 4b 4b Re a > 0, b > 0, |Re ν| < 12 ET II 114(34)

∞

√ K 2ν a x K ν (bx) dx =

∞

√ πb−1 I 2ν a x K ν (bx) dx = 2

ET II 70(22)

8. 0

9. 0

10. 0

6.517 6.51812 6.519

6.517

z

0

J0 ∞

0

π/2

1. 0

π/2

2. 0

#

2

2

2 $ π a a a Kν + L−ν − Lν 4b 2 sin(νπ) 4b 4b 1 Re b > 0, |Re ν| < 2 ET II 147(63) 2

2 a a Iν + Lν 4b 4b Re b > 0, Re ν > − 21 ET II 147(60)

πb−1 4 cos(νπ)

z 2 − x2 dx = sin z

K 2ν (2z sinh x) dx =

MO 48

2 π Jν (z) + Yν2 (z) 8 cos νπ 2

Re z > 0,

− 21 < Re ν <

1 2

MO 45

J 2ν (2z cos x) dx =

π 2 J (z) 2 ν

Re ν > − 12

WH

J 2ν (2z sin x) dx =

π 2 J (z) 2 ν

Re ν > − 12

WA 42(1)a

6.52 Bessel functions combined with x and x2 6.521

1. 0

1

β J ν−1 (β) J ν (α) − α J ν−1 (α) J ν (β) α2 − β 2 α J ν (β) J ν (α) − β J ν (α) J ν (β) = β 2 − α2

x J ν (αx) J ν (βx) dx =

[α = β,

ν > −1]

[α = β,

ν > −1] WH

Bessel functions combined with x and x2

6.522

2.

10

∞

0

x K ν (ax) J ν (bx) dx =

bν aν (b2 + a2 )

673

[Re a > 0,

b > 0,

Re ν > −1] ET II 63(2)

∞ π(ab)−ν a2ν − b2ν x K ν (ax) K ν (bx) dx = 2 sin(νπ) (a2 − b2 ) 0

3.

[|Re ν| < 1,

Re(a + b) > 0] ET II 145(48)

ν a −1 λ x J ν (λx) K ν (μx) dx = μ2 + λ2 + λa J ν+1 (λa) K ν (μa) − μa J ν (λa) K ν+1 (μa) μ 0

4.

[Re ν > −1] ∞

5. 0

∞

6. 0

∞

7. 0

∞

8. 0

∞

9. 0

∞

10. 0

∞

11. 0

∞

12. 0

∞

13. 0

∞

14. 0

∞

15. 0

6.522 1.12

x K 1 (ax) =

π 2a2

[a > 0]

x K 20 (ax) =

1 2a2

[a > 0]

x K 0 (ax) I 0 (bx) = x K 1 (ax) I 1 (bx) =

0

∞

[a > 0,

b > 0]

1 − b2

[a > b > 0]

b a (a2 − b2 )

[a > b > 0]

a2

x2 K 0 (ax) =

π 2a3

[a > 0]

x2 K 1 (ax) =

2 a3

[a > 0]

x2 K 0 (ax) J 1 (bx) = x2 K 1 (ax) J 0 (bx) = x2 K 0 (ax) I 1 (bx) = x2 K 1 (ax) I 0 (bx) =

Notation: 1 =

b a (a2 + b2 )

x K 1 (ax) J 1 (bx) =

2b

[a > 0,

(a2 + b2 )2 2a (a2

[a > b > 0]

2

− b2 ) 2a

(a2

b > 0]

[a > b > 0]

2

+ b2 ) 2b

(a2

ET II 367(26)

[a > b > 0]

2

− b2 )

1 1 (b + c)2 + a2 − (b − c)2 + a2 , 2 = (b + c)2 + a2 + (b − c)2 + a2 2 2

2 x [J μ (ax)] K ν (bx) dx = Γ μ + 12 ν + 1 Γ μ − 12 ν + 1 b−2 − 1 1 −μ 1 2 −2 2 2 −2 2 1 + 4a P 1 + 4a × 1 + 4a2 b−2 2 P −μ b b 1 1 ν ν−1 2

[Re b > 2|Im a|,

2

2 Re μ > |Re ν| − 2]

ET II 138(19)

674

Bessel Functions

∞

2. 0

3.12

∞

0

4.10

6.522

2e2μπi Γ 1 + 12 ν + μ x [K μ (ax)] J ν (bx) dx = 1 b 4a2 + b2 2 Γ 12 ν − μ

2 b−2 ) Q −μ 2 b−2 ) (1 + 4a (1 + 4a × Q −μ 1 1 2ν 2 ν−1 b > 0, Re a > 0, Re 12 ν ± μ > −1 2

ν ν −ν x K 0 (ax) J ν (bx) J ν (cx) dx = r1−1 r2−1 (r2 − r1 ) (r2 + r1 ) = ν 2 1 2 , 2 (2 − 1 ) 2 2 2 2 r1 = a + (b − c) , r2 = a + (b + c) , c > 0, Re ν > −1, Re a > |Im b| ET II 63(6)

∞

0

− 1 x I 0 (ax) K 0 (bx) J 0 (cx) dx = a4 + b4 + c4 − 2a2 b2 + 2a2 c2 + 2b2 c2 2 [Re b > Re a,

5.10

ET II 66(27)a

c > 0]

ET II 16(27)

alternatively, with a and c interchanged ∞ 1 x I 0 (cx) K 0 (bx) J 0 (ax) dx = 2 [Re b > Re c, a > 0] − 21 0 2 ∞ − 1 x J 0 (ax) K 0 (bx) J 0 (cx) dx = a4 + b4 + c4 − 2a2 c2 + 2a2 b2 + 2b2 c2 2 0

6.

alternatively, with a and b interchanged ∞ 1 x J 0 (bx) K 0 (ax) J 0 (cx) dx = 2 2 − 21 0 ∞ x J 0 (ax) Y 0 (ax) J 0 (bx) dx = 0 0

7.12

[Re b > |Im a|,

c > 0]

[Re a > |Im b|,

c > 0]

[0 < b < 2a]

= −2π −1 b−1 b2 − 4a2

− 12

∞

0

[0 < 2a 2|Im a|,

∞

8. 0

9.

8

0

∞

ET II 15(25)

2

2 Re μ > |Re ν| − 3]

ET II 138(20)

x K μ− 12 (ax) K μ+ 12 (ax) J ν (bx) dx 1 −μ− 12 1 2e2μπi Γ 12 ν + μ + 1 −μ+ 12 2 −2 2 2 −2 2 1 + 4a Q 1 + 4a Q b b =− 1 1 1 1 1 ν− 2 ν− 2 b Γ 12 ν − μ b2 + 4a2 2 2 2 b > 0, Re a > 0, Re ν > −1, |Re μ| < 1 + 12 Re ν ET II 67(29)a − 1 x I 12 ν (ax) K 12 ν (ax) J ν (bx) dx = b−1 b2 + 4a2 2 [b > 0,

Re a > 0,

Re ν > −1] ET II 65(16)

Bessel functions combined with x and x2

6.522

∞

10. 0

x J 12 ν (ax) Y

1 2ν

(ax) J ν (bx) dx =0 = −2π −1 b−1 b2 − 4a2

− 12

11.8

[a > 0,

Re ν > −1,

0 0,

Re ν > −1,

2a −1,

0 0,

Re ν > −1,

2a < b]

x I 12 (ν−μ) (ax) K 12 (ν+μ) (ax) J ν (bx) dx = 2−μ a−μ b−1 b2 + 4a2

− 12

[b > 0,

[a > 0,

ET II 52(32) ∞

12.

8

675

∞

Re a > 0,

Re ν > −1,

1 μ b + b2 + 4a2 2

Re(ν − μ) > −2] μ

ν

ET II 66(23)

ν−μ

μ−ν

(cos ψ) (sin ϕ) (sin ψ) (cos ϕ) x J μ (xa sin ϕ) K ν−μ (ax cos ϕ cos ψ) J ν (xa sin ψ) dx = 2 2 2 a 1 − sin ϕ sin ψ π π ET II 64(10) a > 0, 0 < ϕ < , 0 < ψ < , Re μ > −1, Re ν > −1 2 2 x J μ (xa sin ϕ cos ψ) J ν−μ (ax) J ν (xa cos ϕ sin ψ) dx μ

ν

−ν

−μ

−1

= −2π −1 a−2 sin(μπ) (sin ϕ) (sin ψ) (cos ϕ) (cos ψ) [cos(ϕ + ψ) cos(ϕ − ψ)] π a > 0, 0 < ϕ < , 0 < ψ < 12 π, Re ν > −1 ET II 54(39) 2 ∞ 23ν (abc)ν Γ ν + 12 ν+1 x J ν (bx) K ν (ax) J ν (cx) dx = √ 2ν+1 π (22 − 21 ) 0 [Re a > |Im b|, c > 0] ∞ 1 3ν ν 2 (abc) Γ ν + 2 xν+1 I ν (cx) K ν (bx) J ν (ax) dx = √ 2ν+1 π (22 − 21 ) 0 [Re b > |Im a| + |Im c|] ∞ tν−μ−ρ+1 J μ (ct) J ν (bt) K ρ (at) dt 0 μ−ν+ρ−1 1 1+2ν−2ρ 2 1 − x2 22 − x2 x 21+ν−μ−ρ dx = μ ν ρ μ−ν c b a Γ (μ − ν + ρ) 0 (b2 − x2 ) 1 1 1 = (b + c)2 + a2 − (b − c)2 + a2 , 2 = (b + c)2 + a2 + (b − c)2 + a2 2 2 [Re a > |Im b|, c > 0]

676

18.

Bessel Functions

11

∞

0

tμ−ν+ρ+1 J μ (ct) J ν (bt) K ρ (at) dt

ν−μ−ρ−1 1 1+2μ+2ρ 2 1 − x2 22 − x2 x 21+μ−ν+ρ aρ dx = μ ν ν−μ c b Γ (ν − μ − ρ) 0 (c2 − x2 ) 1 1 1 = (b + c)2 + a2 − (b − c)2 + a2 , 2 = (b + c)2 + a2 + (b − c)2 + a2 2 2 [Re a > |Im b|, c > 0]

6.523

6.523

∞

0

−1 2 −1 b ln x 2π −1 K 0 (ax) − Y 0 (ax) K 0 (bx) dx = 2π −1 a2 + b2 + b − a2 a [Re b > |Im a|, Re(a + b) > 0] ET II 145(50)

6.524

∞

1. 0

0 < a < b, 0 − 21

ET II 352(14)

[a > 0, 6.525

Re ν > − 21

b > 0]

ET II 373(9)

1 1 (b + c)2 + a2 − (b − c)2 + a2 , 2 = (b + c)2 + a2 + (b − c)2 + a2 2 2 ∞ − 32 2 2 a + b2 + c2 − 4a2 c2 x2 J 1 (ax) K 0 (bx) J 0 (cx) dx = 2a a2 + b2 − c2 Notation: 1 =

0

[c > 0,

Re b ≥ |Im a|,

Re a > 0] ET II 15(26)

2.10

alternatively, with a and b interchanged ∞ 2b a2 + b2 − c2 2 x J 1 (bx) K 0 (ax) J 0 (cx) dx = [Re a > |Im b|, Re b > 0, 3 (22 − 21 ) 0 ∞ − 32 2 2 a + b2 + c2 − 4a2 b2 x2 I 0 (ax) K 1 (bx) J 0 (cx) dx = 2b b2 + c2 − a2

c > 0]

0

3.10

∞

0

6.526

1. 0

∞

x2 I 0 (cx) K 0 (bx) J 0 (ax) dx =

2

x J 12 ν ax

2b a2 + b2 − c2

−1

J ν (bx) dx = (2a)

3

(22 − 21 )

J 12 ν

b2 4a

[Re b > |Re a|,

c > 0]

[Re a > |Im b|,

c > 0]

ET II 16(28)

[a > 0,

b > 0,

Re ν > −1]

ET II 56(1)

Bessel functions combined with x and x2

6.527

∞

2. 0

x J 12 ν ax2 Y ν (bx) dx −1

= (4a)

∞

3. 0

∞

4. 0

2

x J 12 ν ax

xY

1 2ν

0

1 2ν

− tan

∞

6. 0

∞

7. 0

ax2 J ν (bx) dx = −(2a)−1 H 12 ν

xY

1 2ν

J 12 ν 2 [a > 0,

ET II 140(27)

2

b 4a

[a > 0,

∞

8. 0

1. 2. 3.

Re ν > −1]

Re b > 0,

ax2 K ν (bx) dx

2

x K 12 ν ax

2

x K 12 ν ax

2

2

2 νπ

νπ

π b b b cos H− 12 ν − sin J − 12 ν − H 12 ν 4a sin(νπ) 2 4a 2 4a 4a [a > 0, Re b > 0, |Re ν| < 1] ET II 141(28)

2

2 b b π I 12 ν − L 12 ν J ν (bx) dx = 4a 4a 4a [Re a > 0,

π ⎣ Y ν (bx) dx = cosec(νπ) L− 12 ν 4a

x K 12 ν ax2 K ν (bx) dx π = 8a

∞

# sec

Re ν > −1]

b > 0,

ET II 68(9)

⎡

νπ

2

I 12 ν

b2 4a

b2 4a

− cot(νπ) L 12 ν

νπ

1 − sec K 12 ν π 2

[Re a > 0,

6.527

Re ν > −1]

− tan

2 νπ

b2 b + sec H− 12 ν 4a 2 4a b > 0, Re ν > −1] ET II 109(9)

νπ

2

2 b b π νπ H− 1 ν − Y − 12 ν K ν (bx) dx = 2 4a 4a 8a cos 2 [a > 0, Re b > 0,

=

Y

b2 4a

ET II 61(35) ∞

5.

677

νπ

2

K

1 2ν

b2 4a

1 x2 J 2ν (2ax) J ν− 12 x2 dx = a J ν+ 12 a2 2 0 ∞ 1 x2 J 2ν (2ax) J ν+ 12 x2 dx = a J ν− 12 a2 2 0 ∞ 1 x2 J 2ν (2ax) Y ν+ 12 x2 dx = − a Hν− 12 a2 2 0

b > 0,

b2 4a

⎤ b2 ⎦ 4a

|Re ν| < 1]

ET II 112(25)

2 $ b2 b 1 + π cosec(νπ) L − L2ν 4a 4a [Re a > 0, |Re ν| < 1] ET II 146(52)

− 12 ν

a > 0,

Re ν > − 12

[a > 0,

Re ν > −2]

ET II 355(35)

[a > 0,

Re ν > −2]

ET II 355(36)

ET II 355(33)

678

Bessel Functions

6.528 6.529

∞

0

∞

1. 0

2. 0

x K 14 ν

x2 4

I 14 ν

x2 4

J ν (bx) dx = K 14 ν

b2 4

6.528

b2 I 14 ν 4 [b > 0, ν > −1]

√ √ 2a 1 x J ν 2 ax K ν 2 ax J ν (bx) dx = b−2 e− b 2

[Re a > 0,

MO 183a

b > 0,

Re ν > −1] ET II 70(23)

a

x J λ (2x) I λ (2x) J μ 2 a2 − x2 I μ 2 a2 − x2 dx =

a2λ+2μ+2 2 Γ(λ + 1) Γ(μ + 1) Γ(λ + μ + 2) λ+μ+3 λ+μ+1 ; λ + 1, μ + 1, λ + μ + 1, ; −a4 × 1F 4 2 2 [Re λ > −1, Re μ > −1] ET II 376(31)

6.53–6.54 Combinations of Bessel functions and rational functions 6.531 1.

12

2.

∞

Y ν (bx) π dx = [Eν (ab) + Yν (ab) + 2 cot(πν)(Jν (ab) − Jν (ab))] x + a sin(πν) 0 [Re ν < 1, arg a = π, b > 0] ∞ Y ν (bx) 2 dx = π cot(νπ) [Y ν (ab) + Eν (ab)] + Jν (ab) + 2 [cot(νπ)] [Jν (ab) − J ν (ab)] x−a 0 [b > 0, a > 0, |Re ν| < 1]

ET II 98(9) ∞

3. 0

4.∗

∞

0

6.532 1.12

0

K ν (bx) 1 1 π 2 dx = [cosec(νπ)] I ν (ab) + I −ν (ab) − e− 2 iνπ Jν (iab) − e 2 iνπ J−ν (iab) x+a 2 [Re b > 0, |arg a| < π, |Re ν| < 1] 2

ET II 128(5) ∞

0

5.∗

MC

∞

Jν (bx) π dx = (Jν (ab) − Jν (ab)) x+a sin(πν)

[b > 0,

|arg(a)| < π,

[b > 0,

a > 0,

Re(ν) > −1]

Jν (bx) π dx = (Jν (ab) − Jν (ab)) + Eν (ab) x−a tan(πν) Re(ν) > −1]

J ν (x) 3 − ν 3 + ν a2 1 πIν (a) , ; 1; + dx = F 1 2 x2 + a2 ν2 − 1 2 2 4 2a cos πν 2 [Re a > 0,

Re ν > −1]

6.535

Bessel functions and rational functions

1(b).* PV 0

2.12

∞

0

∞

πν

Jν (x) π tan (Jν (a) − Jν (a)) + Eν (a) dx = x2 − a2 2a 2 ⎡

[Re a > 0,

∞

0

4. 5. 6.

4.

6.535

|Re ν| < 1]

ET II 99(13)

ET II 101(21) ∞

x J 0 (ax) dx = K 0 (ak) x2 + k 2 0 ∞ Y 0 (ax) K 0 (ak) dx = − 2 + k2 x k 0 ∞ J 0 (ax) π [I 0 (ak) − L0 (ak)] dx = 2 + k2 x 2k 0

[a > 0,

Re k > 0]

WA 466(5)

[a > 0,

Re k > 0]

WA 466(6)

[a > 0,

Re k > 0]

WA 467(7)

[Re p > 0,

Re q > −1]

WA 415(3)

[Re p > 0,

Re q > 0]

WA 415(5)

[0 < b < a] [0 < a < b]

⎧

b b2 1 1 ⎪ ⎪ ∞ , ; 2, 2 − 1 [0 0,

νπ

νπ Y ν (bx) π J ν (ab) + tan tan [Jν (ab) − J ν (ab)] − Eν (ab) − Y ν (ab) dx = 2 2 x −a 2a 2 2 [b > 0, a > 0, |Re ν| < 1]

6.533 z J p+q (z) dx = J p (x) J q (z − x) 1. x p 0

z J p (x) J q (z − x) 1 1 J p+q (z) dx = + 2. x z−x p q z 0 ∞ a dx b 3.11 [J 0 (ax) − 1] J 1 (bx) 2 = − 1 + 2 ln x 4 b 0 a2 =− 4b

3b.12

Re ν > −1]

νπ

Y ν (bx) 1 ⎣ 1 π tan I ν (ab) − K ν (ab) dx = − νπ 2 2 x +a 2a 2 a cos 2

⎤ νπ

b sin 2 2 3−ν 3+ν a b ⎦ 2 , ; + 1 F 2 1; 1 − ν2 2 2 4 [b > 0,

3.

679

ET II 21(28)a

ET II 14(16) ∞

x3 J 0 (x) 1 1 dx = K 0 (a) − π Y 0 (a) 4 − a4 x 2 4 0 ∞ x 2 [J ν (x)] dx = I ν (a) K ν (a) 2 2 0 x +a

[a > 0] [Re a > 0,

ET II 340(5)

Re ν > −1]

ET II 342(26)

680

Bessel Functions

6.536

0

6.537 6.538 1.

12

∞

0

∞

0

2.8

∞

6.536

b > 0,

x3 J 0 (bx) dx = ker(ab) x4 + a4

2

x J 0 (bx) 1 dx = − 2 kei(ab) x4 + a4 a

b > 0,

|arg a| <

0

ET II 8(9), MO 46a

π

MO 46a

4

) √ √ * dx a+b 2 ab 2 ab 2 2 2 − (a − b) K (a + b )E J 1 (ax) J 1 (bx) 2 = x 3πab2 a+b a+b [a > 0,

∞

|arg a| < 14 π

x−1 J ν+2n+1 (x) J ν+2m+1 (x) dx = 0

b > 0]

ET II 21(30)

[m = n with m, n integers, ν > −1]

= (4n + 2ν + 2)−1

[m = n,

ν > −1] EH II 64

6.539 1. 2.

π Y ν (b) Y ν (a) − 2 2 J ν (b) J ν (a) a x [J ν (x)] b dx J ν (b) π J ν (a) 2 = 2 Y (a) − Y (b) ν ν a x [Y ν (x)]

b

dx

[J ν (x) = 0

=

for x ∈ [a, b]]

ET II 338(41)

[Y ν (x) = 0 for x ∈ [a, b]] ET II 339(49)

3. a

6.541

b

dx π J ν (a) Y ν (b) = ln x J ν (x) Y ν (x) 2 J ν (b) Y ν (a)

∞

1. 0

2.8

0

x J ν (ax) J ν (bx)

ET II 339(50)

dx = I ν (bc) K ν (ac) x2 + c2 = I ν (ac) K ν (bc)

[0 0,

Re ν > −1]

[0 < a < b,

Re c > 0,

Re ν > −1] ET II 49(10)

∞

dx x1−2n J ν (ax) J ν (bx) 2 x + c2 2 2 p n−1−p 2 2 k *

ν n )

n−1 " " a c /4 b c /4 π 1 1 b I ν (bc) K ν (ac) − = − 2 c 2 a sin(πν) p=0 p! Γ(1 − ν + p) k! Γ(1 − ν + k) k=0

*

ν n−1 n )

" a2 c2 /4 p n−1−p " b2 c2 /4 k 1 1 b I ν (bc) K ν (ac) − = − 2 c 2ν a p!(1 − ν)p k!(1 + ν)k p=0

[0 n − 1,

Re c > 0,

0 < b < a]

6.544

3.

8

Bessel functions and rational functions

∞

0

681

1 c 2ρ−α xα−1 J (cx) J (cx) dx = μ ν ρ (x2 + z 2 ) 2 2 (μ + ν + α)/2 − ρ, 1 + 2ρ − α ×Γ (μ⎛− ν − α)/2 + ρ + 1, (μ + ν − α)/2 + ρ + 1, (ν − μ − α)/2 + ρ + 1 1−α α μ+ν +α μ−ν −α + ρ, 1 − + ρ, ρ; ρ + 1 − ,ρ+ 1 + , 2 2 2 2 ⎞ ν − μ − α 2 2 ⎠ z α−2ρ cz μ+ν μ+ν −α ,ρ + 1 + ;c z ρ+1+ , + 2 2 2 2 ⎛ 1+μ+ν μ+ν ρ − (α + μ + ν)/2, (α + μ + ν) /2 ,1 + Γ 3F 4 ⎝ 2 2 ρ, μ + 1, ν + 1 ⎞ α+μ+ν α+μ+ν ;1 − ρ+ , μ + 1, ν + 1, μ + ν + 1; c2 z 2 ⎠ 2 2

× 3F 4 ⎝

4.∗

Γ (a1 ) . . . Γ (ap ) a 1 , . . . , ap , c > 0, Re z > 0, Re(α + μ + ν) > 0; Re(α − 2ρ) > 1 = Γ b1 , . . . , bq Γ (b1 ) . . . Γ (bq ) ∞ ρ−1 x J (bx) cos 12 (ρ − μ + ν)π Jν (ax) + sin(ax) + sin 12 (ρ + μ − ν)π Yν (ax) 2 + k2 μ x 0 = −k ρ−2 Iμ (kb)Kν (ka) [|Re ν| − Re μ < Re ρ < 4,

6.543

∞

[Re r > 0, 6.544

WA 430(3)

ν J ν (ax) Y ν (bx) − J ν (bx) Y ν (ax) π b dx = − [0 < b < a] ET II 352(16) 2 2 2 a 0 x [J ν (bx)] + [Y ν (bx)] # $ ∞ 1 1 x dx J μ (bx) cos (ν − μ)π J ν (ax) − sin (ν − μ)π Y ν (ax) = I μ (br) K ν (ar) 2 2 2 x + r2 0

6.542

0 0, b > 0,

√ x dx 2 a 1 Jν = J 2ν √ 2 x b x a b

a

√ x dx 2 a 1 iπ 1 1 iνπ 2 4 Kν + = e K 2ν √ e x b x2 a b

a

a ≥ b > 0,

a > 0,

b > 0,

Re μ > |Re ν| − 2]

|Re ν| <

1 2

EI II 357(47)

Re ν > − 12

ET II 57(10)

√ 1 − 1 iνπ 2 a − 1 iπ 2 4 e K 2ν √ e a b Re b > 0, a > 0, |Re ν| < 12

ET II 142(32)

682

Bessel Functions

∞

4. 0

5.12

∞

0

0

|Re ν| <

1 2

ET II 62(38)

√

√ x dx 2 a 1 iπ 2 a − 1 iπ 1 1 i(ν+1)π − 12 i(ν+1)π 2 4 4 √ √ Kν e + e = K K e e 2ν 2ν x b x2 a b b Re b > 0, a > 0, |Re ν| < 12

a

Kν

√ √ x dx 1 2 a i 1 νπi − 12 νπi − 14 πi 2 a 2 4 πi √ √ Jν e e e = K K − e 2ν 2ν x b x2 a b b Re a > 0, b > 0, |Re ν| < 52

a

ET II 70(19)

∞

7. 0

Yν

√

√ x dx 2 π 2 a 2 a Jν K 2ν √ = + Y 2ν √ x b x2 aπ 2 b b a > 0, b > 0,

a

ET II 143(38)

∞

6.

Yν

6.551

∞

8. 0

Kν

Kν

√

√

x dx 3 2 a 2 a 2 3 Yν sin πν kei2ν √ πν ker2ν √ = − cos 2 x b x a 2 2 b b Re a > 0, b > 0, |Re ν| < 52

a

ET II 113(29)

√ x dx 2 a π Kν = K 2ν √ x b x2 a b

a

[Re a > 0,

Re b > 0]

ET II 146(55)

6.55 Combinations of Bessel functions and algebraic functions 6.55110

1. 0

1

x1/2 J ν (xy) dx =

∞

2. 1

Γ 3 + 1ν 2y −3/2 41 21 Γ 4 + 2ν +y −1/2 ν − 12 J ν (y) S −1/2,ν−1 (y) − J ν−1 (y) S 1/2,ν (y) y > 0, Re ν > − 32 √

x1/2 J ν (xy) dx = y −1/2 J ν−1 (y) S 1/2,ν (y) + 12 − ν J ν (y) S −1/2,ν−1 (y) [y > 0]

6.552

∞

1. 0

0

3. 0

J ν (xy)

dx (x2

+

a2 )1/2

= I ν/2

1

2 ay

K ν/2

1

2 ay

[Re a > 0,

ET II 22(2)

y > 0,

Re ν > −1]

ET II 23(11), WA 477(3), MO 44 ∞

2.

ET II 21(1)

Y ν (xy)

dx (x2 +

1/2 a2 )

=−

1 sec 12 νπ K ν/2 12 ay K ν/2 12 ay + π sin 12 νπ I ν/2 12 ay π [y > 0, Re a > 0, |Re ν| < 1] ET II 100(18)

∞

K ν (xy)

dx (x2 + a2 )1/2

2 2 π sec 12 νπ = J ν/2 12 ay + Y ν/2 12 ay 8 [Re a > 0, Re y > 0, 2

|Re ν| < 1] ET II 128(6)

6.561

Bessel functions and powers

1

J ν (xy)

4. 0

1

0

∞

1 ∞

(1 −

1

=

2 π J ν/2 12 y 2

[y > 0,

=

π J 0 12 y Y 0 12 y 2

[y > 0]

1/2 x2 )

dx 1/2

(x2 − 1)

Y ν (xy)

7.

1/2 x2 )

dx

J ν (xy)

6.

(1 −

Y 0 (xy)

5.

dx

683

Re ν > −1]

ET II 102(26)a

π J ν/2 12 y Y ν/2 12 y [y > 0] 2 2 2 π J ν/2 12 y = − Y ν/2 12 y 4

=−

dx (x2 − 1)1/2

ET II 24(23)a

[y > 0] 6.553

∞

−1/2

x 0

I ν (x) K ν (x) K μ (2x) dx =

Γ

1 4

ET II 24(22)a

ET II 102(27)

+ 12 μ Γ 14 − 12 μ Γ 14 + ν + 12 μ Γ 14 + ν − 12 μ 4 Γ 34 + ν + 12 μ Γ 34 + ν − 12 μ |Re μ| < 12 , 2 Re ν > |Re μ| − 12

ET II 372(2)

6.554

∞

1. 0

1

2. 0

x J 0 (xy)

∞

3. 1

∞

4. 0

5.

11

x J 0 (xy)

6.555

0

6.556

∞

0

∞

ET II 7(4)

[y > 0]

ET II 7(6)a

= a−1 e−ay

[y > 0,

Re a > 0]

1 ν √ 2a π J ν (ak) K ν (ak) dx = 2ν (2k) Γ ν + 12 a > 0,

|arg k| >

+ a2 )3/2

ν+1/2

Re a > 0]

= y −1 cos y

1/2

− 1)

xν+1 J ν (ax) (x4 + 4k 4 )

= y −1 sin y

dx (x2

[y > 0,

ET II 7(5)a

dx (x2

= y −1 e−ay

[y > 0]

1/2

(1 − x2 )

x J 0 (xy)

0

1/2

+ x2 ) dx

x J 0 (xy)

∞

dx (a2

a x1/2 J 2ν−1 ax1/2 Y ν (xy) dx = − 2 Hν−1 2y

ET II 7(7)a

π 4,

Re ν > − 12

WA 473(1)

2

a 4y a > 0,

a

a

1/2 dx π √ Y ν/2 J ν a x2 + 1 = − J ν/2 2 2 2 x2 + 1

y > 0,

Re ν > − 12

ET II 111(17)

[Re ν > −1,

a > 0]

MO 46

6.56–6.58 Combinations of Bessel functions and powers 6.561

1. 0

1

1 xν J ν (ax) dx = 2ν−1 a−ν π 2 Γ ν + 12 [J ν (a) Hν−1 (a) − Hν (a) J ν−1 (a)] Re ν > − 12

ET II 333(2)a

684

Bessel Functions

1

2. 0

1

3. 0

1

4. 0

1

5. 0

1

6. 0

1

7. 0

1

8. 0

1

9. 0

1

10. 0

1

11. 0

1

12. 0

13.7

0

1

1 xν Y ν (ax) dx = 2ν−1 a−ν π 2 Γ ν + 12 [Y ν (a) Hν−1 (a) − Hν (a) Y ν−1 (a)] Re ν > − 12

15. 0

ET II 364(2)a

1 xν K ν (ax) dx = 2ν−1 a−ν π 2 Γ ν + 12 [K ν (a) Lν−1 (a) + Lν (a) K ν−1 (a)] Re ν > − 12 xν+1 J ν (ax) dx = a−1 J ν+1 (a)

ET II 367(21)a

[Re ν > −1]

ET II 333(3)a

xν+1 Y ν (ax) dx = a−1 Y ν+1 (a) + 2ν+1 a−ν−2 π −1 Γ(ν + 1)

xν+1 I ν (ax) dx = a−1 I ν+1 (a)

[Re ν > −1]

ET II 339(44)a

[Re ν > −1]

ET II 365(3)a

[Re ν > −1]

ET II 367(22)a

xν+1 K ν (ax) dx = 2ν a−ν−2 Γ(ν + 1) − a−1 K ν+1 (a)

x1−ν J ν (ax) dx = x1−ν Y ν (ax) dx =

a

ν−2

2ν−1 Γ(ν)

− a−1 J ν−1 (a)

aν−2 cot(νπ) − a−1 Y ν−1 (a) 2ν−1 Γ(ν)

x1−ν I ν (ax) dx = a−1 I ν−1 (a) −

ET II 333(4)a

[Re ν < 1]

ET II 339(45)a

aν−2

ET II 365(4)a

2ν−1 Γ(ν)

x1−ν K ν (ax) dx = 2−ν aν−2 Γ(1 − ν) − a−1 K ν−1 (a)

xμ J ν (ax) dx =

2μ Γ

ν+μ+1

aμ+1 Γ

∞

[Re ν < 1]

ET II 367(23)a

2 + a−μ {(μ + ν − 1) J ν (a) S μ−1,ν−1 (a) − J ν−1 (a) S μ,ν (a)} ν−μ+1 2

1 1 1 ∞ μ μ −μ−1 Γ 2 + 2 ν + 2 μ x J ν (ax) dx = 2 a Γ 12 + 12 ν − 12 μ 0

ET II 338(43)a

1 xν I ν (ax) dx = 2ν−1 a−ν π 2 Γ ν + 12 [I ν (a) Lν−1 (a) − Lν (a) I ν−1 (a)] Re ν > − 12

14.

6.561

[a > 0,

Re(μ + ν) > −1]

− Re ν − 1 < Re μ < 12 ,

1 −μ−1 Γ 12 + 12 ν + 12 μ μ μ x Y ν (ax) dx = 2 cot 2 (ν + 1 − μ)π a Γ 12 + 12 ν − 12 μ |Re ν| − 1 < μ < 12 ,

ET II 22(8)a

a>0

EH II 49(19)

a>0

ET II 97(3)a

6.563

Bessel functions and powers

∞

16. 0

∞

17. 0

∞

18. 0

19. 0

6.562

1

a

1

Γ

∞

∞

2. 0

1+μ+ν 2

1+μ−ν Γ 2 [Re (μ + 1 ± ν) > 0,

Γ Y ν (x) dx = ν−μ x

1 2

xμ Y ν (bx)

Re a > 0] EH II 51(27)

1

−1 < Re q < Re ν − 12

Γ q+ J ν (ax) dx = ν−q q−ν+12 2 1 xν−q 2 a Γ ν − 2 q + 12

WA 428(1), KU 144(5)

+

1 1 Γ 2 + 2 μ − ν sin 12 μ − ν π 2ν−μ π |Re ν| < Re(1 + μ − ν) < 32

1 2μ

x2m+n+1/2 K n+1/2 (αx) dx =

0

x K ν (ax) dx = 2

μ−1 −μ−1

1.

μ

685

π 2

n " k=0

(n + k)! γ(2m + n − k + 1, α) k!(n − k)! α2m+n+3/2 2k

WA 430(5) STR

' dx = (2a)μ π −1 sin 12 π(μ − ν) Γ 12 (μ + ν + 1) Γ 12 (1 + μ − ν) S −μ,ν (ab) x+a ( −2 cos 12 π (μ − ν) Γ 1 + 12 μ + 12 ν Γ 1 + 12 μ − 12 ν S −μ−1,ν (ab) b > 0, |arg a| < π, Re (μ ± ν) > −1, Re μ < 32 ET II 98(8)

xν J ν (ax) πk ν dx = [H−ν (ak) − Y −ν (ak)] x+k 2 cos νπ

1 − 2 < Re ν < 32 ,

a > 0,

|arg k| < π

WA 479(7) ∞

3. 0

dx x+a

1 1 −μ μ − ν a2 b 2 μ+ν μ−2 ,1− ; Γ 2 (μ + ν) Γ 2 (μ − ν) b 1 F 2 1; 1 − =2 2 2 4 1 1 1−μ 3 − μ − ν 3 − μ + ν a2 b 2 μ−3 , ; 1; Γ 2 (μ − ν − 1) Γ 2 (μ + ν − 1) ab −2 F 1 2 2 2 4

xμ K ν (bx)

−πaμ cosec[π(μ − ν)] {K ν (ab) + π cos(μπ) cosec[π(ν + μ)] I ν (ab)} [Re b > 0, 6.563

0

∞

x−1 J ν (bx)

|arg a| < π,

Re μ > |Re ν| − 1]

ET II 127(4)

dx πa−μ−1 = (x + a)1+μ sin[( + 1) ⎧ + ν − μ)π] Γ(μ ν+2m ∞ ⎨" m 1 (−1) 2 ab Γ( + ν + 2m) × ⎩ m! Γ(ν + m + 1) Γ ( + ν − μ + 2m) m=0 μ+1−+m ⎫ 1 ∞ 1 ⎬ " ab Γ(μ + m + 1) ( + ν − μ − m)π sin 2 1 1 2 − m! Γ 2 (μ + ν − + m + 3) Γ 2 (μ − ν − + m + 3) ⎭ m=0 ET II 23(10), WA 479 b > 0, |arg a| < π, Re( + ν) > 0, Re( − μ) < 52

686

Bessel Functions

6.564 1.

∞

ν+1

x

0

∞

2. 0

dx J ν (bx) √ = 2 x + a2

x1−ν J ν (bx) √

dx = + a2

x2

2 ν+ 1 a 2 K ν+ 12 (ab) πb

6.564

Re a > 0,

b > 0,

−1 < Re ν <

1 2

ET II 23(15)

π 1 −ν a2 I ν− 12 (ab) − Lν− 12 (ab) 2b Re a > 0,

b > 0,

Re ν > − 21

ET II 23(16)

6.565 1.

∞

−ν

x

2

x +a

1 2 −ν− 2

0

∞

2.

∞

3. 0

5.

6.

ν+1

0

b > 0,

Re ν > − 21

ET II 24(18)

ν√

2

ν+1

b π aeab Γ ν + 32 [Re a > 0,

b > 0,

Re ν > −1]

J ν (bx)x +

μ+1 a2 )

ν−μ μ

dx =

b a K ν−μ (ab) Γ(μ + 1) −1 < Re ν < Re 2μ + 32 ,

2μ

a > 0,

b>0

MO 43

∞

μ x1−ν x2 + a2 Y ν (bx) dx =

7.

Re a > 0,

μ xν+1 x2 + a2 Y ν (bx) dx aμ+1 b−μ−ν−1 2 μ+1 ν π 2 = (ab) csc(π(μ + ν))[cot(πμ)I−μ−ν−1 (ab) + cot(πν))Iμ+ν+1 (ab) 2πΓ(−μ)

a2 b 2 −2ν (ab)μ+1 Γ(−μ − 1)Γ(ν) 1F2 1; μ + 2; 1 − ν; 4 [b > 0, Re a > 0, −1 < Re ν < −2 Re μ] ET II 100(19)

0

WA 477(4), ET II 23(17)

∞

0

12

−ν− 32 xν+1 x2 + a2 J ν (bx) dx =

(x2

0

√ ν−1 πb J ν (bx) dx = ν ab 2 e Γ ν + 12

ET II 24(19) ∞

4.

12

J ν (bx) dx = 2 a

−ν− 12 xν+1 x2 + a2

0

Γ(ν + 1) ab ab Iν Kν b Γ(2ν + 1) 2 2 Re a > 0, b > 0, Re ν > − 21

ν −2ν ν

∞

2μ aμ−ν+1 2 cos((μ − ν)π)K Γ(μ + 1) cot(νπ)I (ab) + (ab) μ−ν+1 μ−ν+1 bμ+1 π

a2 b 2 a2μ+2 bν cot(νπ) 1; ν + 1, μ + 2; F − ν+1 1 2 2 (μ + 1)Γ(ν 4 + 1) Re ν < 1, Re(ν − 2μ) > −3, arg a2 = π, b > 0 MC

μ x1+ν x2 + a2 K ν (bx) dx = 2ν Γ(ν + 1)aν+μ+1 b−1−μ S μ−ν,μ+ν+1 (ab) [Re a > 0,

Re b > 0,

Re ν > −1] ET II 128(8)

6.567

8.

11

Bessel functions and powers

∞

μ+1

(x2 + k 2 )

0

6.566

∞

1. 0

∞

2. 0

0

x2

xν+1 J ν (ax)

1

2

+ 12 ν Γ μ + 1 − 12 − 12 ν

2ν+1 Γ(μ + 1) Γ(ν + 1) a2 k 2 +ν +ν ; − μ, ν + 1; × 1F 2 2 2 4 a2μ+2− Γ 12 ν + 12 − μ − 1

+ 1 1 22μ+3− Γ μ + 2 + ν − 2 2

ν + a2 k 2 ν − ,μ+ 2 − ; × 1 F 2 μ + 1; μ + 2 + 2 2 4 a > 0, − Re ν < Re < 2 Re μ + 72 , Re k > 0

WA 477(1)

dx = 2μ−2 π −1 b1−μ + a2 π (μ − ν + 1) Γ 12 μ + 12 ν − 12 Γ 12 μ − 12 ν − 12 × cos 2

μ+1+ν μ + 1 − ν a2 b 2 × 1 F 2 1; 2 − ,2 − ; 2 2 4 π 1 μ−1 π − πa cosec (μ + ν + 1) cot (μ − ν + 1) I ν (ab) 2 2 π 2 −aμ−1 cosec (μ − ν + 1) K ν (ab) 2 b > 0, Re a > 0, |Re ν| − 1 < Re μ < 52 ET II 100(17)

x2

dx = bν K ν (ab) + b2

a > 0,

Re b > 0,

−1 < Re ν <

3 2

xν K ν (ax)

2 ν−1

x2

dx π b [H−ν (ab) − Y −ν (ab)] = 2 +b 4 cos νπ a > 0,

Re b > 0,

Re ν > − 21

WA 468(9)

∞

4. 0

xμ Y ν (bx)

dx =

aν k +ν−2μ−2 Γ

EH II 96(58) ∞

3.

x−1 J ν (ax)

687

x−ν K ν (ax)

2

dx π [Hν (ab) − Y ν (ab)] = ν+1 x2 + b2 4b cos νπ a > 0,

Re b > 0,

Re ν <

1 2

WA 468(10)

∞

5. 0

x−ν J ν (ax)

x2

dx π = ν+1 [I ν (ab) − Lν (ab)] + b2 2b

a > 0,

Re b > 0,

Re ν > − 25

WA 468(11)

6.567

1. 0

1

μ xν+1 1 − x2 J ν (bx) dx = 2μ Γ(μ + 1)b−(μ+1) J ν+μ+1 (b) [b > 0,

Re ν > −1,

Re μ > −1] ET II 26(33)a

688

Bessel Functions

1

2. 0

μ xν+1 1 − x2 Y ν (bx) dx = b−(μ+1) 2μ Γ(μ + 1) Y μ+ν+1 (b) + 2ν+1 π −1 Γ(ν + 1) S μ−ν,μ+ν+1 (b) [b > 0,

3. 4.

6.567

Re μ > −1,

Re ν > −1]

ET II 103(35)a

1

μ 21−ν S ν+μ,μ−ν+1 (b) [b > 0, Re μ > −1] x1−ν 1 − x2 J ν (bx) dx = bμ+1 Γ(ν) 0 1 μ x1−ν 1 − x2 Y ν (bx) dx = b−(μ+1) 21−ν π −1 cos(νπ) Γ (1 − ν)

ET II 25(31)a

× s μ+ν,μ−ν+1 (b) − 2 cosec(νπ) Γ(μ + 1) J μ−ν+1 (b)

0

μ

[b > 0,

1

5.

1−ν

x 0

2 μ

1−x

K ν (bx) dx = 2

−ν−2 ν

−1

b (μ + 1)

Re μ > −1,

Γ(−ν) 1 F 2

Re ν < 1]

b2 1; ν + 1, μ + 2; 4

ET II 104(37)a

+π2μ−1 b−(μ+1) cosec (νπ) Γ(μ + 1) I μ−ν+1 (b) 6. 7.

8.

9.

10.

11.

12. 13.

1

[Re μ > −1,

Re ν < 1]

π Hν− 12 (b) [b > 0] 2b 0 1 π dx cosec(νπ) cos(νπ) J ν+ 12 (b) − H−ν− 12 (b) x1+ν Y ν (bx) √ = 2b 1 − x2 0 [b > 0, Re ν > −1] 1 π dx cot(νπ) Hν− 12 (b) − Y ν− 12 (b) − J ν− 12 (b) x1−ν Y ν (bx) √ = 2b 1 − x2 0 [b > 0, Re ν < 1] 2 1 ν− 12 √ b xν 1 − x2 J ν (bx) dx = 2ν−1 πb−ν Γ ν + 12 J ν 2 0 b > 0, Re ν > − 12

1 1 b b 1 ν 2 ν− 2 ν−1 √ −ν Jν Yν x 1−x Y ν (bx) dx = 2 πb Γ ν + 2 2 2 0 b > 0, Re ν > − 12

1 ν− 12 √ b b 1 Iν Kν xν 1 − x2 K ν (bx) dx = 2ν−1 πb−ν Γ ν + 2 2 2 0 Re ν > − 12 2

1 1 √ −ν b 1 ν 2 ν− 2 −ν−1 Iν x 1−x I ν (bx) dx = 2 πb Γ ν + 2 2 0

1 ν−1 1 b 1 −ν− 2 − ν sin b xν+1 1 − x2 J ν (bx) dx = 2−ν √ Γ 2 π 0 b > 0, |Re ν| < 12 x1−ν J ν (bx) √

dx = 1 − x2

ET II 129(12)a ET II 24(24)a

ET II 102(28)a

ET II 102(30)a

ET II 24(25)a

ET II 102(31)a

ET II 129(10)a ET II 365(5)a

ET II 25(27)a

6.571

14.

15.

16.

17.12

18.∗

6.568

Bessel functions and powers

b b b b 1 Jν J −ν −Y ν Y −ν x x −1 Y ν (bx) dx = 2 πb Γ ν + 2 2 2 2 2 1 |Re ν| < 12 , b > 0 ET II 103(32)a

∞ 2 ν− 12 b 2ν−1 1 Kν xν x2 − 1 K ν (bx) dx = √ b−ν Γ ν + 2 2 π 1 Re b > 0, Re ν > − 21 ET II 129(11)a

∞ −ν− 12 √ b b 1 − ν Jν Yν x−ν x2 − 1 J ν (bx) dx = −2−ν−1 πbν Γ 2 2 2 1 b > 0, |Re ν| < 12 ET II 25(26)a

∞ ν− 12 2−ν 1 + ν cos b x−ν+1 x2 − 1 J ν (bx) dx = √ b−ν−1 Γ 2 π 1 b > 0, |Re ν| < 12 ET II 25(28) 1 2k k! x(1 − x2 )k I0 (ax)dx = k+1 Ik+1 (a) PBM 2.15.2.6 a 0

∞

∞

1. 0

0

0

ν

2

ν− 12

xν Y ν (bx)

ν−2

dx π = aν−1 J ν (ab) x2 − a2 2

1. 0

−ν

a > 0,

1

xμ Y ν (bx)

π

x2

b > 0,

− 12 < Re ν <

5 2

(μ − ν + 1)

dx π = aμ−1 J ν (ab) + 2μ π −1 aμ−1 cos 2 −a 2 2 μ+ν+1 μ−ν +1 Γ S −μ,ν (ab) ×Γ 2 2 a > 0, b > 0, |Re ν| − 1 < Re μ < 52

ET II (101)(25)

xλ (1 − x)μ−1 J ν (ax) dx =

6.571

√

ET II 101(22) ∞

2.

6.569

689

∞

x2 + a2

12

Γ(μ) Γ(1 + λ + ν)2−ν aν Γ(ν + 1) Γ(1 + λ + μ + ν) λ+1+μ+ν λ+2+μ+ν a2 λ+1+ν λ+2+ν , ; ν + 1, , ;− × 2F 3 2 2 2 2 4 [Re μ > 0, Re(λ + ν) > −1] ET II 193(56)a

μ ab ab dx K 12 (ν±μ) ± x J ν (bx) √ = aμ I 12 (ν∓μ) 2 2 x2 + a 2 Re a > 0, b > 0, Re ν > −1, Re μ < 32

ET II 26(38)

690

Bessel Functions

∞

∞

2.

μ 1 dx x2 + a2 2 − x Y ν (bx) √ 2 2

x + a

ab ab ab ab μ K 12 (μ−ν) − cosec(νπ) I 12 (μ−ν) K 12 (μ+ν) = a cot(νπ) I 12 (μ+ν) 2 2 2 2 Re a > 0, b > 0, Re μ > − 23 , |Re ν| < 1 ET II 104(40)

0

3.

∞

1.

μ dx + x K ν (bx) √ 2 2 x +

a

ab ab ab ab π2 μ a cosec(νπ) J 12 (ν−μ) Y − 12 (ν+μ) − Y 12 (ν−μ) J − 12 (ν+μ) = 4 2 2 2 2 [Re a > 0, Re b > 0] ET II 130(15)

x2 + a2

0

6.572

−μ

x

2

12

x +a

2

12

μ +a

0

6.572

Γ 1+ν−μ dx 2 J ν (bx) √ = W 12 μ, 12 ν (ab) M − 12 μ, 12 ν (ab) ab Γ(ν + 1) x2 + a2 [Re a > 0, b > 0, Re(ν − μ) > −1] ET II 26(40)

∞

2.

x−μ

x2 + a2

12

μ +a

0

K ν (bx) √

dx + a2

x2

1+ν−μ 1−ν−μ Γ 2 2 W 12 μ, 12 ν (iab) W 12 μ, 12 ν (−iab) = 2ab Re b > 0, Re μ + |Re ν| < 1] ET II 130(18), BU 87(6a) Γ

[Re a > 0,

∞

3.

x−μ

x2 + a2

12

0

6.573

∞

1. 0

xν−M+1 J ν (bx)

−a

k +

μ

dx + a2 %

Γ 1+ν+μ ν −μ 1 2 tan π M 12 μ, 12 ν (ab) = − W − 12 μ, 12 ν (ab) ab Γ(ν + 1) 2

$ ν −μ + sec π W 12 μ, 12 ν (ab) 2 Re a > 0, b > 0, |Re ν| < 12 + 12 Re μ ET II 105(42)

Y ν (bx) √

x2

J μi (ai x) dx = 0

i=1 )

ai > 0,

k "

M=

k "

μi

i=1

ai 0,

k " i=1

k + i=1

ai < b < ∞,

aμi i , Γ (1 + μi ) *

0 < Re ν < Re M + 12 k +

3 2

1 2k

−

M=

1 2 k "

ET II 54(42)

μi

i=1

WA 460(16)a, ET II 54(43)

6.576

6.574

Bessel functions and powers

ν +μ−λ+1 ∞ α Γ 2

J ν (αt) J μ (βt)t−λ dt = −ν + μ + λ+1 0 λ ν−λ+1 Γ(ν + 1) 2 β Γ 2

α2 ν +μ−λ+1 ν −μ−λ+1 , ; ν + 1; 2 ×F 2 2 β [Re(ν + μ − λ + 1) > 0, Re λ > −1, 0 < α < β] ν

1.8

2.

691

WA 439(2)a, MO 49

If we reverse the positions of ν and μ and at the same time reverse the positions of α and β, the function on the right hand side of this equation will change. Thus, the right hand side represents α α a function of that is not analytic at = 1. β β For α = β, we have the following equation

ν +μ−λ+1 λ−1 ∞ Γ(λ) Γ α 2

J ν (αt) J μ (αt)t−λ dt = ν +μ+λ+1 ν−μ+λ+1 −ν + μ + λ + 1 0 Γ Γ 2λ Γ 2 2 2 [Re(ν + μ + 1) > Re λ > 0, α > 0] MO 49, WA 441(2)a

If μ − ν + λ + 1 (or ν − μ + λ + 1) is a negative integer, the right hand side of equation 6.574 1 (or 6.574 3) vanishes. 6.575 1.11

∞

0

J ν+1 (αt) J μ (βt)tμ−ν dt = 0 =

∞

2. 0

2

2 ν−μ

[α < β] μ

β α −β Γ(ν − μ + 1)

[α ≥ β]

2ν−μ αν+1 √

[Re(ν + 1) > Re μ > −1]

J ν (x) J μ (x) π Γ(ν + μ) dx = ν+μ ν+μ x 2 Γ ν + μ + 12 Γ ν + 12 Γ μ + 12 [Re(ν + μ) > 0]

6.576 1.

∞

0

xμ−ν+1 J μ (x) K ν (x) dx =

1 2

Γ(μ − ν + 1)

2.11

0

∞

MO 51

aν b ν Γ ν + x−λ J ν (ax) J ν (bx) dx =

[Re μ > −1, 1−λ 2

KU 147(17), WA 434(1)

Re(μ − ν) > −1]

1+λ 2λ (a + b)2ν−λ+1 Γ(ν + 1) Γ 2

1 4ab 1−λ , ν + ; 2ν + 1; ×F ν + 2 2 (a + b)2 [a > 0, b > 0, 2 Re ν + 1 > Re λ > −1]

ET II 370(47)

ET II 47(4)

692

Bessel Functions

ν−λ−μ+1 ν −λ+μ+1 ∞ Γ b Γ 2 2 x−λ K μ (ax) J ν (bx) dx = λ+1 ν−λ+1 2 a Γ(1 + ν) 0

b2 ν−λ+μ+1 ν −λ−μ+1 , ; ν + 1; − 2 ×F 2 2 a [Re (a ± ib) > 0, Re(ν − λ + 1) > |Re μ|] EH II 52(31), ET II 63(4), WA 449(1) ν

3.

∞

4.

−λ

x 0

∞

5.

−λ

x 0

∞

6. 0

8

∞

0

1−λ+μ+ν 1−λ−μ+ν 2−2−λ a−ν+λ−1 bν Γ Γ K μ (ax) K ν (bx) dx = 2 2

Γ(1 − λ) 1−λ−μ−ν 1−λ+μ−ν Γ ×Γ 2 2

b2 1−λ+μ+ν 1−λ−μ+ν , ; 1 − λ; 1 − 2 ×F 2 2 a [Re(a + b) > 0, Re λ < 1 − |Re μ| − |Re ν|] ET II 145(49), EH II 93(36) − 12 λ + 12 μ + 12 ν Γ 12 − 12 λ − 12 μ + 12 ν K μ (ax) I ν (bx) dx = 2λ+1 Γ(ν + 1)a−λ+ν+1

1 1 1 1 1 1 1 1 b2 − λ + 2 μ + 2 ν, 2 − 2 λ − 2 μ + 2 ν; ν + 1; a2 ×F 2 2 [Re (ν + 1 − λ ± μ) > 0, a > b] EH II 93(35) bν Γ

1 2

π(ν − μ − λ) ∞ −λ 2 sin x K μ (ax) I ν (bx) dx π 2 0 Re λ > −1, Re (ν − λ + 1 ± μ) > 0] (see 6.576 5)

x−λ Y μ (ax) J ν (bx) dx = [a > b,

7.

6.577

xμ+ν+1 J μ (ax) K ν (bx) dx = 2μ+ν aμ bν

EH II 93(37)

Γ(μ + ν + 1) (a2 + b2 )μ+ν+1 [Re μ > |Re ν| − 1,

Re b > |Im a|]

ET 137(16), EH II 93(36)

6.577 1.8

∞

0

[a > 0, 2.

12

∞

0

dx = (−1)n cν−μ+2n I μ (ac) K ν (bc) x2 + c2 Re c > 0, 2 + Re μ − 2n > Re ν > −1 − n, n ≥ 0 an integer]

ET II 49(13)

dx = (−1)n cμ−ν+2n I ν (bc) K μ (ac) + c2 Re ν − 2n + 2 > Re μ > −n − 1, n ≥ 0 an integer]

ET II 49(15)

xν−μ+1+2n J μ (ax) J ν (bx) b > a,

xμ−ν+1+2n J μ (ax) J ν (bx)

[b > 0,

a > b,

Re c > 0,

x2

6.578

6.578

Bessel functions and powers

∞

1. 0

∞

2. 0

2−1 aλ bμ c−λ−μ− Γ λ+μ+ν+ 2

x−1 J λ (ax) J μ (bx) J ν (cx) dx = λ+μ−ν + Γ(λ + 1) Γ(μ + 1) Γ 1 − 2

a2 b 2 λ+μ−ν + λ+μ+ν + , ; λ + 1, μ + 1; 2 , 2 × F4 2 2 c c 5 Re(λ + μ + ν + ) > 0, Re < , a > 0, b > 0, c > 0, c > a + b ET II 351(9) 2 x−1 J λ (ax) J μ (bx) K ν (cx) dx =

∞

3. 0

∞

4. 0

∞

5. 0

6.11

7.11

0

8.11

0

2−2 aλ bμ c−−λ−μ +λ+μ+ν +λ+μ−ν Γ Γ Γ(λ + 2 2

1) Γ(μ + 1) a2 +λ+μ−ν +λ+μ+ν b2 , ; λ + 1, μ + 1; − 2 , − 2 × F4 2 2 c c [Re( + λ + μ) > |Re ν|, Re c > |Im a| + |Im b|] ET II 373(8)

xλ−μ−ν+1 J ν (ax) J μ (bx) J λ (cx) dx = 0 Re λ > −1, Re(λ − μ − ν) < 12 ,

c > b > 0,

0
2λ−μ−ν−1 aν bμ Γ(λ) xλ−μ−ν−1 J ν (ax) J μ (bx) J λ (cx) dx = λ c Γ(μ + 1) Γ(ν + 1) Re λ > 0, Re(λ − μ − ν) < 52 , c > b > 0,

0
x1+μ Y μ (ax) J ν (bx) J ν (cx) dx = 0

0 < a < c − b]

[0 |Im b| + |Im c|, Re ν > −1, Re(μ + ν) > −1 WA 452(2), ET II 64(12)

∞

− 1 μ− 1 μ+ 1 1 1 1 xμ+1 I ν (ax) K μ (bx) J ν (cx) dx = √ a−μ−1 bμ c−μ−1 e−(μ− 2 ν+ 4 )πi v 2 + 1 2 4 Q ν− 12 (iv) 2 2π 2acv = b2 − a2 + c2 , Re b > |Re a| + |Im c|, Re ν > −1, Re(μ + ν) > −1 ET II 66(22)

∞

x1−μ J μ (ax) J ν (bx) J ν (cx) dx 1 2 −μ μ− 12 μ−1 (μ− 12 )πi Q 2 −μ1 (cosh u) a (bc) (sinh u) sin[(μ − ν)π]e = ν− 2 π3 1 1 −μ μ− 12 μ−1 2 −μ = √ a (bc) (sin v) P ν− 1 (cos v) 2 2π

[a > b + c]

=0

[0 < a < |b − c|] 2

2

2

2bc cosh u = a − b − c ,

2

2

2

2bc cos v = b + c − a ,

b > 0,

c > 0;

[|b − c| < a −1, Re μ > − 12

694

Bessel Functions

∞

9. 0

J ν (ax) J ν (bx) J ν (cx)x1−ν dx = 0 Δ=

11.11

2 Δ (abc)ν Γ ν + 12 Γ 12

[c2 − (a − b)2 ] [(a + b)2 − c2 ],

[|a − b| < c < a + b]

a > 0,

b > 0,

c > 0,

Re ν > − 12

∞

ET II 67(30)

ν+ 1

1

(ab)−ν−1 cν e−(ν+ 2 )πi Q μ− 21 (u) 2 x K μ (ax) I μ (bx) J ν (cx) dx = √ 1 1 2 2 ν+ 4 2π (u − 1) 2abu = a2 + b2 + c2 , Re a > |Re b| + |Im c|, Re ν > −1, Re(μ + ν) > −1

∞

0

13.12

1 4

2ν−1

(Δ > 0 is equal to the area of a triangle whose sides are a, b, and c.) √ ν ∞ πc Γ(ν + μ + 1) Γ(ν − μ + 1) −ν− 12 ν+1 x K μ (ax) K μ (bx) J ν (cx) dx = P μ− 1 (u) 1 2 ν+ 1 2 0 2 3 (ab)ν+1 (u2 − 1) 2 4 2abu = a2 + b2 + c2 , Re(a + b) > |Im c|, Re (ν ± μ) > −1, Re ν > −1

0

12.8

[0 < c ≤ |a − b| or c ≥ a + b] ν−1

=

10.11

6.578

ν+1

2

xν+1 [J ν (ax)] Y ν (bx) dx = 0 −ν− 12 23ν+1 a2ν b−ν−1 2 1 b − 4a2 = √ πΓ 2 −ν

0 0, a > 0,

=0 −ν− 12 23ν+1 a2ν b−ν−1 2 1 b − 4a2 =− √ πΓ 2 −ν

|Re ν| < 12 , 2a < b,

0 0,

2ν Γ(μ + ν + 1) (sin ϕ)

π > ϕ > 0, 2

xν+1 J ν (ax) K ν (bx) J ν (cx) dx =

3ν

cos α2

2ν+2

aν+2 (cos ψ) π 0 < ψ < , Re ν > −1, 2 ν

1 2

2ν+1

P −μ ν (cos α) Re(μ + ν) > −1 ET II 64(11)

2 (abc) Γ ν + ν+ 12 √ 2 π (a2 + b2 + c2 ) − 4a2 c2 Re b > |Im a|,

c > 0,

Re ν > − 12

ET II 63(8)

6.581

16.

8

Bessel functions and powers

∞

ν+1

x 0

I ν (ax) K ν (bx) J ν (cx) dx =

695

23ν (abc)ν Γ ν + 12

ν+ 12 √ 2 2 π (b − a2 + c2 ) + 4a2 c2 Re b > |Re a| + |Im c|;

Re ν > − 21

ET II 65(18)

6.579

∞

1. 0

x2ν+1 J ν (ax) Y ν (ax) J ν (bx) Y ν (bx) dx =

∞

2. 0

∞

3. 0

∞

4.

a2ν Γ(3ν + 1) 3 a2 1 1 ; F ν + , 3ν + 1; 2ν + 3 2 2 b2 2πb4ν+2 Γ 2 − ν Γ 1 2ν + 2 1 EH II 94(45), ET II 352(15) 0 < a < b, − 3 < Re ν < 2

x2ν+1 J ν (ax) K ν (ax) J ν (bx) K ν (bx) dx Γ ν + 12 Γ 3ν+1 2ν−3 a2ν Γ ν+1 a4 2 2 1 3ν + 1 √ ; 2ν + 1; 1 − 4 F ν + 2, = 2 b πb4ν+2 Γ(ν + 1) 0 < a < b, Re ν > − 31 ET II 373(10) 4

x1−2ν [J ν (ax)] dx = 1−2ν

x 0

Γ(ν) Γ(2ν) 2 2π Γ ν + 12 Γ(3ν)

[Re ν > 0]

a2ν−1 Γ(ν) F [J ν (ax)] [J ν (bx)] dx = 2πb Γ ν + 12 Γ 2ν + 12 2

2

ET II 342(25)

1 a2 1 ν, − ν; 2ν + ; 2 2 2 b ET II 351(10)

6.581

a

1. 0

2.8

xλ−1 J μ (x) J ν (a − x) dx = 2λ

∞ " (−1)m Γ(λ + μ + m) Γ(λ + m) J λ+μ+ν+2m (a) m! Γ(λ) Γ(μ + m + 1) m=0 [Re(λ + μ) > 0, Re ν > −1]

ET II 354(25) a

0

xλ−1 (a − x)−1 J μ (x) J ν (a − x) dx =

∞ 2λ " (−1)m Γ(λ + μ + m) Γ(λ + m) (λ + μ + ν + 2m) J λ+μ+ν+2m (a) aν m=0 m! Γ(λ) Γ(μ + m + 1)

[Re(λ + μ) > 0,

a

3. 0

4. 0

a

Re ν > 0]

ET II 354(27)

Γ μ + 12 Γ ν + 12 μ+ν+ 1 2 J a x (a − x) J μ (x) J ν (a − x) dx = √ μ+ν+ 12 (a) 2π Γ(μ + ν + 1) Re μ > − 12 , Re ν > − 12 μ

ν

ET II 354(28), EH II 46(6)

Γ μ + Γ ν + 32 μ+ν+ 3 μ ν+1 2 J a x (a − x) J μ (x) J ν (a − x) dx = √ μ+ν+ 12 (a) 2π Γ(μ + ν + 2) Re ν > −1, Re μ > − 21 1 2

ET II 354(29)

696

Bessel Functions

5. 0

a

μ

∞

0

6.583

2μ Γ μ + 12 Γ(ν − μ) μ √ a J ν (a) J μ (x) J ν (a − x) dx = π Γ(μ + ν + 1) Re ν > Re μ > − 21

−μ−1

x (a − x)

6.582

∞

−μ

xμ−1 |x − b|

μ−1

x 0

6.582

1 K μ (|x − b|) K ν (x) dx = √ (2b)−μ Γ 12 − μ Γ(μ + ν) Γ(μ − ν) K ν (b) π b > 0, Re μ < 12 , Re μ > |Re ν|

−μ

(x + b)

ET II 355(30)

√ π Γ(μ + ν) Γ(μ − ν) K ν (b) K μ (x + b) K ν (x) dx = 2μ bμ Γ μ + 12 [|arg b| < π, Re μ > |Re ν|]

ET II 374(14)

ET II 374(15)

6.584 1.

8

∞ x−1

0

2.

8

∞

0

∞

3. 0

4. 0

∞

(1) πi πi H (1) axe (ax) − e H ν ν

m d πi (1) −2 r dx = H (ar) ν m+1 2m m! r dr (x2 − r2 ) m = 0, 1, 2, . . . , Im r > 0, a > 0, |Re ν| < Re < 2m + 72

WA 465

1 1 x−1 cos ( − ν)π J ν (ax) + sin ( − ν)π Y ν (ax) m+1 dx 2 2 (x2 + k 2 )

m −2 d (−1)m+1 = m k K ν (ak) 2 · m! k dk m = 0, 1, 2, . . . , Re k > 0, a > 0, |Re ν| < Re < 2m + 72 WA 466(2) x1−ν dx am K ν+m (ak) {cos νπ J ν (ax) − sin νπ Y ν (ax)} = m m+1 2 · m!k ν+m (x2 + k 2 ) m = 0, 1, 2, . . . , Re k > 0, a > 0, −2m −

3 2

< Re ν < 1

WA 466(3)

' ( x−1 cos 12 − 12 ν − μ π J ν (ax) + sin 12 − 12 ν − μ π Y ν (ax) μ+1 dx (x2 + k 2 ) ⎡ ν 1

1 Γ 2 + 12 ν a2 k 2 +ν +ν πk −2μ−2 2 ak ⎣ 1F 2 ; − μ, ν + 1; = 2 sin νπ · Γ(μ + 1) Γ(ν + 1) Γ 12 + 12 ν − μ 2 2 4 ⎤ 1 −ν 1

Γ − 1ν a2 k 2 ⎦ −ν −ν 2 ak 1 2 1 2 1F 2 ; − μ, 1 − ν; − 2 2 4 Γ(1 − ν) Γ 2 − 2 ν − μ WA 407(1) a > 0, Re k > 0, |Re ν| < Re < 2 Re μ + 72

6.591

5.12

Powers and Bessel functions of complicated arguments

∞

0

697

⎤⎧ ⎡ ⎛ ⎞ ⎤ n ⎨ " 1 ⎣ J μj (bn x)⎦ cos ⎣ ⎝ + μj − ν ⎠ π ⎦ J ν (ax) ⎩ 2 j=1 j=1 ⎫ ⎞ ⎤ ⎡ ⎛ n ⎬ x−1 " 1 μj − ν ⎠ π ⎦ Y ν (ax) dx + sin ⎣ ⎝ + ⎭ x2 + k 2 2 j=1 ⎤ ⎡ n + I μj (bn k)⎦ K ν (ak)k −2 = −⎣ ⎡

n +

j=1

⎡ ⎣Re k > 0,

a>

n "

⎛ Re ⎝ +

|Re bj |,

j=1

n "

⎞

⎤

μj ⎠ > |Re ν|⎦

WA 472(9)

j=1

6.59 Combinations of powers and Bessel functions of more complicated arguments 6.591

∞

1. 0

x

K ν (bx) dx =

√

1

2πb−ν−1 aν+ 2 J 1+2ν

√

∞

1

x2ν+ 2 Y ν+ 12

a

x

K ν (bx) dx =

∞

4. 0

1

x2ν+ 2 K ν+ 12

a

∞

5.

1

x−2ν+ 2 J ν− 12

−2ν+ 12

x 0

0

Re b > 0,

Re ν > −1]

x

1 √

1 √

√ 1 K ν (bx) dx = 2πb−ν−1 aν+ 2 K 2ν+1 e 4 iπ 2ab K 2ν+1 e− 4 iπ 2ab

ET II 146(56) [Re a > 0, Re b > 0]

√ √ 1 K ν (bx) dx = 2πbν−1 a 2 −ν K 2ν−1 2ab x √

√ 2ab + cos(νπ) Y 2ν−1 2ab × sin(νπ) J 2ν−1

a

[a > 0,

6.12

Re ν > −1]

ET II 143(41) ∞

0

Re b > 0,

√

√

√ 1 2πb−ν−1 aν+ 2 Y 2ν+1 2ab K 2ν+1 2ab [a > 0,

3.

√

2ab K 1+2ν 2ab ET II 142(35)

0

a

[a > 0,

2.

1

x2ν+ 2 J ν+ 12

Y ν− 12

a

x

Re b > 0,

1

a

ET II 142(34)

√

π ν−1 1 −ν b a 2 sec(νπ) K 2ν−1 2ab √

√ 2 2ab − J 1−2ν 2ab × J 2ν−1

K ν (bx) dx = −

[a > 0, ∞

Re ν < 1]

Re ν < 1]

ET II 143(40)

J ν (bx) dx x−2ν+ 2 J 12 −ν x 1 π i cosec(2νπ)bν−1 a 2 −ν e2νπi J 1−2ν (u) J 2ν−1 (v) − e−2νπi J 2ν−1 (u) J 1−2ν (v) =− 2 √ √ u = 2abeπi/4 , v = 2abe−πi/4 , a > 0, b > 0, − 21 < Re ν < 3 ET II 58(12)

698

Bessel Functions

∞

7. 0

1

x−2ν+ 2 K ν− 12

a

x

Y ν (bx) dx =

6.592

√

√

√ 1 2πbν−1 a 2 −ν Y 2ν−1 2ab K 2ν−1 2ab b > 0, Re a > 0, Re ν > 16

ET II 113(30)

∞ aν− bν Γ 12 μ + 12 − 12 ν b −1 dx = 2ν−+1 x J μ (ax) J ν x 2 Γ(ν + 1) Γ 12 μ + 12 ν − 12 + 1 0

ν +μ− a2 b 2 ν −μ− + 1, + 1; × 0 F 3 ν + 1, 16 2 2 aμ bμ+ Γ 12 ν − 12 μ − 12 + 2μ++1 2 Γ(μ + 1) Γ 12 μ + 12 ν + 12 + 1

μ−ν + ν+μ+ a2 b 2 × 0 F 3 μ + 1, + 1, + 1; 2 2 16 3 a > 0, b > 0, − Re μ + 2 < Re < Re ν + 32

8.

6.592 1.

12

1

0

2.12

0

1

λ

μ−1

x (1 − x)

WA 480(1)

√ Γ(μ) Γ λ + 1 + 12 ν −ν ν Y ν a x dx = 2 a cot(νπ) Γ(1 + ν) Γ λ + 1 + μ + 12 ν

a2 1 1 × 1 F 2 λ + 1 + 2 ν; 1 + ν, λ + 1 + μ + 2 ν; − 4 Γ(μ) Γ λ + 1 − 12 ν ν −ν −2 a cosec(νπ) Γ(1 − ν) Γ λ + 1 + μ − 12 ν

a2 × 1 F 2 λ − 12 ν + 1; 1 − ν, λ + 1 + μ − 12 ν; − 4 Re λ > −1 + 12 |Re ν|, Re μ > 0 ET II 197(76)a

√ xλ (1 − x)μ−1 K ν a x dx =2

+2

ν−1 −ν

a

Γ(ν) Γ(μ) Γ λ + 1 − 12 ν a2 1 1 1 F 2 λ + 1 − 2 ν; 1 − ν, λ + 1 + μ − 2 ν; 4 Γ λ + 1 + μ − 12 ν

Γ(−ν) Γ λ + 1 + 12 ν Γ(μ) 1 a2 1 a 1 F 2 λ + 1 + 2 ν; 1 + ν, λ + 1 + μ + ν; 2 4 Γ λ + 1 + μ + 12 ν

−1−ν ν

2ν−1 = ν Γ(μ) G 21 13 a

a2 4

!ν ! −λ !2 ! ν, 0, ν − λ − μ 2 OB 159 (3.16)

Re λ > −1 + 12 |Re ν|,

Re μ > 0

ET II 198(87)a

6.592

3.

11

Powers and Bessel functions of complicated arguments

∞

1

∞

4. 1

5. 6. 7.12

8. 9.

1

∞

λ

μ−1

x (x − 1)

√ J ν a x dx = 22λ a−2λ G 20 13

a2 4

! !0 ! ! −μ, λ + 1 ν, λ − 1 ν Γ(μ) 2 2 a > 0, 0 < Re μ < 34 − Re λ

√ xλ (x − 1)μ−1 K ν a x dx = Γ(μ)22λ−1 a−2λ G 30 13

√ 1 1 x− 2 (1 − x)− 2 J ν a x dx = π J 1 ν

2

699

ET II 205(36)a

! a !! 0 4 ! −μ, 12 ν + λ, − 12 ν + λ 2

[Re a > 0,

Re μ > 0]

ET II 209(60)a

1 a [Re ν > −1] ET II 194(59)a 2 2 0

2 1 √ 1 − 12 − 12 a x (1 − x) I ν a x dx = π I 1 [Re ν > −1] ET II 197(79) ν 2 2 0 1 a a

a

√ 1 1 1√ + I − ν2 K ν2 x− 2 (1 − x)− 2 K ν a x dx = π sec (νπ) I ν2 2 2 2 2 0 [|Re ν| < 1] ET II 198(85)a ∞

2 √ 1 1 a x− 2 (x − 1)− 2 K ν a x dx = K ν2 [Re a > 0] ET II 208(56)a 2 1 # 1 a 2 a 2 $ √ − 12 − 12 x (1 − x) Y ν a x dx = π cot(νπ) J ν2 − cosec(νπ) J − ν2 2 2 0 [|Re ν| < 1]

10. 1

∞

11. 1

1

1

∞

14. 1

15. 1

1 2

Re ν +

3 4

ET II 205(34)a

√ 1 x− 2 ν (x − 1)μ−1 J −ν a x dx = Γ(μ)2μ a−μ [cos(νπ) J ν−μ (a) − sin(νπ) Y ν−μ (a)] a > 0, 0 < Re μ < 12 Re ν + 34 √ 1 x− 2 ν (x − 1)μ−1 K ν a x dx = Γ(μ)2μ a−μ K ν−μ (a) [Re a > 0,

∞

13.

0 < Re μ <

ET II 205(35)a

∞

12.

√ 1 x− 2 ν (x − 1)μ−1 J ν a x dx = Γ(μ)2μ a−μ J ν−μ (a) a > 0,

ET II 195(68)a

√ 1 x− 2 ν (x − 1)μ−1 Y ν a x dx = 2μ a−μ Y ν−μ (a) Γ(μ) a > 0,

0 < Re μ <

1 2

ET II 209(59)a

Re ν +

3 4

ET II 206(40)a

√ 1 (1) a x dx = 2μ a−μ H ν−μ (a) Γ(μ) x− 2 ν (x − 1)μ−1 H (1) ν [Re μ > 0,

∞

Re μ > 0]

Im a > 0]

ET II 206(45)a

Im a < 0]

ET II 207(48)a

√ 1 (2) a x dx = 2μ a−μ H ν−μ (a) Γ(μ) x− 2 ν (x − 1)μ−1 H (2) ν [Re μ > 0,

700

Bessel Functions

1

16. 0

17. 0

1

6.593

√ 1 22−ν a−μ s μ+ν−1,μ−ν (a) x− 2 ν (1 − x)μ−1 J ν a x dx = Γ(ν) [Re μ > 0]

ET II 194(64)a

√ 1 22−ν a−μ cot(νπ) s μ+ν−1,μ−ν (a) x− 2 ν (1 − x)μ−1 Y ν a x dx = Γ(ν) −2μ a−μ cosec(νπ) J μ−ν (a) Γ(μ) [Re μ > 0,

6.593 1. 2.

Re ν < 1]

ET II 196(75)a

2 √ a 1 b > 0, Re ν > − 12 x J 2ν−1 a x J ν (bx) dx = ab−2 J ν−1 2 4b 0

2

2 ∞ √ √ a a πa − Lν−1 x J 2ν−1 a x K ν (bx) dx = 2 I ν−1 4b 4b 4b 0 Re b > 0, Re ν > − 21

6.594 1.

∞√

∞

0

√ √ √ 1 x I 2ν−1 a x J 2ν−1 a x K ν (bx) dx = π2−ν a2ν−1 b−2ν− 2 J ν− 12 ν

[Re b > 0, ∞

2. 0

a2 2b

ET II 58(15)

ET II 144(44)

Re ν > 0]

ET II 148(65)

√ √ xν I 2ν−1 a x Y 2ν−1 a x K ν (bx) dx

√ −ν−1 2ν−1 −2ν− 1 2 cosec(νπ) π2 a b

2

2

2 a a a + cos(νπ) J ν− 12 + sin(νπ) Y ν− 12 × H 12 −ν 2b 2b 2b [Re b > 0, Re ν > 0] ET II 148(66) ∞ √ √ xν J 2ν−1 a x K 2ν−1 a x K ν (bx) dx 0

2

2 a a 2 −ν−2 2ν−1 −2ν− 12 − Y 12 −ν =π 2 a b cosec(νπ) H 12 −ν 2b 2b [Re b > 0, Re ν > 0] ET II 148(67) =

3.

6.595 1. 0

∞

ν+1

x )

J ν (cx)

n + i=1

zi = x2 + b2i ,

zi−μi J μi (ai zi ) dx = 0 ai > 0,

Re bi > 0,

n "

ai < c;

Re

i=1

n " 1 1 μi − n+ 2 2 i=1

* > Re ν > −1

EH II 52(33), ET II 60(26)

2. 0

∞

n +

n +

−μi bi J μi (ai bi ) xν−1 J ν (cx) zi−μi J μi (ai zi ) dx = 2ν−1 Γ(ν)c−ν i=1 i=1 ) * n n " " 1 3 zi = x2 + b2i , ai > 0, Re bi > 0, n+ > Re ν > 0 ai < c, Re μi + 2 2 i=1 i=1 EH II 52(34), ET II 60(27)

6.596

6.596

Powers and Bessel functions of complicated arguments

∞

1. 0

701

x2μ+1 2μ Γ(μ + 1) J ν a x2 + z 2 dx = μ+1 ν−μ−1 J ν−μ−1 (az) ν a z (x2 + z 2 )

1 1 ν− > Re μ > −1 a > 0, Re 2 4

√ ∞ a

a

J ν a t2 + 1 π √ Y ν2 [Re ν > −1, dt = − J ν2 2 2 2 t2 + 1 0 ∞

x2μ+1 2μ Γ(μ + 1) K ν a x2 + z 2 dx = K ν−μ−1 (az) ν aμ+1 z ν−μ−1 (x2 + z 2 ) 0

WA 457(5)

2. 3.

4.8

0

5.8

6.6

∞

J ν (bx)

' √ ( J μ−1 a x2 + z 2 (x2 + z 2 )

1 1 2 μ+ 2

[a > 0, xν+1 dx =

Re μ > −1]

WA 457(6)

Re(μ + 2) > Re ν > −1]

' √ ( ∞ J μ a x2 + z 2 ν−1 2ν−1 Γ(ν) J μ (az) J ν (bx) x dx = μ bν zμ (x2 + z 2 ) 0 [Re(μ + 2) > Re ν > 0,

ET II 59(19)

b > a > 0] WA 459(12)

√ ∞ J μ a x2 + z 2 ν+1 J ν (bx) dx μ x (x2 + z 2 ) 0

=

8.8

MO 46

=0

7.8

aμ−1 z ν K ν (bz) 2μ−1 Γ(μ) [a < b,

a > 0]

bν aμ

√

a2 − b 2 z

μ−ν−1

[0 < a < b] J μ−ν−1 z a2 − b2

[a > b > 0]

[Re μ > Re ν > −1] WA 415(1) √ √ μ−ν−1 ∞

K μ a x2 + z 2 ν+1 a2 + b 2 bν J ν (bx) x dx = μ K μ−ν−1 z a2 + b2 μ a z (x2 + z 2 ) 0 π a > 0, b > 0, Re ν > −1, |arg z| < KU 151(31), WA 416(2) 2

ν ∞

μ 1 u π 2 2 − 2 ν+1 2 2 · μ x −y J ν (ux) K μ v x − y x dx = exp −iπ μ − ν − 2 2 v 0 *μ−ν−1 )√

2 2 u +v (2) · H μ−ν−1 y u2 + v 2 y 1α 1α 2 1 Re μ < 1, Re ν > −1, u > 0, v > 0, y > 0; x − y 2 2 = e 2 απi y 2 − n2 2 if x < y

702

9.

Bessel Functions

8

∞

0

− μ v x2 + y 2 x2 + y 2 2 xν+1 dx J ν (ux) H (2) μ uν = μ v ⎡ ⎣ Re μ > Re ν > −1,

u > 0,

6.597

*μ−ν−1 )√

v 2 − u2 (2) H μ−ν−1 y v 2 − u2 y [u < v]

v > 0,

y > 0;

arg

v 2 − u2 = 0, for v > u;

⎤ 2 μ − ν − 1 1 σ ⎦ arg v − u2 = −πσ for v < u, where σ = or σ = 2 2

10.8

MO 43

∞

0

xν−1 2ν−1 Γ(ν) J μ (az) J μ (γz) J ν (bx) J μ a x2 + z 2 J μ γ x2 + z 2 dx = μ μ bν zμ (x2 + z 2 ) z a > 0; b > a + γ; γ > 0, Re 2μ + 52 > Re ν > 0 WA 459(14)

n

+ −μ x J μ (ak x) J μ ak t2 + x2 (t2 + x2 )−nμ dt = 2ν−1 b−ν Γ(ν) 0 k=1 k=1 ) *

n + 1 1 x > 0, ai > 0, b > > Re ν > 0 ak ; Re nμ + n + MO 43 2 2 k=1 ∞ 2 √ 2 Jμ a + x2 2ν−2 Γ ν − 12 8 √ Hν (2a) 12. x dx = Re ν > 12 WA 457(8) 2 + x2 )ν ν+1 π (a 2a 0 ∞ 1 − 1 μ −1 6.597 tν+1 J μ b t2 + y 2 2 t2 + y 2 2 t2 + β 2 J ν (at) dt 0 1 − 1 μ = β ν J μ b y 2 − β 2 2 y 2 − β 2 2 K ν (aβ)

11.

8

∞

J ν (bt)tν−1

n +

[a ≥ b, 6.598

1

0

Re β > 0,

−1 < Re ν < 2 + Re μ]

EH II 95(56)

√ √ − 1 (ν+μ+1) μ ν x 2 (1 − x) 2 J μ a x J ν b 1 − x dx = 2aμ bν a2 + b2 2 J ν+μ+1 a2 + b 2 [Re ν > −1,

Re μ > −1]

EH II 46a

6.61 Combinations of Bessel functions and exponentials 6.611 1.

∞

e

0

2. 0

−αx

J ν (bx) dx =

b−ν

√ ν α2 + b2 − α √ α2 + b2

[Re ν > −1,

Re (α ± ib) > 0] EH II 49(18), WA 422(8)

∞

− 1 e−αx Y ν (bx) dx = α2 + b2 2 cosec(νπ) # −ν ν $ 2 1 1 ν 2 2 −ν 2 2 2 × b α +b α +b +α cos(νπ) − b +α [Re α > 0,

b > 0,

|Re ν| < 1]

MO 179, ET II 105(1)

6.612

Bessel functions and exponentials

∞

3.

e−αx K ν (bx) dx =

sin(νθ) π b sin(νπ) sin θ

703

π as b → ∞ 2 ET II 131(22)

ν −ν π cosec(νπ) −ν ν 2 2 2 2 b α+ α −b = √ −b α −b +α 2 α2 − b2 [|Re ν| < 1, Re(α + b) > 0]

0

cos θ =

4.8

∞

0

α ; b

with θ →

ET I 197(24), MO 180

√ ν b−ν α − α2 − b2 −αx √ e I ν (bx) dx = α2 − b2

[Re ν > −1,

Re α > |Re b|] MO 180, ET I 195(1)

) √ √ 2ν *& ν % ∞ α + α2 + b2 α2 + b2 − α i (1,2) −αx √ cos(νπ) − 1± e H ν (bx) dx = sin(νπ) b2ν bν α2 + b2 0

5.

(2) [−1 < Re ν < 1; a plus sign corresponds to the function H (1) ν , a minus sign to the function H ν ].

∞

6. 0

∞

7. 0

9.12

10.

10

1 (2) e−αx H 0 (bx) dx = √ α2 + b2

1−

α 2i ln + π b )

% 1+

α 2i ln + π b

1+

∞

b

α dα

a

∞ 0

dk J 1 (kα)e

−k|β|

= a

b

|β|

1− α2 + β 2

α 2

*&

MO 180, ET I188(54, 55)

b

[Re α > |Im b|] *& α 2 1+ b

MO 180, ET I 188(53)

[Re α > |Im b|]

MO 180, ET I 188(53)

[Re α > |Im b|]

MO 47, ET I 187(44)

√ −2 α + α2 + b2 −αx e Y 0 (bx) dx = √ ln b π α2 + b2 0 ∞ α arccos b e−αx K 0 (bx) dx = √ b2 − α2 0 α2 1 α + = √ ln −1 b b2 α2 − b2

8.

1 (1) e−αx H 0 (bx) dx = √ 2 α + b2

)

%

[0 < a < b] [0 ≤ b < a]

WA 424, ET II 131(22) MO 48

dα (see 3.241 6)

6.612 1.

e−2αx J 0 (x) Y 0 (x) dx =

− 1 K α α2 + 1 2 1

π (α2 + 1) 2 ∞ 1 1 e−2αx I 0 (x) K 0 (x) dx = K 1 − α2 2 2 ) 0 1 * 1 2 1 K 1− 2 = 2α α 0

2.

∞

[Re α > 0]

ET II 347(58)

[0 < α < 1] [1 < α < ∞] ET II 370(48)

704

Bessel Functions

∞

3.

e

−αx

0

5. 6.

2 1 α + b2 + γ 2 J ν (bx) J ν (γx) dx = √ Q ν− 12 2bγ π γb Re (α ± ib ± iγ) > 0, γ > 0, Re ν > − 12

WA 426(2), ET II 50(17)

2 2b 2 e−αx [J 0 (bx)] dx = √ K √ π α2 + 4b2 α2 + 4b2 0

2 ∞ 2α + b2 K √α2b+b2 − 2 α2 + b2 E √α2b+b2 √ e−2αx J12 (bx) dx = πb2 α2 + b2 0 ∞ r2 e−3x I l (x) I m (x) I n (x) dx = r1 g + 2 + r3 π g 0 where √

3−1 2 1 11 Γ2 Γ g= 3 96π 24 24

4.

6.612

∞

MO 178

WA 428(3)

and (lmn) 000 100 110 111 200 210 211 220 221 222 300 310 311 320 321 322 330 331 332 333 400 410 411 420 421 422 430 431

r1 1 1 5/12 − 1/8 10/3 3/8 − 2/3 73/36 − 15/16 5/8 35/2 − 79/36 − 11/4 319/48 − 125/36 35/16 50/3 − 35/3 35/9 − 35/16 994/9 − 515/16 − 9/2 12907/120 − 229/16 35/3 2641/48 − 1505/36

r2 0 0 − 1/2 3/4 2 − 9/4 2 − 29/6 21/8 − 27/20 21 − 85/6 21/2 − 119/8 269/30 − 213/40 − 1046/25 148/5 − 1012/105 1587/280 542/3 − 879/8 357/5 − 13903/10 1251/40 − 1024/35 − 28049/200 118051/1050

r3 0 − 1/3 0 0 −2 1/3 0 0 0 0 −13 4 − 2/3 − 1/3 0 0 0 0 0 0 −92 115/3 −12 −6 1 0 1/3 0

(lmn) 432 433 440 441 442 443 444 500 510 511 520 521 522 530 531 532 533 540 541 542 543 544 550 551 552 553 554 555

r1 525/32 − 595/72

6025/36 − 29175/224 2975/48 − 539/32 77/8 9287/12 − 189029/180 275/4 2897/16 − 937/12 509/8 3589/18 − 1329/8 2555/36 − 2233/48 18471/32 − 1390/3 7777/32 − 5621/72 1155/32 197045/108 − 12023/8 1683/2 − 5159/16 24563/312 − 9251/208

r2 − 4617/112 8809/420 − 620161/1470 131379/400 − 31231/200 119271/2800 − 186003/7700 3005/2 − 138331/50 5751/10 − 15123/20 27059/30 − 4209/28 − 1993883/3075 297981/700 − 187777/1050 164399/1400 − 28493109/19600 286274/245 − 1715589/2800 4550057/23100 − 560001/6160 − 101441689/22050 18569853/4900 − 5718309/2695 2504541/3080 − 1527851/77000 12099711/107800

r3 0 0 0 0 0 0 0 − 2077/3 348 −150 − 229/3 24 0 0 − 4/3 0 0 − 1/3 0 0 0 0 0 0 0 0 0 0

6.616

6.613 6.614

Bessel functions and exponentials

11

∞

e

−xz

0

∞

e

1.

−αx

0

J ν+ 12

x2 2

dx =

705

π Γ(ν + 1) πi √ D −ν−1 ze 4 i D −ν−1 ze− 4 π

[Re ν > −1]

2

2

2 √ π b b b b I 12 (ν−1) − I 12 (ν+1) J ν b x dx = exp − 3 4 α 8α 8α 8α 1 −b2 /4α = e α

MO 122

[ν = 0] MO 178

∞

2. 0

√

e e−αx Y 2ν 2 bx dx = √ αb

b − 12 α

b Γ(ν + 1) M 12 ,ν − cosec(νπ) W cot(νπ) Γ(2ν + 1) α |Re ν| < 1]

ET I 188(50)a

[Re α > 0,

1 b ∞ √

e2 α b −αx e K 2ν 2 bx dx = √ Γ(ν + 1) Γ(1 − ν) W − 12 ,ν α 2 αb 0

Re ν > −1]

ET I 197(20)a

1 b 2 α

4.

$ b 1 2 ,nu α

[Re α > 0,

∞ √

Γ(ν + 1) b e M − 12 ,ν e−αx I 2ν 2 bx dx = √ Γ(2ν + 1) α αb 0

3.

#

ET I 199(37)a [Re α > 0, |Re ν| < 1]

2 √ b π b b b K1 − K0 5. e−αx K 1 b x dx = exp MO 181 3 8 α 8α 8α 8α 0

∞ √ √ β2 + γ2 2βγ 1 exp − [Re ν > −1] 6.615 e−αx J ν 2β x J ν 2γ x dx = I ν α α α 0

∞

2

2

MO 178

6.616 1. 2. 3.

1 e−αx J 0 b x2 + 2γx dx = √ exp γ α − α2 + b2 α2 + b2 0 ∞

1 e−αx J 0 b x2 − 1 dx = √ exp − α2 + b2 α2 + b2 1 √ ∞ 2 2

eiα r +x (1) itx 2 2 e H 0 r α − t dt = −2i √ 2 2 −∞ r + x 0 ≤ arg α2 − t2 < π, 0 ≤ arg α < π;

∞

e −∞

5.3

1

−1

MO 179

r and x are real

MO 49

r and x are real

MO 49

√

∞

4.

MO 179

−itx

(2) H0

2 2

e−iα r +x r α2 − t2 dt = 2i √ 2 2 r +x 2 2 −π < arg α − t ≤ 0,

−π < arg α ≤ 0,

−1/2 e−ax I 0 b 1 − x2 dx = 2 a2 + b2 sinh a2 + b2 [a > 0,

b > 0]

706

Bessel Functions

6.8

∞

0

6.617

∞

1. 0

∞

2. 0

6.617

∞

" P n (x) e−xy J 0 y 1 − x2 /(α + y) dy = n! n+1 α n=0

K q−p (2z sinh x) e(p+q)x dx =

K 0 (2z sinh x) e−2px dx = −

π 4

π2 [J p (z) Y q (z) − J q (z) Y p (z)] 4 sin[(p − q)π] [Re z > 0, −1 < Re(p − q) < 1] # J p (z)

∂ Y p (z) ∂ J p (z) − Y p (z) ∂p ∂p

MO 44

$

[Re z > 0] 6.618

∞

1.

e

−αx2

0

2

2 √ π b b I 12 ν J ν (bx) dx = √ exp − 8α 8α 2 α

[Re α > 0,

MO 44

b > 0,

Re ν > −1] WA 432(5), ET II 29(8)

2

2

2 √ ∞ νπ

π b νπ 1 b b −αx2 √ exp − tan I 12 ν + sec K 12 ν e Y ν (bx) dx = − 2 α 8α 2 8α π 2 8α 0 [Re α > 0, b > 0, |Re ν| < 1]

2.

∞

3. 0

∞

4. 0

5.

0

∞

2

2 νπ √π 2 b b 1 √ exp K 12 ν e−αx K ν (bx) dx = sec 4 2 8α 8α α [Re α > 0,

2

2 √ π b b −αx2 I 12 ν e I ν (bx) dx = √ exp 2 α 8α 8α

WA 432(6), ET II 106(3)

|Re ν| < 1] EH II 51(28), ET II 132(24)

[Re ν > −1,

Re α > 0]

EH II 92(27)

2

e−αx J μ (bx) J ν (bx) dx

Γ μ+ν+1 2 α b =2 Γ(μ + 1) Γ(ν + 1)

b2 ν +μ+1 ν +μ+2 ν +μ+1 , , ; μ + 1, ν + 1, ν + μ + 1; − × 3F 3 2 2 2 α [Re(ν + μ) > −1, Re α > 0] EH II 50(21)a −ν−μ−1 − ν+μ+1 ν+μ 2

6.62–6.63 Combinations of Bessel functions, exponentials, and powers 6.621

Notation: 1 1 = (a + b)2 + z 2 − (a − b)2 + z 2 , 2

2 =

1 (a + b)2 + z 2 + (a − b)2 + z 2 2

6.621

Bessel functions, exponentials, and powers

∞

1.

e 0

−αx

μ−1

J ν (bx)x

dx = =

b ν Γ(ν + μ) 2α μ α Γ(ν + 1)

b ν Γ(ν + μ) 2α αμ Γ(ν + 1)

b ν 2

F

707

ν+μ ν +μ+1 b2 , ; ν + 1; − 2 2 2 α

WA 421(2)

1 −μ b2 2 b2 ν −μ+1 ν −μ 1+ 2 , + 1; ν + 1; − 2 F α 2 2 α

Γ(ν + μ)

WA 421(3)

b2 ν +μ 1−μ+ν , ; ν + 1; 2 2 2 α + b2

= F ν+μ (α2 + b2 ) Γ(ν + 1) [Re(ν + μ) > 0,

Re (α + ib) > 0,

Re (α − ib) > 0] WA 421(3)

− 1 μ 1 2 2 −2 α α = α2 + b2 2 Γ(ν + μ) P −ν + b μ−1 [α > 0,

b > 0,

Re(ν + μ) > 0] ET II 29(6)

∞

2. 0

e−αx Y ν (bx)xμ−1 dx = cot νπ

b ν

ν+μ

(α2 + b2 )

− cosec νπ

Γ(ν + μ)

2

b −ν 2

F

Γ(ν + 1)

b2 ν +μ ν −μ+1 , ; ν + 1; 2 2 2 α + b2

Γ(μ − ν)

(α2 + b2 )μ−ν Γ(1 − ν)

F

b2 μ−ν 1−ν −μ , ; 1 − ν; 2 2 2 α + b2

[Re μ ≥ |Re ν|, − 1 μ 1 2 2 2 −2 = − Γ(ν + μ) b2 + α2 2 Q −ν μ−1 α α + b π [α > 0, b > 0,

Re (α ± ib) > 0] WA 421(4)

Re μ > |Re ν|] ET II 105(2)

∞

3. 0

∞

4. 0

5.10

0

xμ−1 e−αx K ν (bx) dx =

√ π(2b)ν Γ(μ + ν) Γ(μ − ν) F μ + ν, ν + (α + b)μ+ν Γ μ + 12 [Re μ > |Re ν|,

xm+1 e−αx J ν (bx) dx = (−1)m+1 b−ν

dm+1 dαm+1

) √

ν * α2 + b2 − α √ α2 + b2 [b > 0,

∞

e−zx J 1 (ax) J 1/2 (bx) x−3/2 dx 1 = a

2 πb

#

1 2

1 α−b 1 ;μ + ; 2 2 α+b Re(α + b) > 0]

ET II 131(23)a, EH II 50(26)

Re ν > −m − 2]

ET II 28(3)

$ 1 a2 2 2 2 2 arcsin +z a − 1 + b − 1 − b 2 2 [arg a > 0, arg b > 0, arg z > 0]

708

6.

10

7.10

8.10

9.10

10.10

11.12

12.10

Bessel Functions

14.10

2 b − b2 − 21 πb 0 [arg a > 0, arg b > 0, arg z > 0] ∞ 2 2 1 a2 − 1 1 e−zx J 1 (ax) J 1/2 (bx) x1/2 dx = a πb 22 − 21 0 [arg a > 0, arg b > 0, arg z > 0] ∞ 2 21 b2 − 21 e−zx J 1 (ax) J 3/2 (bx) x1/2 dx = π b3/2 a (22 − 21 ) 0 [arg a > 0, arg b > 0, arg z > 0]

∞ 1 1 1 −zx −1/2 2 2 2 − 1 a − 1 a arcsin e J 1 (ax) J 3/2 (bx) x dx = √ a 2π b3/2 a 0 [arg a > 0, arg b > 0, arg z > 0] )

* ∞ z 1 2a2 1 1 −zx −1/2 2 2 2 1 a − 1 + e J 1 (ax) J 5/2 (bx) x dx = √ − 3a arcsin 5/2 2 2 a 2π b a a − 1 0 [arg a > 0, arg b > 0, arg z > 0] ∞ e−zx J 1 (ax) J 5/2 (bx) x−3/2 dx 0 ⎡

4 1 ⎣ 1 5a2 21 1 7a 41 2 2 = √ − a − z − 8 4 8 2π b5/2 a a2 − 21 ⎤

4 1 2 1 3a 3 1 ⎦ + 22 1 a2 − 21 + arcsin a2 z 2 + a2 b 2 − − 2 1 a 2 2 8

∞

e

0

13.10

6.621

−zx

−1/2

J 1 (ax) J 1/2 (bx) x

1 dx = a

[arg a > 0, ∞

arg b > 0,

e−zx J 1 (ax) J 5/2 (bx) x−5/2 dx ⎧ 2 5/2 2 5/2

2 b2 z2 1 ⎨ 2 b − b − 1 1 3a 1 2 + za − − arcsin =√ 15 a 8 2 2 2π b5/2 a ⎩ ⎫ 2 3a2 z2 2 z 3 a2 1 ⎬ b − + − 1 + + z1 a2 − 21 2 8 6 4 3 a2 − 21 ⎭

[arg a > 0, arg b > 0, ∞ 22 − b2 2 2 3/2 a b e−zx J 2 (ax) J 3/2 (bx) x1/2 dx = 2 π (2 − 21 ) 42 0 [arg a > 0, arg b > 0, ) 2 3/2 * ∞ b − 21 2 b3/2 2 b2 − 21 −zx −1/2 − + e J 2 (ax) J 3/2 (bx) x dx = π a2 3 b 3b3 0 [arg a > 0, arg b > 0,

arg z > 0]

arg z > 0]

arg z > 0]

arg z > 0]

6.623

15.

10

Bessel functions, exponentials, and powers

∞

0

16.10

∞

18.10

19.10

e−zx J 3 (ax) J 1/2 (bx) x−1/2 dx # $ 2 ' ( 2 1 2 2 2 − 2 122 − 16b2 + 42 − 3a2 b 3a − − 4b + 12z b = 2 1 1 πb 3a3 [arg a > 0, arg b > 0, arg z > 0] e−zx J 3 (ax) J 3/2 (bx) x1/2 dx

& 2 3/2 * b − 21 a 22 − a2 b2 − 21 + − 2 = b 3b2 (2 − 21 ) 32 [arg a > 0, arg b > 0, arg z > 0] ) * ∞ 3/2 4b2 2b2 − 21 − 41 2b −zx −1/2 2 2 − 8z e J 3 (ax) J 3/2 (bx) x dx = 2 − b π 3a3 b4 0 [arg a > 0, arg b > 0, arg z > 0] ∞ e−zx J 3 (ax) J 3/2 (bx) x−3/2 dx 0 # 2 $ 2 b3/2 24b 821 a2 41 42 4 2 2 2 2 − 2 b − + − + a − + 4z − b = 1 π 3a3 5 b 5 5b b 5b3 [arg a > 0, arg b > 0, arg z > 0] ∞ e−zx J 3 (ax) J 3/2 (bx) x−5/2 dx 0 ⎧

2 b3/2 ⎨ 2 4 2 4z 3 a =− b z + − π 3a3 ⎩ 5 3 32 2 12 2 4 2 241 a4 21 a2 21 61 2 2 2 + 2 − b a + b − 1 − 2 + 2 + + + 15⎫ 5 3 5b 16b4 24b4 30b4

⎬ b a6 arcsin − 16b3 2 ⎭ 0

17.10

2 3/2 b π

%

4 a3

)

2 − 3

[arg a > 0, 6.622 1.

∞

J 0 (x) − e−αx

0

2. 3.8

709

dx = ln 2α x

arg b > 0,

arg z > 0]

[α > 0]

NT 66(13)

∞ i(u+x)

π e (1) J 0 (x) dx = i H 0 (u) u + x 2 0 μ− 1 ∞ Q ν− 12 (cosh α) 1 2 2 e−(μ− 2 )πi e−x cosh α I ν (x)xμ−1 dx = 1 π 0 sinhμ− 2 α [Re(μ + ν) > 0,

MO 44

Re (cosh α) > 1] WA 388(6)a

6.623 1. 0

∞

e

−αx

(2b)ν Γ ν + 12 J ν (bx)x dx = √ ν+ 1 π (α2 + b2 ) 2 ν

Re ν > − 12 ,

Re α > |Im b| WA 422(5)

710

Bessel Functions

2α(2b)ν Γ ν + 32 e J ν (bx)x dx = √ ν+ 3 0 π (α2 + b2 ) 2 √ ν ∞ α2 + b2 − α dx −αx = e J ν (bx) x νbν 0 [Re ν > 0;

2.

3.

6.624 1. 2. 3.

∞

−αx

6.624

ν+1

[Re ν > −1,

Re α > |Im b|]

∞

e−tz(z

2

e−tz(z

2

−1/2

−1)

0

∞

5. 0

∞

6. 0

∞

7. 0

WA 422(7)

−1 2

−1)

I −μ (t)tν dt =

I μ (t)tν dt =

Γ(−ν − μ) (z 2

− 1)

1 2ν

P μν (z)

Γ(ν + μ + 1) (z 2

− 12 (ν+1)

− 1)

(2b)ν Γ ν + J ν (bx)xν √ dx = eπx − 1 π

1 2

1

1. 0

MO 181

EH II 57(7)

[Re(ν + μ) < 0]

EH II 57(8)

[Re(ν + μ) > −1]

EH II 57(9)

P −μ ν (z)

e−t cos θ J μ (t sin θ) tν dt = Γ(ν + μ + 1) P −μ ν (cos θ)

MO 181

[Re (ν ± μ) > −1]

Re(ν + μ) > −1,

0 ≤ θ < 12 π

EH II 57(10)

∞ "

1

n=1

(n2 π 2 + b2 )ν+ 2

1

[Re ν > 0,

xλ−ν−1 (1 − x)μ−1 e±iαx J ν (αx) dx =

|Im b| < π]

WA 423(9)

1 2−ν αν Γ(λ) Γ(μ) ; λ + μ, 2ν + 1; ±2iα λ, ν + 2F 2 Γ(λ + μ) Γ(ν + 1) 2

[Re λ > 0,

1

Re μ > 0]

ET II 194(58)a

1 ; μ + 2ν + 1; ±2iα ν + 1F 1 2 0 ET II 194(57)a Re μ > 0, Re ν > − 12

1 1 (2α)ν Γ ν + 2 Γ(μ) 1 ; μ + 2ν + 1; ±2α ν + xν (1 − x)μ−1 e±αx I ν (αx) dx = √ 1F 1 π Γ(μ + 2ν + 1) 2 0 Re μ > 0, Re ν > − 12

3.

(cf. 6.611 1)

∞

4.

2.

WA 422(6)

% ) * &

2 α α 1 α √ + xe−αx K 0 (bx) dx = 2 ln −1 −1 α − b2 b b α2 − b2 0 ∞ √ −αx π 1 xe K ± 12 (bx) dx = 2b α + b 0 ∞ −1/2 2 Γ(ν − μ + 1) −iμπ μ e−tz(z −1) K μ (t)tν dt = e Q ν (z) − 1 (ν+1) 0 (z 2 − 1) 2

6.625

Re α > |Im b|]

1

xν (1 − x)μ−1 e±iαx J ν (αx) dx =

(2α)ν Γ(μ) Γ ν + 2 √ π Γ(μ + 2ν + 1)

BU 9(16a), ET II 197(77)a

6.626

Bessel functions, exponentials, and powers

1

4.

λ−1

x 0

1

5. 0

μ−1 ±αx

(1 − x)

e

711

1 ν α Γ (λ + ν) Γ(μ) I ν (αx) dx = 2 Γ(ν + 1) Γ(λ + μ + ν) 1 × 2 F 2 ν + , λ + ν; 2ν + 1, μ + λ + ν; ±2α 2 [Re μ > 0, Re(λ + ν) > 0] ET II 197(78)a

xμ−κ (1 − x)2κ−1 I μ−κ

1 1 1 Γ(2κ) xz e− 2 xz dx = √ ez/2 z −κ− 2 M κ,μ (z) 2 π Γ(1 + 2μ) Re κ − 12 − μ < 0, Re κ > 0

BU 129(14a)

!1 ∞ ! − λ, 0 (2α)λ Γ(μ) 21 2 ! √ 2α x−λ (x − 1)μ−1 e−αx I ν (αx) dx = G 23 ! π 1 −μ, ν − λ,1−ν − λ 0 < Re μ < 2 + Re λ, Re α > 0

6.

! 1 ∞ ! 0, − λ √ ! 2 2α x−λ (x − 1)μ−1 e−αx K ν (αx) dx = Γ(μ) π(2α)λ G 30 23 ! −μ, ν − λ, −ν − λ 1

7.

[Re μ > 0, Re α > 0] ∞ (2α)ν−μ Γ 2 − μ + ν Γ(μ) −ν μ−1 −αx √ x (x − 1) e I ν (αx) dx = π Γ(1 − μ + 2ν) 1

1 − μ + ν; 1 − μ + 2ν; −2α × 1F 1 2 0 < Re μ < 12 + Re ν, Re α > 0 1

8.

∞

9. 1

10.12

x−ν (x − 1)μ−1 e−αx K ν (αx) dx =

1

1

x−μ− 2 (x − 1)μ−1 e−αx K ν (αx) dx =

11.3

1

−1

1 − x2

−1/2

1.11

0

∞

Re α > 0]

ET II 208(53)a

Re α > 0]

ET II 207(51)a

−1/2 2 sinh a − a a2 + b2 xe−ax I 1 b 1 − x2 dx = sinh a2 + b2 b [a > 0,

6.626

ET II 207(49)a

√ 1 π Γ(μ)(2α)− 2 e−α W −μ,ν (2α) [Re μ > 0,

ET II 208(55)a

√ 1 1 π Γ(μ)(2α)− 2 μ− 2 e−α W − 12 μ,ν− 12 μ (2α) [Re μ > 0,

∞

ET II 207(50)a

xλ−1 e−αx J μ (bx) J ν (cx) dx =

b > 0]

∞ " Γ(λ + μ + ν + 2m) b μ cν 2−ν−μ α−λ−μ−ν Γ(ν + 1) m! Γ(μ + m + 1) m=0

m 2 c b2 × F −m, −μ − m; ν + 1; 2 − 2 b 4α [Re(λ + μ + ν) > 0, Re (α ± ib ± ic) > 1]

EH II 48(15)

712

Bessel Functions

∞

2.

e 0

3.

∞

5.10

6.627 6.628

∞

∞

0

∞

0

3.

4.8

J ν (bx) J μ (bx)x

Γ ν + μ + 12 bν+μ √ dx = π3 π2 cosν+μ ϕ cos(ν − μ)ϕ × dϕ ν+μ 2 2 2 α2 + b2 cos2 ϕ 0 (α + b cos ϕ) Re α > |Im b|, Re(ν + μ) > − 12

e−2αx J 0 (bx) J 1 (bx)x dx =

K

√ b α2 +b2

−E

√ b α2 +b2

WA 427(1)

WA 427(2)

[Re α > Re b] WA 428(5)

1 a μ−ν−2n−1 ρ ν xν−μ+2n e−zx J μ (αx) J ν (ρx) dx = √ π 2 a 0 ∞ Γ ν +n+q+ 1 " ν − μ + n + 12 q 1 2 × Γ μ − ν − n + 12 q=0 q! Γ ν + q + 12

q 1 /ρ z2 dx −2q 2ν+2q 2 √ ρ + ×a x 2 1 − x2 0 1 − x 1 2 2 2 2 μ > ν + 2n, n = 0, 1, . . . , ν > − 21 where 1 = 2 (a + ρ) + z − (a − ρ) + z

1.

2.

ν+μ

√ 2πb α2 + b2 #

$

∞ 1 b α 1 b − K e−2αx I 0 (bx) I 1 (bx)x dx = E 2 2 2πb α − b α α α 0 0

4.

−2αx

6.627

πea K ν (a) x−1/2 −x e K ν (x) dx = √ x+a a cos(νπ)

|arg a| < π,

|Re ν| <

1 2

e−x cos β J −ν (x sin β) xμ dx = Γ(μ − ν + 1) P νμ (cos β) π 0<β< , 2

ET II 368(29)

Re(μ − ν) > −1

WA 424(3), WH

∞

∞

sin μπ Γ (μ − ν + 1) sin(μ + ν)π π 0 1 1 ν × Q μ (cos β + 0 · i) e 2 νπi + Q νμ (cos β − 0 · i) e− 2 νπi π Re(μ + ν) > −1, 0 < β < WA 424(4) 2

1 u xu π B(2ν, 2μ − 2ν + 1) e 2 ixu dx = 22(ν−μ) e 2 (μ−ν)i e 2 (1 − x)2ν−1 xμ−ν J μ−ν 1 M ν,μ (u) 2 Γ(μ − ν + 1) uν+ 2 0

0

e−x cos β Y ν (x sin β) xμ dx = −

MO 118a

e−x cosh α I ν (x sinh α) xμ dx = Γ(ν + μ + 1) P −ν μ (cosh α) Re(μ + ν) > −1,

|Im α| < 12 π

WA 423(1)

6.631

Bessel functions, exponentials, and powers

∞

5. 0

e−x cosh α K ν (x sinh α) xμ dx =

713

sin μπ Γ(μ − ν + 1) Q νμ (cosh α) sin(ν + μ)π [Re(μ + 1) > |Re ν|]

∞

6. 0

e−x cosh α I ν (x)xμ−1 dx =

WA 423(2)

μ− 12 ν− 12

(cosh α) Q cos νπ sin(μ + ν)π π (sinh α)μ− 12 2 [Re(μ + ν) > 0,

Re (cosh α) > 1] WA 424(6)

∞

7.

e

−x cosh α

0

μ−1

K ν (x)x

dx =

1 2 −μ ν− 12

P (cosh α) π Γ(μ − ν) Γ(μ + ν) 1 2 (sinh α)μ− 2 [Re μ > |Re ν|,

Re (cosh α) > −1] WA 424(7)

6.6298

∞

0

x−1/2 e−xα cos ϕ cos ψ J μ (αx sin ϕ) J ν (αx sin ψ) dx α > 0,

6.631

∞

1. 0

2

xμ e−αx J ν (bx) dx =

1 (cos ϕ) P −ν = Γ μ + ν + 12 α− 2 P −μ 1 (cos ψ) ν− 12 μ− 2 π 1 π 0 < ϕ < , 0 < ψ < , Re(μ + ν) > − ET II 50(19) 2 2 2

1

1 1 2ν + 2μ + 2 1 2ν+1 α 2 (μ+ν+1) Γ(ν +

bν Γ

1

1)

1F 1

b2 ν +μ+1 ; ν + 1; − 2 4α

1 2ν + 2μ +

1 2

BU 8(15)

2

2 b b 1 1 M = exp − 1 μ, 2 ν μ 2 8α 4α bα 2 Γ(ν + 1) [Re α > 0, Γ

Re(μ + ν) > −1]

EH II 50(22), ET II 30(14), BU 14(13b)

∞

2.

2

xμ e−αx Y ν (bx) dx

2 1 b ν −μ = −α− 2 μ b−1 sec π exp − 2 8α %

2 Γ 12 + 12 μ + 12 ν ν −μ b sin π M 12 μ, 12 ν +W × Γ(1 + ν) 2 4α

0

[Re α > 0, 3.

12

∞

x e 0

4.11

μ −αx2

0

∞

1 1 K ν (bx) dx = α− 2 μ b−1 Γ 2

2

xν+1 e−αx J ν (bx) dx =

bν (2α)ν+1

1+ν +μ 2

2 b exp − 4α

Re μ > |Re ν| − 1,

1 1 2 μ, 2 ν

b > 0]

b2 4α

&

ET II 106(4)

2

2 1−ν+μ b b Γ exp W − 12 μ, 12 ν 2 8α 4α [Re α > 0, Re μ > |Re ν| − 1] ET II 132(25)

[Re α > 0,

Re ν > −1] WA 431(4), ET II 29(10)

714

5.

12

6.

Bessel Functions

b2 [Re α > 0, Re ν > 0] x e J ν (bx) dx = 2 b γ ν, 4α 0 ∞ 2 b2 bν ν +1 π − xν+1 e±iαx J ν (bx) dx = exp ±i (2α)ν+1 2 0 4α α > 0, −1 < Re ν < 12 ,

8.

∞

ν−1 −αx2

ν−1 −ν

2

2

2 2 b b b 1 1 1 1 I − I xe−αx J ν (bx) dx = exp − 3 2 ν− 2 2 ν+ 2 8α 8α 8α 2 8α 0 [Re α > 0, Re ν > −2] ) * 1 n " 2 1 eα − e−α xn+1 e−αx I n (2αx) dx = I r (2α) 4α 0 r=−n

7.

6.632

∞

9. 1

√ πb

∞

∞

10. 0

)

2

x1−n e−αx I n (2αx) dx =

ET II 30(11)

b>0

ET II 30(12)

ET II 29(9)

[n = 0, 1, . . .] *

ET II 365(8)a

[n = 1, 2, . . .]

ET II 367(20)a

n−1 " 1 eα − e−α I r (2α) 4α r=1−n

√ 2 1 n! e−x x2n+μ+1 J μ 2x z dx = e−z z 2 μ Lμn (z) 2

[n = 0, 1, . . . ;

n + Re μ > −1] BU 135(5)

6.632

∞

0

1 − 1 1 x− 2 exp − x2 + a2 − 2ax cos ϕ 2 x2 + a2 − 2ax cos ϕ 2 K ν (x) dx 1

= πa− 2 sec(νπ) P ν− 12 (− cos ϕ) K ν (a) |arg a| + |Re ϕ| < π, |Re ν| < 12 ET II 368(32) 6.633

∞

1.

λ+1 −αx2

x

e

0

2 aβ 1 a + β2 I J p (ax) J p (βx)x dx = 2 exp − p 2 42 22 0 π Re p > −1, |arg | < , a > 0, β > 0 KU 146(16)a, WA 433(1) 4

∞ 1 1 − 3 ν− 1 1 2ν+1 −αx2 2 2 W 12 ν, 12 ν x e J ν (x) Y ν (x) dx = − √ α exp − 2α α 2 π 0 Re α > 0, Re ν > − 21 ET II 347(59)

2.

3.

m μ+ν+λ+2 ∞ " 2 Γ m + 12 ν + 12 μ + 12 λ + 1 b2 bμ cν α− − J μ (bx) J ν (cx) dx = ν+μ+1 2 Γ(ν + 1) m=0 m! Γ(m + μ + 1) 4α

2 c × F −m, −μ − m; ν + 1; 2 b [Re α > 0, Re(μ + ν + λ) > −2, b > 0, c > 0] EH II 49(20)a, ET II 51(24)a

∞

2

e−

x2

6.637

Bessel functions, exponentials, and powers

∞

4.

xe

−αx2

0

1 exp I ν (bx) J ν (cx) dx = 2α

b 2 − c2 4α

Jν

Re ν > −1]

ET II 63(1)

2

xλ−1 e−αx J μ (bx) J ν (bx) dx

+ 12 μ + 12 ν α b =2 Γ(μ + 1) Γ(ν + 1) μ 1 ν μ ν +μ+λ b2 ν + + , + + 1, ; μ + 1, ν + 1, μ + ν + 1; − × 3F 3 2 2 2 2 2 2 α [Re(ν + λ + μ) > 0, Re α > 0] WA 434, EH II 50(21)

0

−ν−μ−1 − 12 (ν+λ+μ) ν+μ Γ

6.634

[Re α > 0, ∞

5.

bc 2α

715

∞

0

1

x2

xe− 2a [I ν (x) + I −ν (x)] K ν (x) dx = aea K ν (a)

2λ

[Re a > 0,

−1 < Re ν < 1] ET II 371(49)

6.635

∞

1. 0

∞

2. 0

∞

3. 0

α

x−1 e− x J ν (bx) dx = 2 J ν α

√ √

2αb K ν 2αb

x−1 e− x Y ν (bx) dx = 2 Y ν α

√

x−1 e− x −βx J ν (γx) dx = 2 J ν

√

2αb K ν 2αb

[Re α > 0,

b > 0]

ET II 30(15)

[Re α > 0,

b > 0]

ET II 106(5)

# # 12 $ 12 $ √ √ Kν 2α β2 + γ2 − β 2α β2 + γ2 + β [Re α > 0,

Re β > 0,

γ > 0] ET II 30(16)

√ ∞ 1 1

1

√ 2 1 1 1 − 12 −α x x e J ν (bx) dx = √ Γ ν + 12 D −ν− 12 2− 2 αe 4 πi b− 2 D −ν− 12 2− 2 αe− 4 πi b− 2 πb 0 Re α > 0, b > 0, Re ν > − 12

6.636

ET II 30(17)

6.637

∞

− 12

∞

− 12

1.

β 2 + x2

0

2. 0

β 2 + x2

1 exp −α β 2 + x2 2 J ν (γx) dx # # $ $ 1 1 1 2 1 2 β α + γ 2 2 − α K 12 ν β α + γ2 2 + α = I 12 ν 2 2 [Re α > 0, Re β > 0, γ > 0, Re ν > −1] ET II 31(20)

1 exp −α β 2 + x2 2 Y ν (γx) dx # νπ

$ 1 1 2 K 12 ν β α + γ2 2 + α = − sec 2 # 2 #

$ νπ

$ 1 1 1 2 1 2 1 2 2 2 2 1 1 K β α +γ I 2ν β α +γ + α + sin −α × π 2ν 2 2 2 [Re α > 0, Re β > 0, γ > 0, |Re ν| < 1] ET II 106(6)

716

Bessel Functions

∞

3.

x2 + β 2

− 12

0

6.641

1 exp −α x2 + β 2 2 K ν (γx) dx

νπ

2 2 1 1 1 1 1 2 2 2 2 K 12 ν β α+ α −γ β α− α −γ = sec K 12 ν 2 2 2 2 [Re α > 0, Re β > 0, Re(γ + β) > 0, |Re ν| < 1] ET II 132(26)

6.64 Combinations of Bessel functions of more complicated arguments, exponentials, and powers

6.641 6.642 1.

10

∞√

xe−αx J ± 14 x2 dx =

0

∞

−1 −αx

x

e

0

2.12

∞

0

6.643

∞

μ− 12 −αx

e

0

2.

3.

∞

∞

0

MC MI 44, EH II 91(26)

2

2 √ Γ μ + ν + 12 b b exp − α−μ M μ,ν J 2ν 2b x dx = b Γ(2ν + 1) 2α α Re μ + ν + 12 > 0

1

√ Γ μ + ν + 2 −1 1 b exp xμ− 2 e−αx I 2ν 2b x dx = Γ(2ν + 1)

BU 14(13a), MI 42a

2

∞

MI 47a

2

2 √ b 1 b α−n−ν−1 Lνn xn+ 2 ν e−αx J ν 2b x dx = n!bν exp − α α

− 12 −αx

√ Y 2ν b x dx = −

exp − b2 8α π

[n + ν > −1]

b2 8α

1 + Kν π 1

MO 178a

b2 8α

sin(νπ) I ν |Re ν| < 2

12 m− 12

∞ √ 1 1 Γ(m + 1) 1 1 W − 12 (m+1),− 12 m x 2 m e−αx K m 2 x dx = exp 2α α 2α α 0 x

0

6.

[Re α > 0]

b b2 α−μ M −μ,ν 2α α 0 1 Re μ + ν + 2 > 0 MI 45

∞ √ Γ μ + ν + 12 Γ μ − ν + 12 1 b2 b2 exp α−μ W −μ,ν xμ− 2 e−αx K 2ν 2b x dx = 2b 2α α 0 Re μ + ν + 12 > 0 , (cf. 6.631 3)

4.

5.

MI 42

√ √ 2 dx = 2 K ν 2 α Y ν 2 α Yν x

√ √ 2 (1,2) −1 −αx dx = 2 H (1,2) x e Hν α Kν α ν x

x

1.

2

2 √ πα α α H∓ 14 − Y ∓ 14 4 4 4

e

α

cos(νπ)

MI 44 MI 48a

6.647

Bessel functions of complicated arguments, exponentials, and powers

6.644

∞

e

−βx

0

J 2ν

717

√ a2 β a2 b 1 Jν 2a x J ν (bx) dx = exp − 2 2 2 2 2 β +b β +b β + b2 Re β > 0, b > 0, Re ν > − 12

ET II 58(17)

6.645

∞

1.

x2 − 1

1

∞

2. 1

3.3

x2 − 1

1 1 e−αx J ν β x2 − 1 dx = I 12 ν α2 + β 2 − α K 12 ν α2 + β 2 + α 2 2

− 12

12 ν

e−αx J ν β x2 − 1 dx =

MO 179a

− 1 ν− 1 2 ν 2 β α + β 2 2 4 K ν+ 12 α2 + β 2 π MO 179a

−1/2 −ax 2 cosh a2 + b2 − cosh a 1 − x2 e I 1 b 1 − x2 dx = b −1 1

[a > 0, 6.646 1.

2.

3.7

b > 0]

√ ν

exp − α2 + b2 b 2 √ √ e J ν b x − 1 dx = α2 + b2 α + α2 + b2 1 [Re ν > −1] EF 89(52), MO 179 √ 1 ν

ν ∞

exp − α2 − b2 x − 1 2 −αx 2 b √ √ e I ν b x − 1 dx = x+1 α2 − b2 α + α2 − b2 1 [Re ν > −1, α > b] MO 180 12 ν

∞

x−b Γ(ν + 1) ν −bs x e e−px K ν a x2 − b2 dx = Γ(−ν, bx) − y ν ebs Γ(−ν, by) ν x+b 2sa b 2 1/2 where x = p − s, y = p + s, s = p − a2 [Re(p + a) > 0, |Re(ν)| < 1] .

∞

x−1 x+1

12 ν

−αx

ME 39a

6.647 1.

∞

1

x−λ− 2 (b + x)

0

λ− 12

e−αx K 2μ

x(b + x) dx =

2. 0

∞

1

1

(a + x)− 2 x− 2 e−x cosh t K ν

1 21 αb 1 e Γ 2 − λ + μ Γ 12 − λ − μ W λ,μ (z1 ) W λ,μ (z2 ) b

z1 = 12 b α + α2 − 1 ,

|arg b| < π,

x(a + x) dx =

Re α > −1,

z2 = 12 b α − α2 − 1 ET II 377(37) Re λ + |Re μ| < 12

νπ 1 1 t 1 −t 1 sec e 2 a cosh t K 12 ν ae K 12 ν ae 2 2 4 4 [−1 < Re ν < 1] ET II 377(36)

718

3.

Bessel Functions

11

a

0

1

1

6.648

12

x(a − x) dx 1 1 −(a/2) sinh t 2 Γ 2 + λ + μ Γ 2 − λ + μ

xλ− 2 (a − x)−λ− 2 e−x sinh t I 2μ =e

∞

e

x

−∞

a + bex aex + b

6.648

K 2ν

1 t 1 −t ae ae M −λ,μ 2 2 a [Γ(2μ + 1)]2 ET II 377(32) Re μ > |Re λ| − 12 M λ,μ

1 a2 + b2 + 2ab cosh x 2 dx = 2 K ν+ (a) K ν− (b) [Re a > 0,

6.649

∞

1. 0

0

∞

3. 0

ET II 379(45)

π2 [J ν (z) Y μ (z) − J μ (z) Y ν (z)] 4 sin[(ν − μ)π] [Re z > 0, −1 < Re(ν − μ) < 1] MO 44

∞

2.

K μ−ν (2z sinh x) e(ν+μ)x dx =

Re b > 0]

∞

4. 0

J ν+μ (2x sinh t) e(ν−μ)t dt = K ν (x) I μ (x) Re(ν − μ) < 32 , Y ν−μ (2x sinh t) e−(ν+μ)t dt =

K 0 (2z sinh x) e−2νx dx = −

π 4

Re(ν + μ) > −1,

x>0

EH II 97(68)

1 {I μ (x) K ν (x) − cos[(ν − μ)π] I ν (x) K μ (x)} sin[π(μ − ν)] |Re(ν − μ)| < 1, Re(ν + μ) > − 12 , x > 0 EH II 97(73) # $ ∂ Y ν (z) ∂ J ν (z) J ν (z) − Y ν (z) ∂ν ∂ν

6.65 Combinations of Bessel and exponential functions of more complicated arguments and powers 6.651

∞

1. 0

1

1

2

xλ+ 2 e− 4 α

x2

Iμ

1

2 2 4α x

3 1 = √ 2λ+1 b−λ− 2 2π

J ν (bx) dx

2 ! b !! 1 − μ, 1 + μ 21 G 23 2α2 ! h, 12 , k h=

π |arg α| < , 4 2. 0

∞

b > 0,

3 4

+ 12 λ + 12 ν,

k=

3 4

− 23 − Re(2μ + ν) < Re λ < 0

+ 1 λ − 12 ν 2 ET II 68(8)

1 1 2 2 xλ+ 2 e− 4 α x K μ 14 α2 x2 J ν (bx) dx !

π λ+1 −λ− 3 12 b2 !! 1 − μ, 1 + μ 2 2 b = G 23 2 2α2 ! h, 12 , k

h= |arg α| <

π , 4

3 4

+ 12 λ + 12 ν,

k=

Re (λ + ν ± 2μ) > − 23

3

4

+ 12 λ − 12 ν ET II 69(15)

6.652

Bessel and exponential functions and powers

∞

3. 0

∞

4.

2

1

x2μ−ν+1 e− 4 αx I μ

2

1

x2μ+ν+1 e− 4 α

2 4 αx

Kμ =

∞

5. 0

2

1

J ν (bx) dx

bν−2μ−1 1 b2 1 1 μ−ν+ 12 − 12 1F 1 +μ + μ; − μ + ν; − =2 (πα) Γ 2 2 2 2α Γ 12 − μ + ν Re α > 0, b > 0, Re ν > 2 Re μ + 12 > − 21 ET II 68(6)

x2

0

1

719

x2μ+ν+1 e− 2 αx I μ

1

2 2 4α x

J ν (bx) dx

√ μ −2μ−2ν−2 ν Γ (1 + 2μ + ν) π2 α b F 1 1 1 + 2μ + ν; μ + ν + Γ μ + ν + 32 |arg α| < 14 π, Re ν > −1, Re(2μ + ν) > −1, b > 0

1

2 2 αx

b2 3 ;− 2 2 2α

ET II 69(13)

K ν (bx) dx

2

2 1 b b 2μ− 2 −μ− 32 − 12 μ− 12 ν− 14 1 W k,m α Γ(2μ + ν + 1) Γ μ + 2 exp = √ b π 8α 4α 1 2k = −3μ − ν − , 2m = μ + ν + 12 2 1 Re α > 0, Re μ > − 2 , Re (2μ + ν) > −1 ET II 146(53)

∞

6. 0

1

2

xe− 4 αx J 12 ν

1 2 bc2 αc2 2 2 −2 1 J J bx (cx) dx = 2 α + b exp − ν 4 2ν α2 + b2 α2 + b2 [c > 0, Re α > |Im b|, Re ν > −1]

1

ET II 56(2)

∞

7. 0

∞

8.

1

2

xe− 4 αx I 12 ν

2

1

x1−ν e− 4 α

x2

0

∞

9.

1

2

x−ν−1 e− 4 α

1

2 J ν (bx) dx = 4 αx

Iν

x2

0

6.652

∞

2ν −

x e 0

x2 8

+αx

1

2 2 4α x

I ν+1

Iν

J ν (bx) dx =

2 2 4α x

x2 8

− 12

2 1 b πα b−1 exp − 2 2α [Re α > 0,

1

b > 0,

Re ν > −1] ET II 67(3)

2 bν−1 b2 b exp − 2 D −2ν π α 4α α |arg α| < 14 π, b > 0,

J ν (bx) dx =

Re ν > − 12

2 ν b b2 b exp − 2 D −2ν−3 π 4α α |arg α| < 14 π, Re ν > −1,

ET II 67(1)

b>0

ET II 67(2)

2

α Γ(4ν + 1) e 2 dx = 4ν W − 32 ν, 12 ν α2 ν+1 2 Γ(ν + 1) α Re ν + 14 > 0

MI 45

720

Bessel Functions

6.653 1.

∞

0

6.653

1 2 ab dx 1 a + b2 I ν = 2 I ν (a) K ν (b) exp − x − 2 2x x x = 2 K ν (a) I ν (b) [Re ν > −1]

∞

2. 0

6.655

[0 < b < a] WA 482(2)a, EH II 53(37), WA 482(3)a

zw dx 1 2 1 z + w2 K ν = 2 K ν (z) K ν (w) exp − x − 2 2x x x |arg z| < π, |arg w| < π, arg(z + w) < 14 π

WA 483(1), EH II 53(36)

√ √ 1 β2 dx = 4πα− 2 K 2ν β α x e Kν ME 39 8x 0

∞ 2 2 − 1 α x α β √ Jν J ν (γx) dx = γ −1 e−βγ J 2ν (2α γ) x β 2 + x2 2 exp − 2 2 2 2 β +x β +x 0 Re β > 0, γ > 0, Re ν > − 12

6.654

[0 < a < b]

∞

2

− 12 − β 8x −αx

ET II 58(14)

6.656 1.

∞

0

∞

2. 0

1 e−(ξ−z) cosh t J 2ν 2(zξ) 2 sinh t dt = I ν (z) K ν (ξ)

Re ν > − 12 ,

1 1 e−(ξ+z) cosh t K 2ν 2(zξ) 2 sinh t dt = K ν (z) K ν (ξ) sec(νπ) 2 |Re ν| < 12 ,

Re(ξ − z) > 0

EH II 98(78)

2 1 1 Re z 2 + ξ 2 ≥ 0 EH II 98(79)

6.66 Combinations of Bessel, hyperbolic, and exponential functions Bessel and hyperbolic functions 6.661

1. 0

∞

π cosec sinh(ax) K ν (bx) dx = 2

νπ 2

sin ν arcsin ab √ b 2 − a2 [Re b > |Re a|,

∞ π cos ν arcsin ab νπ

cosh(ax) K ν (bx) dx = 0 2 b2 − a2 cos 2

2.

|Re ν| < 2] ET II 133(32)

[Re b > |Re a|,

|Re ν| < 1] ET II 134(33)

6.662

Notation: 1 1 = (b + c)2 + a2 − (b − c)2 + a2 , 2

2 =

1 (b + c)2 + a2 + (b − c)2 + a2 2

6.663

1.

10

Bessel, hyperbolic, and exponential functions

∞

0

K(k) cosh(βx) K 0 (αx) J 0 (γx) dx = √ u+v

721

# $ 1 2 2 2 2 2 2 2 2 2 u= (α + β + γ ) − 4α β + α − β − γ 2 # $ 1 2 2 2 2 2 2 2 2 2 (α + β + γ ) − 4α β − α + β + γ v= 2 k 2 = v(u + v)−1

[Re α > |Re β|,

γ > 0]

ET II 15(23)

alternatively, with a = γ, b = β, c = α, ∞ K(k) cosh(bx) K 0 (cx) J 0 (ax) dx = 2 2 − 21 0 k2 = 2.10

∞

0

22 − c2 , 22 − 21

[Re c > |Re b|,

a > 0]

K(k) snu dn u sinh(βx) K 1 (αx) J 0 (γx) dx = a−1 u E(k) − K(k) E(u) + cn u # $−1 12 2 cn 2 u = 2γ 2 α2 + β 2 + γ 2 − 4α2 β 2 − α2 + β 2 + γ 2 # − 12 $ 2 2 1 1 − α2 − β 2 − γ 2 α + β 2 + γ 2 − 4α2 β 2 k2 = 2 [Re α > |Re β|, γ > 0] ET II 15(24)

alternatively, with a = γ, b = β, c = α, ∞ K(k) snu dn u −1 u E(k) − K(k) E(u) + sinh(bx) K 1 (cx) J 0 (ax) dx = c cn u 0 2 2 2 − c a , k 2 = 22 [Re c > |Re b|, cn 2 u = 2 2 − c2 2 − 21 6.663 1.

∞

0

0

0

4. 0

Y μ+ν (2z cosh t) cosh[(μ − ν)t] dt =

WA 484(1), EH II 54(39)

π [J μ (z) J ν (z) − Y μ (z) Y ν (z)] 4 [z > 0]

∞

3.

1 K μ (z) K ν (z) 2 [Re z > 0]

∞

2.

K ν±μ (2z cosh t) cosh [(μ ∓ ν) t] dt =

J μ+ν (2z cosh t) cosh[(μ − ν)t] dt = −

EH II 96(64)

π [J μ (z) Y ν (z) + J ν (z) Y μ (z)] 4 [z > 0]

∞

a > 0]

1 J μ+ν (2z sinh t) cosh[(μ − ν)t] dt = [I ν (z) K μ (z) + I μ (z) K ν (z)] 2 Re(ν + μ) > −1, |Re(μ − ν)| < 32 ,

EH II 97(65)

z>0

EH II 97(71)

722

Bessel Functions

∞

5. 0

6.664

∞

1. 0

∞

6.664

1 J μ+ν (2z sinh t) sinh[(μ − ν)t] dt = [I ν (z) K μ (z) − I μ (z) K ν (z)] 2 Re(ν + μ) > −1, |Re(μ − ν)| < 32 ,

J 0 (2z sinh t) sinh(2νt) dt =

sin(νπ) 2 [K ν (z)] π

|Re ν| < 34 ,

z>0

z>0

EH II 97(72)

EH II 97(69)

cos(νπ) 2 [K ν (z)] |Re ν| < 34 , z > 0 EH II 97(70) π 0 ∞ 1 1 ∂ K ν (z) ∂ I ν (z) 2 I ν (z) − K ν (z) − cos(νπ) [K ν (z)] Y 0 (2z sinh t) sinh(2νt) dt = π ∂ν ∂ν π 0 |Re ν| < 34 , z > 0 EH II 97(75) ∞ 2 ' ( π Jν2 (z) + Nν2 (z) K 0 (2z sinh t) cosh 2νt dt = [Re z > 0] MO 44 8 0

∞ 1 1 1 Γ +μ−ν Γ − μ − ν W ν,μ (iz) W ν,μ (−iz) K 2μ (z sinh 2t) coth2ν t dt = 4z 2 2 0 π |arg z| ≤ , |Re μ| + Re ν < 12 2

2. 3.

4. 5.

MO 119

∞

6. 0

6.665

Y 0 (2z sinh t) cosh(2νt) dt = −

cosh(2μx) K 2ν (2a cosh x) dx =

∞

0

1 K μ+ν (a) K μ−ν (a) 2

sech x cosh(2λx) I 2μ (a sech x) dx =

[Re a > 0] Γ 12 + λ + μ Γ 12 − λ + μ

ET II 378(42)

M λ,μ (a) M −λ,μ (a) 2 2a [Γ(2μ + 1)] ET II 378(43) |Re λ| − Re μ < 12

Bessel, hyperbolic, and algebraic functions ∞ ∞ 2" xν+1 sinh(αx) cosech(πx) J ν (βx) dx = (−1)n−1 nν+1 sin(nα) K ν (nβ) 6.666 π n=1 0 [|Re α| < π, Re ν > −1] ET II 41(3), WA 469(12)

6.667 1.

2.

3

√

a2 − x2 sinh t I 2ν (x) 1 t 1 −t π √ ae I ν ae dx = I ν 2 2 a2 − x2 0 2 Re ν > − 12 ET II 365(10) √ a cosh a2 − x2 sinh t K 2ν (x) π2 √ cosec(νπ) I −ν aet I −ν ae−t − I ν aet I ν ae−t dx = 4 a2 − x2 0 |Re ν| < 12 ET II 367(25)

a

cosh

6.669

Bessel, hyperbolic, and exponential functions

723

Exponential, hyperbolic, and Bessel functions 6.668

Notation: 1 1 = (b + c)2 + a2 − (b − c)2 + a2 , 2

1.10

2.12

6.669

∞

1 (b + c)2 + a2 + (b − c)2 + a2 2 1

−1

e−αx sinh(βx) J 0 (γx) dx = (αβ) 2 r1−1 r2−1 (r2 + r1 ) 2 (r2 − r1 ) 2 0 r1 = γ 2 + (β − α)2 , r2 = γ 2 + (β + α)2 , [Re α > |Re β|, γ > 0] alternatively, with a = γ, b = β, c = α, ∞ 1 e−cx sinh(bx) J 0 (ax) dx = 2 2 0 2 − 1 [Re c > |Re b|, a > 0] ∞ 1 1 −1 e−αx cosh(βx) J 0 (γx) dx = (αβ) 2 r1−1 r2−1 (r2 + r1 ) 2 (r2 − r1 ) 2 0 r1 = γ 2 + (β − α)2 , r2 = γ 2 + (β + α)2 , [Re α > |Re β|, γ > 0] alternatively, with a = γ, b = β, c = α, ∞ 2 e−cx cosh(bx) J 0 (ax) dx = 2 2 − 21 0 [Re c > |Re b|, a > 0]

∞

1. 0

∞

2. 0

3.12

1

2 =

0

ET II 12(52)

ET II 12(54)

2λ 1 Γ 12 − λ + μ 1 −β cosh x x M −λ,μ α2 + β 2 2 − β coth e J 2μ (α sinh x) dx = 2 α Γ(2μ + 1) 1 × W λ,μ α2 + β 2 2 + β Re β > |Re α|, Re(μ − λ) > − 12 BU 86(5b)a, ET II 363(34)

2λ 1 x coth e−β cosh x Y 2μ (α sinh x) dx 2

sec[(μ + λ)π] =− α2 + β 2 + β W −λ,μ α2 + β 2 − β W λ,μ α

tan[(μ + λ)π] Γ 12 − λ + μ W λ,μ α2 + β 2 + β M −λ,μ α2 + β 2 − β − α Γ(2μ + 1) Re β > |Re α|, Re λ < 12 − |Re μ| ET II 363(35)

2ν √ 1 1 x e− 2 (a1 +a2 )t cosh x coth K 2μ (t a1 a2 sinh x) dx 2 Γ 12 + μ − ν Γ 12 − μ − ν W ν,μ (a1 t) W ν,μ (a2 t) = √ 2t a1 a2 √ √ 2 1 ± 2μ Re ν < Re , Re t ( a1 + a2 ) > 0 BU 85(4a) 2

∞

724

4.

12

Bessel Functions

∞

e

− 12 (a1 +a2 )t cosh x

0

6.

x 2ν Γ 12 + μ − ν √ W ν,μ (a1 t) M ν,μ (a2 t) coth I 2μ (t a1 a2 sinh x)dx = √ 2 1 t a1 a2 Γ(1 + 2μ) Re 2 + μ − ν > 0, Re μ > 0, a1 > a2 BU 86(5c)

√ Γ 12 + μ + ν Γ 12 + μ − ν xy ds = M ν,μ (x) M −ν,μ (y) √ cosh s cosh s xy [Γ(1 + 2μ)]2 −∞ Re ±ν + 12 + μ > 0 BU 83(3a)a 1 1

√ ∞ Γ 2 +μ+ν Γ 2 +μ−ν x+y xy ds = e2νs− 2 tanh s J 2μ M ν,μ (x) M ν,μ (y) √ 2 cosh s cosh s xy [Γ(1 + 2μ)] −∞ Re ∓ν + 12 + μ > 0 BU 84(3b)a

5.

6.671

∞

e2νs−

x−y 2

tanh s

I 2μ

6.67–6.68 Combinations of Bessel and trigonometric functions 6.671

∞

1. 0

∞

2. 0

sin ν arcsin ab √ J ν (ax) sin bx dx = a2 − b 2

[b < a]

= ∞ or 0

[b = a]

aν cos νπ = √ √2 ν b 2 − a2 b + b 2 − a2

[b > a]

J ν (ax) cos bx dx =

cos ν arcsin √ a2 − b 2

b a

[b = a] ν

−a = √ 2 b − a2 b

3. 0

∞

WA 444(4)

[b < a]

= ∞ or 0

[Re ν > −2]

sin νπ √2 + b2

− a2

ν

[b > a] [Re ν > −1]

Y ν (ax) sin(bx) dx

νπ 1 b 2 2 −2 a −b = cot sin ν arcsin 2 a νπ 1 1 2 2 −2 b −a = cosec 2# 2 $ 1 ν 1 −ν × a−ν cos(νπ) b − b2 − a2 2 − aν b − b2 − a2 2

WA 444(5)

[0 < b < a, |Re ν| < 2]

[0 < a < b,

|Re ν| < 2] ET I 103(33)

6.671

Bessel and trigonometric functions

∞

4. 0

Y ν (ax) cos(bx) dx

tan νπ b 2 cos ν arcsin = 1 a (a2 − b2 ) 2 # νπ 1 1 ν − = − sin b2 − a2 2 a−ν b − b2 − a2 2 + cot(νπ) 2 $ 1 −ν ν +a b − b2 − a2 2 cosec(νπ)

0

∞

6.

∞

7. 0

8. 0

K ν (ax) cos(bx) dx

J 0 (ax) sin(bx) dx = 0

[0 < b < a]

1 = √ 2 b − a2

[0 < a < b] ET I 99(1)

∞

J 0 (ax) cos(bx) dx = √

1 a2 − b 2

[0 < b < a]

=∞

[a = b]

=0

[0 < a < b]

∞ 1 b J 2n+1 (ax) sin(bx) dx = (−1)n √ T 2n+1 2 − b2 a a 0 =0

ET I 43(1)

∞ 1 b J 2n (ax) cos(bx) dx = (−1)n √ T 2n 2 − b2 a a 0

[0 0, b > 0, |Re ν| < 2, ν = 0] ET I 105(48)

9.

[0 < a < b,

$ νπ # 2 2 1 1 ν 1 −ν π 2 2 −2 −ν 2 2 ν 2 2 b +a a = b+ b +a sec +a b+ b +a 4 2 [Re a > 0, b > 0, |Re ν| < 1] ET I 49(40)

0

|Re ν| < 1]

K ν (ax) sin(bx) dx =

[0 < b < a,

ET I 47(29) ∞

5.

725

=0

[0 < b < a] [0 < a < b] ET I 43(2)

726

Bessel Functions

∞

11. 0

0

∞

0

14.8

∞

0

[0 < a < b]

α + β

∞

2. 0

α2 +1 β2

π K 0 (βx) cos αx dx = 2 α2 + β 2

0

∞

4. 0

5. 0

WA 425(11)a, MO 48 WA 425(10)a, MO 48

J ν (ax) J ν (bx) sin(cx) dx

2

2

2

J ν (x) J −ν (x) cos(bx) dx =

1 P 1 2 ν− 2

1 2 b −1 2

[Re ν > −1,

0 < c −1,

b − a < c −1,

b + a < c,

0 < a < b] 0 < a < b]

0 < a < b] ET I 102(27)

[0 0]

[α > 0]

=0

ET I 47(28)

b +a −c 1 = √ P ν− 12 2 ab

2ab2 b + a2 − c2 cos(νπ) =− √ Q ν− 12 − 2ab π ab

[0 < a < b]

[0 0, ∞

0

6.672 1.

Y 0 (ax) cos(bx) dx = 0 1 = −√ 2 b − a2

13.

2 arcsin ab Y 0 (ax) sin(bx) dx = √ π a2 − b 2 ) * b2 1 2 b − = √ ln −1 π b 2 − a2 a a2

ET I 103(31) ∞

12.

6.672

2

π K ν (ax) K ν (bx) cos(cx) dx = √ sec(νπ) P ν− 12 4 ab

1 K ν (ax) I ν (bx) cos(cx) dx = √ Q ν− 12 2 ab

2 a + b2 + c2 (2ab)−1 Re(a + b) > 0, c > 0,

|Re ν| <

1 2

ET I 50(51)

2

a2 + b 2 + c 2ab Re a > |Re b|,

c > 0,

Re ν > − 12

ET I 49(47)

∞

2

1 P 1 1 − 2a2 2 ν− 2 1 = cos(νπ) Q ν− 12 2a2 − 1 π

sin(2ax) [J ν (x)] dx =

[0 < a < 1, [a > 1,

Re ν > −1]

Re ν > −1] ET II 343(30)

6.673

Bessel and trigonometric functions

∞

6. 0

1 Q 1 1 − 2a2 π ν− 2 1 = − sin(νπ) Q ν− 12 2a2 − 1 π

cos(2ax) [J ν (x)]2 dx =

0

sin(2ax) J 0 (x) Y 0 (x) dx = 0 K

=− ∞

8. 0

1−a

[0 < a < 1]

1 −2 2

[a > 1]

πa

1 K 0 (ax) I 0 (bx) cos(cx) dx = K c2 + (a + b)2

%

& √ 2 ab c2 + (a + b)2 [Re a > |Re b|,

∞

9. 0

0 < a < 1, Re ν > − 12 a > 1, Re ν > − 12 ET II 344(32)

∞

7.

727

1 K(a) π

1 1 =− K πa a

cos(2ax) J 0 (x) Y 0 (x) dx = −

c > 0]

ET II 348(60)

ET I 49(46)

[0 < a < 1] [a > 1] ET II 348(61)

∞

10.

2

1 K 1 − a2 π 1 2 K 1− 2 = πa a

cos(2ax) [Y 0 (x)] dx =

0

[0 < a < 1] [a > 1] ET II 348(62)

6.673

∞

1. 0

νπ

νπ − Y ν (ax) sin sin(bx) dx J ν (ax) cos 2 2 =0 =

2. 0

2aν

1 √ b 2 − a2

# $ 2 1 ν 1 ν 2 2 2 2 2 b+ b −a + b− b −a

[0 < b < a,

|Re ν| < 2]

[0 < a < b,

|Re ν| < 2] ET I 104(39)

∞

νπ

νπ + J ν (ax) sin cos(bx) dx Y ν (ax) cos 2 2 =0 =−

2aν

1 √ b 2 − a2

# $ 2 1 ν 1 ν 2 2 2 2 2 b+ b −a + b− b −a

[0 < b < a,

|Re ν| < 1]

[0 < a < b,

|Re ν| < 1] ET I 48(32)

3. 0

π/2

[cos x I 0 (a cos x) + I 1 (a cos x)] dx =

ea − 1 a

728

6.674

Bessel Functions

a

1. 0

a

a

0

a

0

0

0

6.675

sin(a − x) J 2n (x) dx = a J 2n+1 (a) + (−1)n 2n cos a − J 0 (a) − 2 )

a

(−1)m J 2m (a)

n−1 "

ET II 334(10)

*

(−1)m J 2m+1 (a)

ET II 334(11) n "

*

(−1)m J 2m (a)

m=1

[n = 0, 1, 2, . . .] z

z

ET II 336(22)

sin(z − x) J 0 (x) dx = z J 1 (z)

WA 415(2)

cos(z − x) J 0 (x) dx = z J 0 (z)

WA 415(1)

∞

2

2

2

2 √ √ νπ νπ a π a a a a 1 1 1 1 − J − sin − J cos J ν a x sin(bx) dx = 3 ν− 2 ν+ 2 2 2 8b 4 8b 8b 4 8b 4b 2 [a > 0, b > 0, Re ν > −4] ET I 110(23)

∞

0

√ J ν a x cos(bx) dx =−

0

*

cos(a − x) J 2n+1 (x) dx = a J 2n+1 (a) + (−1)n (2n + 1) cos a − J 0 (a) − 2

2.

3.

n "

[n = 0, 1, 2, . . .] )

0

ET II 336(23)

[n = 0, 1, 2, . . .]

cos(a − x) J 2n (x) dx = a J 2n (a) − (−1)n 2n sin a − 2

1.

[Re ν > −1]

m=0

7. 8.

)

ET II 335(21) [n = 0, 1, 2, . . .] ) * n a " n m sin(a − x) J 2n+1 (x) dx = a J 2n+2 (a) + (−1) (2n + 1) sin a − 2 (−1) J 2m+1 (a)

0

0

(−1)n J ν+2n+1 (a)

m=0

6.

ET II 334(12)

m=1

4.

5.

cos(a − x) J ν (x) dx = a J ν (a) − 2ν

∞ "

n=0

3.

(−1)n J ν+2n+2 (a) [Re ν > −1]

0

∞ "

n=0

2.

sin(a − x) J ν (x) dx = a J ν+1 (a) − 2ν

6.674

∞

√ a π 3

4b 2

sin

νπ a2 − 8b 4

√ 1 J 0 a x sin(bx) dx = cos b

a2 4b

2

2 a νπ a2 a J 12 ν− 12 + cos J 12 ν+ 12 − 8b 8b 4 8b [a > 0, b > 0, Re ν > −2] ET I 53(22)a

[a > 0,

b > 0]

ET I 110(22)

6.677

Bessel and trigonometric functions

∞

4. 0

6.676

∞

1. 0

∞

2. 0

3. 4.

a2 4b

[a > 0,

√ √ 1 J ν a x J ν b x sin(cx) dx = J ν c

√ √ 1 J ν a x J ν b x cos(cx) dx = J ν c

ab 2c

ab 2c

∞

cos

sin

b > 0]

a2 + b 2 νπ − 4c 2 [a > 0, b > 0,

2

5. 0

c > 0,

c > 0,

b > 0]

ET I 111(31)

b > 0]

ET I 54(29)

a

√ √ 1 K0 ax Y 0 ax cos(bx) dx = − K 0 2b 2b

√ Re a > 0, b > 0 ∞ √

√

a

a π2 1 1 H0 −Y0 K0 axe 4 πi K 0 axe− 4 πi cos(bx) dx = 8b 2b 2b 0

∞

1.

J 0 b x2 − a2 sin(cx) dx = 0

√ cos a c2 − b2 √ = c2 − b 2

a

√ ∞

exp −a b 2 − c2 √ J 0 b x2 − a2 cos(cx) dx = 2 2 a b √− c − sin a c2 − b2 √ = c2 − b 2

3.6

0

∞

cos z a2 − β 2 2 2 J 0 a x + z cos βx dx = a2 − β 2 =0

ET I 54(30)

ET I 54(31)

[0 < c < b] [0 −1] ET I 54(27)

[Re a > 0, b > 0] 6.677

Re ν > −2] ET I 111(29)a

νπ a2 + b 2 − 4c 2 [a > 0, b > 0,

[Re a > 0, ∞

ET I 53(21)

√ √ a 1 K0 [Re a > 0, J 0 a x K 0 a x sin(bx) dx = 2b 2b 0 ∞ a √ √ π a

I0 − L0 J0 ax K 0 ax cos(bx) dx = 4b 2b 2b 0

6.

√ 1 J 0 a x cos(bx) dx = sin b

729

[0 < c < b] [0 0]

[0 < a < β,

z > 0] MO 47a

730

Bessel Functions

∞

4. 0

6.678

1 Y 0 a x2 + z 2 cos βx dx = sin z a2 − β 2 a2 − β 2

1 exp −z β 2 − a2 = − β 2 − a2

[0 < β < a,

z > 0]

[0 < a < β,

z > 0] MO 47a

∞

5. 0

π K 0 a x2 + β 2 cos(γx) dx = exp −β a2 + γ 2 2 a2 + γ 2 [Re a > 0, Re β > 0,

γ > 0] ET I 56(43)

√ a

sin a b2 + c2 √ J 0 b a2 − x2 cos(cx) dx = [b > 0] b 2 + c2 0 √ ∞

cosh a b2 − c2 √ J 0 b x2 − a2 cos(cx) dx = [0 < c < b, b 2 − c2 0

6. 7.

=0

8.

ET I 57(49)

i exp −iβ α2 + γ 2 (2) H 0 α β 2 − x2 cos(γx) dx = α2 + γ 2 −π < arg β 2 − x2 ≤ 0,

α > 0,

γ>0

∞

∞

0

∞

1. 0

∞

0

√ π √ 1 π sin K 0 2 x + Y 0 2 x sin(bx) dx = 2 2b b

ET I 59(59)

[b > 0]

ET I 58(58)

ET I 111(34)

x sin(bx) dx = −i [I ν−ib (a) K ν+ib (a) − I ν+ib (a) K ν−ib (a)] J 2ν 2b sinh 2 [a > 0, b > 0, Re ν > −1] ET I 115(59)

2.

4.

a > 0]

γ>0

0

3.

[0 0,

9.

6.679

a > 0]

∞

exp iβ α2 + γ 2 (1) H 0 α β 2 − x2 cos(γx) dx = −i α2 + γ 2 0 π > arg β 2 − x2 ≥ 0,

6.678

MO 48a, ET I 57(47)

x cos(bx) dx = I ν−ib (a) K ν+ib (a) + I ν+ib (a) K ν−ib (a) J 2ν 2a sinh 2 a > 0, b > 0, Re ν > − 12

x π cos(bx) dx = − [J ν+ib (a) Y ν−ib (a) + J ν−ib (a) Y ν+ib (a)] J 2ν 2a cosh 2 2 0 ∞ x 2 sin(bx) dx = sinh(πb) [K ib (a)]2 J 0 2a sinh 2 π 0

ET I 59(64)

∞

[a > 0,

b > 0]

ET I 59(63)

ET I 115(58)

6.681

Bessel and trigonometric functions

∞

5. 0

6. 0

x cos(bx) dx = [I ib (a) + I −ib (a)] K ib (a) J 0 2a sinh 2 [a > 0,

∞

∞

7. 0

731

Y 0 2a sinh

x 2

cos(bx) dx = −

b > 0]

ET I 59(62)

b > 0]

ET I 59(65)

2 2 cosh(πb) [K ib (a)] π

[a > 0, x 2 π 2 2 cos(bx) dx = [J ib (a)] + [Y ib (a)] K 0 2a sinh 2 4

[Re a > 0, 6.681

1.

π 2

0

2.

π 2

0

3.

π 2

0

4.

0

π 2

π

5. 0

π

6. 0

7.

0

π 2

π

8. 0

0

10.

π 2

0

11. 0

Re ν > − 12

π J ν+μ (a) J ν−μ (a) 2

cos(2μx) Y 2ν (2a cos x) dx =

π [cot(2νπ) J ν+μ (a) J ν−μ (a) − cosec(2νπ) J μ−ν (a) J −μ−ν (a)] 2 ET II 361(24) |Re ν| < 12

cos(2μx) I 2ν (2a cos x) dx =

π I ν−μ (a) I ν+μ (a) 2

cos(νx) K ν (2a cos x) dx =

π I 0 (a) K ν (a) 2

ET II 361(23)

Re ν > − 12

ET I 59(61)

[Re ν < 1]

WA 484(3)

J 0 (2z cos x) cos 2nx dx = (−1)n πJn2 (z).

MO 45

J 0 (2z sin x) cos 2nx dx = πJn2 (z). cos(2nx) Y 0 (2a sin x) dx =

WA 43(3), MO 45

π J n (a) Y n (a) 2

[n = 0, 1, 2, . . .]

ET II 360(16)

sin(2μx) J 2ν (2a sin x) dx = π sin(μπ) J ν−μ (a) J ν+μ (a) ET II 360(13)

cos(2μx) J 2ν (2a sin x) dx = π cos(μπ) J ν−μ (a) J ν+μ (a) Re ν > − 12 J ν+μ (2z cos x) cos[(ν − μ)x] dx =

ET II 360(14)

π J ν (z) J μ (z) 2 [Re(ν + μ) > −1]

π 2

ET I 59(66)

cos(2μx) J 2ν (2a cos x) dx =

[Re ν > −1] π

9.

b > 0]

cos[(μ − ν)x] I μ+ν (2a cos x) dx =

π I μ (a) I ν (a) 2

MO 42

[Re(μ + ν) > −1] WA 484(2), ET II 378(39)

732

12.12

Bessel Functions

π 2

0

cos[(μ − ν)x] K μ+ν (2a cos x) dx =

6.682

π cosec[(μ + ν)π] [I −μ (a) I −ν (a) − I μ (a) I ν (a)] 2 [|Re(μ + ν)| < 1]

13.8

π 2

0

K ν−m (2a cos x) cos[(m + ν)x] dx = (−1)m

ET II 378(40)

π I m (a) K ν (a) 2 [|Re(ν − m)| < 1]

6.682 1.7

π 2

WA 485(4)

π J ν (x) 2x 0 [ν may be zero, a natural number, one half, or a natural number plus one half; x > 0]

2.

π 2

0

6.683 1.

π 2

0

2.

π 2

0

1

J ν− 12 (x sin t) sinν+ 2 t dt =

z

√ 1 z −ν Jν2 J ν (z sin x) sinν x cos2ν x dx = 2ν−1 π Γ ν + 2 2 Re ν > − 12

3. 0

4.

π 2

0

0

0

Re μ > −1]

π 2

WA 407(4)

WA 410(1)

∞ 1" J ν z cos2 x J μ z sin2 x sin x cos x dx = (−1)k J ν+μ+2k+1 (z) z

J μ (z sin θ) (sin θ)

1−μ

2ν+1

(cos θ)

dθ =

k=0

Re μ > −1]

(see also 6.513 6)

J μ (z sin θ) (sin θ)

1−μ

J μ (a sin θ) (sin θ)

μ+1

dθ =

WA 407(2)

Hμ− 12 (z) 2z π

(cos θ)

2+1

WA 414(1)

s μ+ν,ν−μ+1 (z) 2μ−1 z ν+1 Γ(μ) [Re ν > −1]

π 2

5.

6.

[Re ν > Re μ > −1]

z1ν z2μ J ν+μ+1 z12 + z22 J ν (z1 sin x) J μ (z2 cos x) sinν+1 x cosμ+1 x dx = ν+μ+1 (z12 + z22 )

[Re ν > −1,

MO 42a

μ−ν Γ 2 2

J μ (z) J ν (z sin x) I μ (z cos x) tanν+1 x dx = μ+ν Γ +1 2 z ν

[Re ν > −1, π 2

MO 42a

WA 407(3)

dθ = 2 Γ( + 1)a−−1 J +μ+1 (a) [Re > −1,

Re μ > −1] WA 406(1),

EH II 46(5)

6.686

Bessel and trigonometric functions

π 2

ν

2ν

J ν (2z sin θ) (sin θ) (cos θ)

7. 0

733

dθ

∞ 1 " (−1)m z ν+2m Γ ν + m + 12 Γ ν + 12 = 2 m=0 m! Γ(ν + m + 1) Γ(2ν + m + 1) √ 1 2 = z −ν π Γ ν + 12 [J ν (z)] 2

Re ν > − 21

EH II 47(10)

8.

π 2

ν−1

z J ν (z sin θ) (sin θ)ν+1 (cos θ)−2ν dθ = 2−ν √ Γ π

1 − ν sin z 2 −1 < Re ν < 12 EH II 68(39) 1 Γ 2 + ν J 2ν+ 12 (z) J ν z sin2 θ J ν z cos2 θ (sin θ)2ν+1 (cos θ)2ν+1 dθ = 2ν+ 3 √ 2 Γ(ν + 1) 2 z 1 WA 409(1) Re ν > − 2 Γ μ + 12 Γ ν + 12 J μ+ν+ 12 (z) √ J μ z sin2 θ J ν z cos2 θ sin2μ+1 θ cos2ν+1 θ dθ = √ 2 π Γ(μ + ν + 1) 2z WA 417(1) Re μ > − 12 , Re ν > − 12

0

9.

π 2

0

10.

π 2

0

6.684 1.8

π

(sin x)

2ν

0

π

2.

(sin x) 0

2ν

Jν α2 + β 2 − 2αβ cos x √ 1 J ν (α) J ν (β) ν

ν dx = 2 π Γ ν + 2 αν βν α2 + β 2 − 2αβ cos x Re ν > − 12

Yν α2 + β 2 − 2αβ cos x √ 1 J ν (α) Y ν (β)

ν dx = 2ν π Γ ν + 2 αν βν α2 + β 2 − 2αβ cos x |α| < |β|,

6.685 6.686

π 2

0

∞

1. 0

2. 0

∞

sec x cos(2λx) K 2μ (a sec x) dx =

π W λ,μ (a) W −λ,μ (a) 2a

Re ν > − 21

[Re a > 0]

2

2 √ π ν+1 b b sin ax2 J ν (bx) dx = − √ sin − π J 12 ν 8a 4 8a 2 a [a > 0, b > 0,

2

2 √ 2 π b ν+1 b − π J 12 ν cos ax J ν (bx) dx = √ cos 2 a 8a 4 8a [a > 0, b > 0,

ET II 362(27)

ET II 362(28)

ET II 378(41)

Re ν > −3] ET II 34(13)

Re ν > −1] ET II 38(38)

734

Bessel Functions

∞

3.

∞

4.

cos ax2 Y ν (bx) dx

√ νπ

π = √ sec 4 a 2

2

2

2 3ν + 1 ν −1 b b b b2 − π J 12 ν + cos + π Y 12 ν × sin 8a 4 8a 8a 4 8a [a > 0, b > 0, −1 < Re ν < 1] ET II 107(8)

0

sin ax2 Y ν (bx) dx

√ νπ

π = − √ sec 4 a 2

2

2

2 3ν + 1 b ν −1 b b b2 − π J 12 ν − sin + π Y 12 ν × cos 8a 4 8a 8a 4 8a [a > 0, b > 0, −3 < Re ν < 3] ET II 107(7)

0

∞

b2 1 sin ax2 J 1 (bx) dx = sin b 4a 0

2 ∞ 2 b 2 6. cos ax J 1 (bx) dx = sin2 b 8a 0

2 ∞ 1 b 7. sin2 ax2 J 1 (bx) dx = cos 2b 8a 0

2 ∞ π π x K 2ν xei 4 K 2ν xe−i 4 dx 6.687 cos 2a 0 5.

= 6.688

π 2

1. 0

0

3.

π J ν (μz sin t) cos (μx cos t) dt = J ν2 2

Γ

1 4

[a > 0,

b > 0]

ET II 19(16)

[a > 0,

b > 0]

ET II 20(20)

[a > 0,

b > 0]

ET II 19(17)

√ π + ν Γ 14 − ν π √ W 14 ,ν aei 2 W 8 a a > 0, |Re ν| < 14

1 4 ,ν

π

ae−i 2

ET II 372(1)

√ √ x2 + z 2 + x x2 + z 2 − x J ν2 μ μ 2 2 [Re ν > −1,

π 2

2.

6.687

MO 46 Re z > 0] − 1 ν− 1 1 1√ ν+1 (sin x) cos (β cos x) J ν (α sin x) dx = 2− 2 παν α2 + β 2 2 4 J ν+ 12 α2 + β 2 2

[Re ν > −1] π 2

0

ET II 361(19)

π cos [(z − ζ) cos θ] J 2ν 2 zζ sin θ dθ = J ν (z) J ν (ζ) 2 Re ν > − 12

EH II 47(8)

6.69–6.74 Combinations of Bessel and trigonometric functions and powers

6.691

0

∞

x sin(bx) K 0 (ax) dx =

− 3 πb 2 a + b2 2 2

[Re a > 0,

b > 0]

ET I 105(47)

6.693

Bessel and trigonometric functions and powers

6.692 1.

∞

0

0

∞

0

2.8

∞

0

− 1 1 3 x K ν (ax) I ν (bx) sin(cx) dx = − (ab)− 2 c u2 − 1 2 Q 1ν− 1 (u), u = (2ab)−1 a2 + b2 + c2 2 2 Re a > |Re b|, c > 0, Re ν > − 32 ET I 106(54)

∞

2.

6.693 1.

735

− 1 3 π x K ν (ax) K ν (bx) sin(cx) dx = (ab)− 2 c u2 − 1 2 Γ 32 + ν Γ 32 − ν P −1 (u) ν− 12 4 u = (2ab)−1 a2 + b2 + c2 Re(a + b) > 0, c > 0, |Re ν| < 32 ET I 107(61)

1 b dx = sin ν arcsin J ν (ax) sin bx x ν a aν sin νπ 2 = √ ν ν b + b 2 − a2

b dx 1 = cos ν arcsin J ν (ax) cos bx x ν a aν cos νπ 2 = √ ν ν b + b 2 − a2

[b ≤ a] [b ≥ a] [Re ν > −1]

WA 443(2)

[b ≤ a] [b ≥ a]

[Re ν > 0] WA 443(3)

∞

3. 0

Y ν (ax) sin(bx)

dx x

νπ

b 1 sin ν arcsin = − tan ν 2 a

[0 < b < a, |Re ν| < 1] $ νπ # 1 ν 2 1 −ν 1 −ν 2 2 2 ν 2 2 sec a cos(νπ) b − b − a −a b− b −a = 2ν 2 [0 < a < b, |Re ν| < 1]

ET I 103(35) ∞

4. 0

5. 0

dx x√2 b cos ν arcsin ab a2 − b2 sin ν arcsin ab − = ν 2− ν (ν 2 − 1) 1 √ νπ ν −a cos 2 b + ν b2 − a2 = √ ν ν (ν 2 − 1) b + b2 − a2

J ν (ax) sin(bx)

[0 0]

[0 < a < b,

Re ν > 0] ET I 99(6)

∞

dx J ν (ax) cos(bx) 2 x a cos (ν + 1) arcsin ab a cos (ν − 1) arcsin ab + = 2ν(ν − 1) 2ν(ν + 1) νπ aν sin 2 aν+2 sin νπ 2 = − √ √ ν−1 ν+1 2 2 2 2ν(ν − 1) b + b − a 2ν(ν + 1) b + b − a2

[0 1]

[0 < a < b,

Re ν > 1] ET I 44(6)

736

6.

12

Bessel Functions

∞

0

7.

8. 9.

J 0 (ax) sin x

dx π = x 2 = arccosec a

[0 < a < 1] [a > 1] WH

∞

dx π = x 2 0 = arcsin b π =− 2 ∞ dx [J 0 (x) − cos ax] = ln 2a x 0 z ∞ 2" dx = J ν (x) sin(z − x) (−1)k J ν+2k+1 (z) x ν 0

10. 0

6.694

J 0 (x) sin bx

[b > 1] 2 b <1 [b < −1] NT 66(13)

[Re ν > 0]

WA 416(4)

k=0

z

J ν (x) cos(z − x)

∞ 1 dx 2" = J ν (z) + (−1)k J ν+2k (z) x ν ν k=1

WA 416(5) [Re ν > 0] * √ √ 2 ∞ J 1 (ax) b 2a + b 2 2ab 2 2ab 2 2 2 − (2a − b) K (4a + b )E sin(bx) dx = − x 2 12πa2 2a + b 2a + b 0 [a > 0, b > 0]

)

6.69412

ET I 102(22)

6.695

∞

1. 0

∞

2. 0

[a > 0,

Re b > 0,

u > a]

π e−ab cos ax J (ux) dx = I 0 (bu) 0 b2 + x2 2 b

[a > 0,

Re b > 0,

−a 0,

Re b > 0,

0 < γ < a] ET II 10(36)

∞

4.

x2

0

x cos(αx) J 0 (γx) dx = cosh(αβ) K 0 (βγ) + β2

[α > 0,

Re β > 0,

α < γ] ET II 11(45)

6.696

0

∞

[1 − cos(ax)] J 0 (bx)

a

dx = arccosh x b =0

[0 < b < a] [0 < a < b] ET II 11(43)

6.698

6.697 1.

Bessel and trigonometric functions and powers

∞

sin[a(x + b)] J 0 (x) dx = 2 x+b −∞

a

0

cos bu √ du 1 − u2

[0 ≤ a ≤ 1]

= π J 0 (b)

∞

2. 0

∞

3. 0

4. 5.12 6.12

7.

737

WA 463(2)

[1 ≤ a < ∞]

WA 463(1), ET II 345(42)

π sin(x + t) J 0 (t) dt = J 0 (x) x+t 2

[x > 0]

WA 475(4)

cos(x + t) π J 0 (t) dt = − Y 0 (x) x+t 2

[x > 0]

WA 475(5)

∞

|x| sin[α(x + β)] J 0 (bx) dx = 0 [0 ≤ α < b] WA 464(5), ET II 345(43)a −∞ x + β ∞ 2 2 sin[a(x + b)] J n+ 12 (x) dx = π J n+ 12 (b) [2 ≤ a, n = 0, 1, . . .] ET II 346(45) x+b −∞ ∞ sin[a(x + b)] J n+ 12 (x) J −n− 12 (x) dx = π J n+ 12 (b) J −n− 12 (b) x+b −∞

[2 ≤ a, n = 0, 1, . . .] √ ∞ Γ(μ + ν) π a2 J μ+ν− 12 [a(z − ζ)] J μ [a(z + x)] J ν [a(ζ + x)] · dx = 1 μ (ζ + x)ν Γ μ + 12 Γ ν + 12 (z − ζ)μ+ν− 2 −∞ (z + x)

ET II 346(46)

[Re(μ + ν) > 0] 6.698

1. 0

∞√

x J ν+ 14 (ax) J −ν+ 14 (ax) sin(bx) dx =

b 2 cos 2ν arccos 2a √ πb 4a2 − b2

=0 2. 0

∞√

x J ν− 14 (ax) J −ν− 14 (ax) cos(bx) dx =

WA 463(3)

[0 < b < 2a] [0 < 2a < b]

2 cos 2ν arccos √ πb 4a2 − b2

b 2a

ET I 102(26)

=0

[0 0, b > 0,

Re ν <

5 4

ET I 106(56)

4. 0

∞√

x I − 14 −ν

√ 2ν

π −2ν b + a2 + b2 1 1 √ ax K − 14 +ν ax cos(bx) dx = a 2 2 2b a2 + b 2 Re a > 0, b > 0, Re ν < 34

ET I 50(49)

738

6.699

Bessel Functions

∞

1. 0

0

0

1+λ −(2+λ)

Γ

2 Γ ν−λ 2

∞

xλ K μ (ax) sin(bx)dx =

∞

0

∞

5. 0

2+λ−μ Γ 2λ b Γ 2+μ+λ 2 2 a2+λ

6. 0

∞

ET I 45(13)

b2 2+μ+λ 2+λ−μ 3 , ; ;− 2 2 2 2 a [Re (−λ ± μ) < 2, Re a > 0, b > 0]

F

1+λ−μ μ+λ+1 λ λ−1 −λ−1 Γ x K μ (ax) cos(bx) dx = 2 a Γ 2 2

b2 μ+λ+1 1+λ−μ 1 , ; ;− 2 ×F 2 2 2 a [Re (−λ ± μ) < 1, Re a > 0,

ET I 106(50)

−ν− 12 √ ν ν 2 π2 b a − b2 x sin(ax) J ν (bx) dx = Γ 12 − ν ν

=0

xλ J ν (ax) cos(bx) dx 2λ a−(1+λ) Γ 1+λ+ν 1 + λ + ν 1 + λ − ν 1 b2 2 ν−λ+1 , ; ; 2 F = 2 2 2 a Γ 2 0 < b < a, − Re ν < 1 + Re λ < 32 a ν −(ν+1+λ) b Γ (1 + λ + ν) cos π2 (1 + λ + ν) a2 1+λ+ν 2+λ+ν 2 = F , ; ν + 1; 2 Γ(ν + 1) 2 2 b 0 < a < b, − Re ν < 1 + Re λ < 32

4.

2 + λ + ν 2 + λ − ν 3 b2 , ; ; 2 F x J ν (ax) sin(bx) dx = 2 a b 2 2 2 a 0 < b < a, − Re ν − 1 < 1 + Re λ < 32

ν Γ (ν + λ + 1) 1+λ+ν 1 a b−(ν+λ+1) sin π = 2 Γ(ν + 1) 2

a2 2+λ+ν 1+λ+ν , ; ν + 1; 2 ×F 2 2 b 0 < a < b, − Re ν − 1 < 1 + Re λ < 32 λ

ET I 100(11) ∞

2.

3.

2+λ+ν

6.699

0 0]

−ν− 12

0 < b < a,

|Re ν| <

1 2

0 < a < b,

|Re ν| <

1 2

ET II 36(29)

6.711

Bessel and trigonometric functions and powers

∞

7.

xν+1 sin(ax) J ν (bx) dx

−ν− 32 sin(νπ) ν 3 2 b Γ ν+ a − b2 = −2 a √ π 2 −ν− 32 3 2 21+ν b − a2 = − √ abν Γ ν + π 2

0

739

1+ν

0 < b < a,

− 23 < Re ν < − 21

0 < a < b,

− 23 < Re ν < − 21

ET II 32(3)

∞

8. 0

√ xν+1 cos(ax) J ν (bx) dx = 21+ν πabν

−ν− 32 a2 − b 2 Γ − 12 − ν

=0

0 −1]

1

ET II 334(9)a

1 [cos aJν (a) + sin a J ν+1 (a)] 2ν +1 0 Re ν > − 12

∞ − 3 −ν √ 3 + ν b b 2 + a2 2 x1+ν K ν (ax) sin(bx) dx = π(2a)ν Γ 2 0 Re a > 0, b > 0,

xν cos(ax) J ν (ax) dx =

∞

12. 0

Re ν > − 23

ET I 105(49)

b > 0,

Re μ > − 21

ET I 49(41)

∞

13.

xν Y ν−1 (ax) sin(bx) dx = 0

√ −ν− 12 2ν πaν−1 b 2 1 b − a2 = Γ 2 −ν

0

−μ− 12 1√ 1 2 μ μ b + a2 x K μ (ax) cos(bx) dx = π(2a) Γ μ + 2 2 Re a > 0,

ET II 335(20)

0 < b < a,

|Re ν| <

0 < a < b,

|Re ν| <

1 2 1 2

ET I 104(36)

∞

14. 0

xν Y ν (ax) cos(bx) dx = 0 = −2

ν√

πa

ν

−ν− 12 b 2 − a2 Γ 12 − ν

0 < b < a,

|Re ν| <

1 2

0 < a < b,

|Re ν| <

1 2

ET I 47(30)

6.711

1. 0

∞

xν−μ J μ (ax) J ν (bx) sin(cx) dx = 0

[0 < c < b − a,

−1 < Re ν < 1 + Re μ] ET I 103(28)

740

Bessel Functions

∞

2. 0

xν−μ+1 J μ (ax) J ν (bx) cos(cx) dx = 0 [0 < c 0,

c Γ(ν) Γ(μ + 1) 0 < c 0,

a > 0,

xν−μ−2 J μ (ax) J ν (bx) sin(cx) dx = 2ν−μ−1 aμ b−ν [0 < a,

4.

6.712

a > 0,

ET I 47(26)

Γ 32 − ν a 3 3 2 − ν, − 2ν; 2 − ν; a F sin(2ax) J ν (x) Y ν (x) dx = − 2 2 2 Γ 2ν − 12 Γ(2 − ν) 0 < Re ν < 32 , 0 < a < 1

ET II 348(63)

ρ2 a2 2z 2 Γ (ν) aμ ρ−ν − − μ−ν+3 2 Γ (μ + 1) ν − 1 μ + 1 3 0 2 ∞ μ −ν 2 a Γ (ν) a ρ ρ − − 2z 2 cos (zx)xν−μ−3 J μ (ax) J ν (ρx) dx = μ−ν+3 2 Γ (μ + 1) ν − 1 μ + 1 0 arg sin (zx)xν−μ−4 J μ (ax) J ν (ρx) dx = z

√ −ν− 12 π(2a)ν 2 b + 2ab x [J ν (ax) cos(ax) + Y ν (ax) sin(ax)] sin(bx) dx = 1 Γ 2 −ν 0 b > 0, −1 < Re ν < 12 √ ∞ −ν− 12 π(2a)ν 2 ν b + 2ab x [Y ν (ax) cos(ax) − J ν (ax) sin(ax)] cos(bx) dx = − 1 Γ 2 −ν 0

∞

ν

0

xν [J ν (ax) cos(ax) − Y ν (ax) sin(ax)] sin(bx) dx =0 =

4. 0

ET I 104(40)

ET I 48(35) ∞

3.

√ −ν− 12 2ν πbν 2 1 b − 2ab Γ 2 −ν

−1 < Re ν < −1 < Re ν < 12

0 < b < 2a, 2a < b,

1 2

ET I 104(41) ∞

xν [J ν (ax) sin(ax) + Y ν (ax) cos(ax)] cos(bx) dx =0

√ −ν− 12 π(2a)ν 2 b − 2ab = − 1 Γ 2 −ν

0 < b < 2a, 0 < 2a < b,

|Re ν| <

1 2

|Re ν| <

1 2

ET I 48(33)

6.715

Bessel and trigonometric functions and powers

6.713 1.

∞

0

x1−2ν sin(2ax) [J ν (x)]2 − [Y ν (x)]2 dx =

∞

2. 0

∞

3. 0

6.714 1.

∞

0

2.

sin(2νπ) Γ

741

3

− ν Γ 32 − 2ν a 3 3 F − ν, − 2ν; 2 − ν; a2 π Γ(2 − ν) 2 2 0 < Re ν < 34 , 0 < a < 1 ET II 348(64) 2

x2−2ν sin(2ax) [J ν (x) J ν−1 (x) − Y ν (x) Y ν−1 (x)] dx

sin(2νπ) Γ 32 − ν Γ 52 − 2ν a 5 3 F − ν, − 2ν; 2 − ν; a2 =− π Γ(2 − ν) 2 2 1 5 < Re ν < , 0 < a < 1 ET II 348(65) 2 4 x2−2ν sin(2ax) [J ν (x) Y ν−1 (x) + Y ν (x) J ν−1 (x)] dx

Γ 32 − ν a 5 3 2 − ν, − 2ν; 2 − ν; a F =− 2 2 Γ 2ν − 32 Γ(2 − ν) 1 5 0 < a < 1 ET II 349(66) 2 < Re ν < 2 ,

sin(2ax) [xν J ν (x)]2 dx a−2ν Γ 12 + ν 1 1 + ν, ; 1 − ν; a2 F = √ 2 2 π Γ(1 − ν) 2

a−4ν−1 Γ 12 + ν 1 1 1 F = + ν, + 2ν; 1 + ν; 2 2 2 a 2 Γ (1 + ν) Γ 12 − 2ν

|Re ν| <

0 < a < 1,

|Re ν| <

a > 1,

1 2

1 2

ET II 343(31) ∞

2

cos(2ax) [xν J ν (x)] dx

a−2ν Γ(ν) 1 1 1 F ν + , ; 1 − ν; a2 = √ 2 2 2 πΓ 2 − ν Γ(−ν) Γ 12 + 2ν 1 1 2 F + ν, + 2ν; 1 + ν; a + 2 2 2π Γ 12 − ν sin(νπ)a−4ν−1 Γ 12 + 2ν 1 1 1 + ν, + 2ν; 1 + ν; 2 =− F 2 2 a Γ(1 + ν) Γ 12 − ν 0

0 < a < 1, a > 1,

− 14 < Re ν <

− 41 < Re ν <

1 2

1 2

ET II 344(33)

6.715 1. 0

2.

0

∞

∞

xν π sin(x + b) J ν (x) dx = sec(νπ)bν J −ν (b) x+b 2 xν π cos(x + b) J ν (x) dx = − sec(νπ)bν Y −ν (b) x+b 2

|arg b| < π,

|Re ν| <

1 2

|arg b| < π,

|Re ν| <

1 2

ET II 340(8)

ET II 340(9)

742

6.716

Bessel Functions

a

1. 0

xλ sin(a − x) J ν (x) dx = 2aλ+1

6.716

∞ " (−1)n Γ(ν − λ + 2n) Γ(ν + λ + 1) (ν + 2n + 1) J ν+2n+1 (a) Γ(ν − λ) Γ(ν + λ + 3 + 2n) n=0

[Re(λ + ν) > −1]

a

2. 0

xλ cos(a − x) J ν (x) dx =

ET II 335(16)

λ+1

J ν (a) a + 2aλ+1 λ+ν+1 ∞ " (−1)n Γ (ν − λ + 2n − 1) Γ(ν + λ + 1) (ν + 2n) J ν+2n (a) × Γ(ν − λ) Γ(ν + λ + 2n + 2) n=1 [Re(λ + ν) > −1]

6.718

∞

sin[a(x + b)] J ν+2n (x) dx = πb−ν J ν+2n (b) ν (x + b) x −∞ 1 ≤ a < ∞,

6.717

∞

1. 0

∞

2. 0

∞

3. 0

ET II 335(26)

n = 0, 1, 2, . . . ;

xν sin(ax) J ν (cx) dx = bν−1 sinh(ab) K ν (bc) x2 + b2 0 < a ≤ c,

Re b > 0,

−1 < Re ν <

0 < c ≤ a,

x1−ν π sin(ax) J ν (cx) dx = b−ν e−ab I ν (bc) x2 + b2 2

Re b > 0,

ET II 345(44)

3 2

−1 < Re ν <

Re b > 0,

xν+1 cos(ax) J ν (cx) dx = bν cosh(ab) K ν (bc) x2 + b2 0 < a ≤ c,

Re ν > − 32

1 2

ET II 33(8)

ET II 37(33)

Re ν > − 21

ET II 33(9) ∞

4.

0 < c ≤ a,

−ν

x π cos(ax) J ν (cx) dx = b−ν−1 e−ab I ν (bc) 2 +b 2

x2

0

Re b > 0,

Re ν > − 23

ET II 37(34)

6.719 1.

6

a

∞ " sin(bx) √ J ν (x) dx = π (−1)n J 2n+1 (ab) J 12 ν+n+ 12 12 a J 12 ν−n− 12 12 a a2 − x2 n=0

a

2 cos(bx) π √ J ν (x) dx = J 0 (ab) J 12 ν 12 a +π (−1)n J 2n (ab) J 12 ν+n 12 a J 12 ν−n 12 a 2 a2 − x2 n=1

0

2. 0

[Re ν > −2]

[Re ν > −1] 6.721

1. 0

ET II 335(17)

∞ "

√ x J 14 a2 x2 sin(bx) dx = 2−3/2 a−2 πb J 14

∞√

b2 4a2

ET II 336(27)

[b > 0]

ET I 108(1)

6.722

Bessel and trigonometric functions and powers

2. 0

3. 0

5.

6.

2 2

x J − 14 a x

cos(bx) dx = 2

−3/2 −2

a

√ πb J − 14

√ x Y 14 a2 x2 sin(bx) dx = −2−3/2 πba−2 H 14

∞√

b2 4a2

[b > 0]

b 4a2

√ x Y − 14 a2 x2 cos(bx) dx = −2−3/2 πba−2 H− 14

∞√

[b > 0] 6.722

1. 0

2.12

0

3.12

0

4.12

0

∞√

2 2

x K 18 +ν a x

2 2

I 18 −ν a x

ET I 51(1)

2

b2 4a2 0

2 ∞ √ 2 2 √ b2 b −5/2 −2 3 I 14 − L 14 x K 14 a x sin(bx) dx = 2 π ba 2 2 4a 4a 0 π |arg a| < , b > 0 4

2

2 ∞ √ 2 2 √ b b 1 − L x K − 14 a x cos(bx) dx = 2−5/2 π 3 ba−2 I − 14 −4 2 4a 4a2 0

4.

∞√

743

ET I 108(7) ET I 52(7)

ET I 109(11)

ET I 52(10)

5

2

2 √ b b −3/2 Γ 8−ν M −ν, 18 W ν, 18 sin(bx) dx = 2πb 5 2 8a 8a2 Γ 4 π 5 Re ν < , |arg a| < , b > 0 8 4 ET I 109(13)

∞√

x J − 18 −ν a2 x2 J − 18 +ν a2 x2 cos(bx) dx

2 −πi/2

2 −πi/2 2 b e b e iπ/8 e W−nu,−1/8 Wnu,−1/8 = b3 π 8a2 8a2

2 πi/2

2 πi/2 b e b e W−nu,−1/8 + e−iπ/8 Wnu,−1/8 2 8a 8a2 2 a > 0, Im b = 0 MC

∞√

x J 18 −ν a2 x2 J 18 +ν a2 x2 sin(bx) dx

2 πi/2

2 πi/2 2 −3/2 πi/8 b e b e b e W −ν, 18 W ν, 18 = π 8a2 8a2 ⎤

2 −πi/2 πi b2 e− 2 ⎦ b e W −ν, 18 + e−iπ/8 W ν, 18 2 8a 8a2 2 ET I 108(6) a > 0, b > 0

∞√

x K 18 −ν a2 x2 I − 18 −ν a2 x2 cos(bx) dx =

√

3

2

2 b b 8−ν M −ν,− 18 W ν,− 18 2πb 3 2 8a 8a2 Γ 4 3 π Re ν < 8 , |arg a| < 4 , b > 0 ET I 52(12) −3/2 Γ

744

Bessel Functions

6.723 6.724

∞

0

∞

1. 0

∞

2. 0

6.723

1 x J ν x2 sin(νπ) J ν x2 − cos(νπ) Y ν x2 J 4ν (4ax) dx = J ν a2 J −ν a2 4 ET II 375(20) [a > 0, Re ν > −1]

x2λ J 2ν

x2λ J 2ν

a

x

a

x

sin(bx) dx

√ 2ν πa Γ(λ − ν + 1)b2ν−2λ−1 1 a2 b 2 0 F 3 2ν + 1, ν − λ, ν − λ + ;

= 1 2 16 42ν−λ Γ(2ν + 1) Γ ν − λ + 2

a2 b 2 3 a2λ+2 Γ(ν − λ − 1)b , λ − ν + 2, λ + ν + 2; + 2λ+3 0F 3 2 Γ(ν + λ + 2) 2 16 − 45 < Re λ < Re ν, a > 0, b > 0 ET I 109(15) cos(bx) dx

√ 2ν 2ν−2λ−1 Γ λ − ν + 12 a2 b 2 1 =4 πa b 0 F 3 2ν + 1, ν − λ + , ν − λ; Γ(2ν + 2 16 1) Γ(ν − λ) 1 2 2 Γ ν − λ − b 3 3 a 1 2 ,λ − ν + ,ν + λ+ ; +4−λ−1 a2λ+1 0F 3 3 2 2 2 16 Γ ν +λ+ 2 3 − 4 < Re λ < Re ν − 12 , a > 0, b > 0 ET I 53(14) λ−2ν

6.725

∞

1. 0

∞

2. 0

∞

3. 0

4. 0

∞

√ sin(bx) √ J ν a x dx = − x

√ cos(bx) √ J ν a x dx = x

π sin b

π cos b

a2 νπ π − − 8b 4 4

a2 νπ π − − 8b 4 4

√ 1 x 2 ν J ν a x sin(bx) dx = 2−ν aν b−ν−1 cos

√ 1 x 2 ν J ν a x cos(bx) dx = 2−ν b−ν−1 aν sin

a2 J ν2 8b [Re ν > −3,

b > 0] ET I 110(27)

2

a 8b [Re ν > −1,

J 12 ν

a > 0,

a > 0,

b > 0] ET I 54(25)

νπ a2 − 4b 2 −2 < Re ν < 12 ,

a > 0,

ET I 110(28)

νπ a2 − 4b 2 −1 < Re ν < 12 ,

b>0

a > 0,

b>0

ET I 54(26)

6.727

Bessel and trigonometric functions and powers

6.726 1.

∞

0

− 1 ν x x2 + b2 2 J ν a x2 + b2 sin(cx) dx

1 ν− 34 π −ν −ν+ 3 2 2 c a − c2 2 a b J ν− 32 b a2 − c2 = 2 =0

∞

2.

x +b J ν a x2 + b2 cos(cx) dx

1 ν− 1 π −ν −ν+ 1 2 2 a b a − c2 2 4 J ν− 12 b a2 − c2 = 2

=0

∞

3.

2

∞

∞

x2 + b2

0

5. 0

6.727 1.9 0

2.12

3. a

1 2

0 < a < c,

Re ν >

1 2

ET I 111(37)

0 < c < a, 0 < a < c,

b > 0,

Re ν > − 21

b > 0,

Re ν > − 21

∓ 12 ν

K ν a x2 + b2 cos(cx) dx

± 1 ν− 1 π ∓ν 1 ∓ν 2 a b2 a + c2 2 4 K ±ν− 12 b a2 + c2 = 2 [Re a > 0, Re b > 0, c is real] ET I 56(45)

− 1 ν x2 + a2 2 Y ν b x2 + a2 cos(cx) dx

1 ν− 1 aπ (ab)−ν b2 − c2 2 4 Y ν− 12 a b2 − c2 = 2

1 ν− 1 2a (ab)−ν c2 − b2 2 4 K ν− 12 a c2 − b2 =− π

0 < c < b,

a > 0,

Re ν > − 21

0 0,

Re ν > − 21

ET I 56(41) a

a a

cos(cx) π √ J ν b a2 − x2 dx = J 12 ν b2 + c2 − c J 12 ν b 2 + c2 + c 2 2 2 a2 − x2 [Re ν > −1, c > 0, a > 0] ET I 113(48)

∞

a

a

sin(cx) π √ c − c2 − b2 J − 12 ν c + c2 − b 2 J ν b x2 − a2 dx = J 12 ν 2 2 2 x2 − a2 [0 0, Re ν > −1]

∞

a

a

cos(cx) π √ c − c2 − b2 Y − 12 ν c + c2 − b 2 J ν b x2 − a2 dx = − J 12 ν 2 2 2 x2 − a2 [0 0, Re ν > −1]

a

Re ν >

ET I 55(37)

2 2 K ±ν a x + b sin(cx) dx

− 1 ν− 3 π ν ν+ 3 2 a b 2 c a + c2 2 4 K −ν− 32 b a2 + c2 = 2 [Re a > 0, Re b > 0, c > 0] ET I 113(45)

x x +b

0 < c < a,

12 ν

2

0

4.11

1 2 −2ν

2

0

745

ET I 113(49)

ET I 58(54)

746

4.

8

Bessel Functions

a

0

6.728

∞

1.

∞

2.

∞

3. 0

∞

4. 0

∞

5. 0

a −x

12 ν

cos x I ν a2 − x2 dx =

√ 2ν+1 πa ν+1 2 Γ ν + 32 Re ν > − 12

WA 409(2)

x sin ax2 J ν (bx) dx

x cos ax2 J ν (bx) dx

2

2

2

2 √ πb νπ b νπ b b b − J 12 ν+ 12 + sin − J 12 ν− 12 = 3/2 cos 8a 4 8a 8a 4 8a 8a [a > 0, b > 0, Re ν > −2] ET II 38(39)

0

2

2

2

2

2 √ πb νπ νπ b b b b − J 12 ν− 12 − sin − J 12 ν+ 12 = 3/2 cos 8a 4 8a 8a 4 8a 8a [a > 0, b > 0, Re ν > −4] ET II 34(14)

0

2

6.728

∞

6. 0

1 b2 J 0 (bx) sin ax2 x dx = cos 2a 4a

[a > 0,

b > 0]

MO 47

b2 1 sin J 0 (bx) cos ax2 x dx = 2a 4a

[a > 0,

b > 0]

MO 47

νπ b2 − 4a 2 a > 0,

b > 0,

xν+1 sin ax2 J ν (bx) dx =

xν+1 cos ax2 J ν (bx) dx =

bν cos ν+1 2 aν+1

bν 2ν+1 aν+1

sin

−2 < Re ν <

1 2

ET II 34(15)

νπ b2 − 4a 2 a > 0,

b > 0,

−1 < Re ν <

1 2

ET II 38(40)

6.729

∞

1.

2

x sin ax 0

2. 0

∞

1 cos J ν (bx) J ν (cx) dx = 2a

1 sin x cos ax2 J ν (bx) J ν (cx) dx = 2a

b 2 + c2 νπ bc − Jν 4a 2 2a [a > 0, b > 0,

c > 0,

ET II 51(26)

b 2 + c2 νπ bc − Jν 4a 2 2a [a > 0, b > 0,

Re ν > −2]

c > 0,

Re ν > −1] ET II 51(27)

6.735

6.731 1.11

Bessel and trigonometric functions and powers

∞

2.10

x sin ax2 J ν bx2 J 2ν (2cx) dx

1 bc2 ac2 = √ Jν sin 2 2 2 2 2 b 2 − a2

b −2a

b −2a 1 bc ac Jν cos = √ a2 − b 2 a2 − b 2 2 a2 − b 2

0

747

[0 < a < b,

Re ν > −1]

[0 −1] ET II 356(41)a

∞

x cos ax2 J ν bx2 J 2ν (2cx) dx

1 bc2 ac2 = √ Jν cos 2 2 2 2 2 b 2 − a2

b −2a b −2 a bc ac 1 Jν sin = √ a2 − b 2 a2 − b 2 2 a2 − b 2

0

0 < a < b,

Re ν > − 21

0 − 21

ET II 356(42)a

6.73212 6.733

∞

x3 cos

0

∞

sin

1. 0

∞

2. 0

x2 Y 1 (x) K 1 (x) dx = −a3 K 0 (a) 2a

[a > 0]

ET II 371(52)

a

√ √ dx [sin x J 0 (x) + cos x Y 0 (x)] = π J0 a Y0 a 2x x

[a > 0] a

√ √ dx [sin x Y 0 (x) − cos x J 0 (x)] = π J0 cos a Y0 a 2x x

ET II 346(51)

[a > 0] ET II 347(52) a

√ √ πa K 0 (x) dx = J1 3. x sin a K1 a [a > 0] ET II 368(34) 2x 2 0 ∞ a

√ √ πa K 0 (x) dx = − Y1 4. x cos a K1 a [a > 0] ET II 369(35) 2x 2 0 ∞ √ dx 6.734 cos a x K ν (bx) √ x 0

a a a a π D −ν− 12 − √ + D ν− 12 − √ D −ν− 12 √ = √ sec(νπ) D ν− 12 √ 2 b 2b 2b 2b 2b Re b > 0, |Re ν| < 12 ET II 132(27) 6.735

∞

∞

1. 0

∞

2. 0

3.

0

∞

√ √ x1/4 sin 2a x J − 14 (x) dx = πa3/2 J 34 a2

[a > 0]

ET II 341(10)

√ √ x1/4 cos 2a x J 14 (x) dx = πa3/2 J − 34 a2

[a > 0]

ET II 341(12)

√ √ x1/4 sin 2a x J 34 (x) dx = πa3/2 J − 14 a2

[a > 0]

ET II 341(11)

748

Bessel Functions

∞

4. 0

6.736 1.

11

∞

0

√ √ x1/4 cos 2a x J − 34 (x) dx = πa3/2 J 14 a2

6.736

[a > 0]

ET II 341(13)

√ √ π 2 π

J 0 a − sin a2 − Y 0 a2 x−1/2 sin x cos 4a x J 0 (x) dx = −2−3/2 π cos a2 − 4 4

ET II 341(18) [a > 0] ∞ √ √ π 2 π

J 0 a + cos a2 − Y 0 a2 x−1/2 cos x cos 4a x J 0 (x) dx = −2−3/2 π sin a2 − 4 4 0

2.

∞

3. 0

∞

4. 0

∞

5. 0

0

∞

0

∞

2. 0

4.

[a > 0]

π π J 0 a2 cos a2 − 2 4

ET II 341(16)

[a > 0] ET II 342(20)

√ √ π π

J 0 a2 − cos a2 − Y 0 a2 x−1/2 sin x cos 4a x Y 0 (x) dx = 2−3/2 π 3 sin a2 − 4 4 ET II 347(55)

√ x−1/2 cos x cos 4a x Y 0 (x) dx

√

sin a x2 + b2 b b π 2 2 2 2 √ a− a −c a+ a −c J − 12 ν J ν (cx) dx = J 12 ν 2 2 2 x2 + b2 [a > 0, Re b > 0, c > 0, a > c, Re ν > −1] ET II 35(19) √

cos a x2 + b2 b b π √ a − a2 − c2 Y − 12 ν a + a 2 − c2 J ν (cx) dx = − J 12 ν 2 2 2 x2 + b2 [a > 0, Re b > 0, c > 0, a > c, Re ν > −1] ET II 39(44)

√ a a

cos b a2 − x2 π √ J ν (cx) dx = J 12 ν b2 + c2 − b J 12 ν b 2 + c2 + b 2 2 2 a2 − x2 0 ET II 39(47) [c > 0, Re ν > −1] √ √ a cos a2 − x2 πa2ν+1

[Re ν > −1] xν+1 √ I ν (x) dx = ET II 365(9) 3 a2 − x2 0 2ν+1 Γ ν + 2

3.

ET II 342(22)

√ π 2 π

J 0 a + sin a2 − Y 0 a2 = −2−3/2 π 3 cos a2 − 4 4 [a > 0] ET II 347(56)

1.

√ x−1/2 cos x sin 4a x J 0 (x) dx =

[a > 0]

π π cos a2 + J 0 a2 2 4

[a > 0] ∞

6.

6.737

√ x−1/2 sin x sin 4a x J 0 (x) dx =

a

6.741

Bessel and trigonometric functions and powers

√ a b2 + x2 √ x J ν (cx) dx b2 + x2 0

π 1 +ν ν 2 2 − 14 − 12 ν b 2 c a −c J −ν− 12 b a2 −c2 = 2

5.

∞

749

ν+1 sin

0 < c < a, 0 < a < c,

=0

Re b > 0,

−1 < Re ν <

1 2

Re b > 0,

−1 < Re ν <

1 2

ET II 35(20)

√ ∞

2 2 − 1 − 1 ν π 1 +ν ν 2 ν+1 cos a x + b √ b 2 c a − c2 4 2 Y −ν− 12 b a2 − c2 x J ν (cx) dx = − 2 x2 + b2 0

6.

1 0 < c < a, Re b > 0, −1 < Re ν < 2

1 1 2 1 +ν ν 2 − − ν = b 2 c c − a2 4 2 K ν+ 12 b c2 − a2 π 1 0 < a < c, Re b > 0, −1 < Re ν < 2 ET II 39(45)

6.738 a

− 1 ν− 3 π ν+ 3 a 2 b 1 + b2 2 4 J ν+ 32 a 1 + b2 xν+1 sin b a2 − x2 J ν (x) dx = 1. 2 0 [Re ν > −1] ET II 335(19) ∞

2. xν+1 cos a x2 + b2 J ν (cx) dx 0

− 1 ν− 3 π ν+ 3 ν 2 ab 2 c a − c2 2 4 cos(πν) J ν+ 32 b a2 − c2 − sin (πν) Y ν+ 32 b a2 − c2 = 2 0 < c < a, Re b > 0, −1 < Re ν < − 21 =0

0 < a < c,

−1 < Re ν < − 21

Re b > 0,

ET II 39(43)

√ √ t

√t

√ t −1/2 cos b t − x 2 2 2 2 √ J 2ν a x dx = π J ν x a + b + b Jν a +b −b 2 2 t−x 0 Re ν > − 12 EH II 47(7)

6.739 6.741 1.

2.

a

a

cos (μ arccos x) π √ J 12 (ν−μ) J ν (ax) dx = J 12 (μ+ν) 2 2 2 1 − x2 0 [Re(μ + ν) > −1, 1

a

cos [(ν + 1) arccos x] π a √ J ν+ 12 J ν (ax) dx = cos a 2 2 1 − x2 0

3. 0

1

1

cos [(ν − 1) arccos x] √ J ν (ax) dx = 1 − x2

a

π sin J ν− 12 a 2

[Re ν > −1, a

a > 0]

a > 0]

ET II 41(54)

ET II 40(53)

2 [Re ν > 0,

a > 0]

ET II 40(52)a

750

Bessel Functions

6.751

6.75 Combinations of Bessel, trigonometric, and exponential functions and powers 6.751 1.

2.

3.10

1 1 (b + c)2 + a2 − (b − c)2 + a2 , 2 = (b + c)2 + a2 + (b − c)2 + a2 2 2

∞ 1 1 1 1 e− 2 ax sin(bx) I 0 b + b 2 + a2 ax dx = √ √ 2 2 2b b + a2 0 [Re a > 0, b > 0] ET I 105(44)

∞ 1 1 a 1 ax dx = √ √ e− 2 ax cos(bx) I 0 √ 2 2 2 2b a + b b + a2 + b2 0 [Re a > 0, b > 0] ET I 48(38) 1/2 2 ∞ (b2 + c2 − a2 ) + 4a2 b2 + b2 + c2 − a2 e−bx cos(ax) J 0 (cx) dx = √ 2 0 2 (b2 + c2 − a2 ) + 4a2 b2 Notation: 1 =

[c > 0] alternatively, with a and b interchanged, ∞ 2 − b2 −ax e cos(bx) J 0 (cx) dx = 2 2 2 2 − 1 0 6.752 10

∞

e

1.

0

2.10

0

∞

−ax

dx = arcsin J 0 (bx) sin(cx) x

ET II 11(46)

[c > 0]

2c

a2 + (c + b)2 + a2 + (c − b)2

= arcsin

c 2

ET I 101(17) [Re a > |Im b|, c > 0] b b − b2 − 21 dx = (1 − r) = , e−ax J 1 (cx) sin(bx) x c c c2 a2 b2 = − , c > 0 1 − r2 r2 ET II 19(15)

Notation: For integrals 6.752 3–6.752 5 we deﬁne the auxiliary functions 1 1 (a) ≡ 1 (a, ρ, z) = (a + ρ)2 + z 2 − (a − ρ)2 + z 2 2 1 2 (a) ≡ 1 (a, ρ, z) = (a + ρ)2 + z 2 + (a − ρ)2 + z 2 2 when a ≥ 0, ρ ≥ 0, and z ≥ 0. ∞ √ π 10 e−zx J ν+1/2 (ax) J ν+1 (ρx) x dx 3. 2 0

a ρ2 − 21 2ν+2 1 ρ =a 2 (2 − 2 ) 2 1 ρ − 1 1 2 ν+1 2 − a2 ρ = aν+1/2 2ν+2 2 2 2 2 − 1 2 −ν−3/2 −ν−1

[Re z > |Im a| + |Im ρ|]

6.753

4.

10

5.10

Combinations of Bessel, trigonometric, and exponential functions and powers

751

∞ π dx e−zx J ν+1/2 (ax) J ν (ρx) √ 2 0 x

1/ 2 1 1 1 ν+1/2 ν =a ρ d 2ν 2 2 1 − a /2 2 0 2 a/ 2 dx ν > − 12 , Re z > |Im a| + |Im ρ| x2ν √ = a−ν−1/2 ρν 2 1−x 0

2

2 − a2 a − ∞ 2 − 2 a 2 1 ρ a dx −zx + arcsin e sin(ax) J 1 (ρx) 2 = x 2aρ 2 2 0 [Re z > |Im a| + |Im ρ|]

6.753 1.8

sin (xa sin ψ) −xa cos ϕ cos ψ ϕ ν e J ν (xa sin ϕ) dx = ν −1 tan sin(νψ) x 2 0 π π Re ν > −1, a > 0, 0 < ϕ < , 0 < ψ < ET II 33(10) 2 2 ∞ cos (xa sin ψ) −xa cos ϕ cos ψ ϕ ν e J ν (xa sin ϕ) dx = ν −1 tan cos(νψ) x 0 2 π Re ν > 0, a > 0, 0 < ϕ, ψ < 2

2.

3.8

∞

ET II 38(35) ∞

0

ν

2(2a) Γ(ν + 32 )R xν+1 e−sx sin(bx) J ν (ax) dx = − √ π

−2ν−3

b cos(ν + 32 )ϕ + s sin(ν + 32 )ϕ

Re ν > − 23 ,

2 R4 = s2 + a2 − b2 + 4b2 s2 , 8

∞

4.

0

xν+1 e−sx cos(bx) J ν (ax) dx =

Re s > |Im a| + |Im b|, ϕ = arg s2 + a2 − b2 − 2ibs

2(2a)ν √ Γ(ν + 32 )R−2ν−3 s cos(ν + 32 )ϕ − b sin(ν + 32 )ϕ , π Re ν > −1,

5.10

∞

0

6. 0

∞

2 R4 = s2 + a2 − b2 + 4b2 s2 ,

Re s > |Im a| + |Im b|, 2 2 2 ϕ = arg s + a − b − 2ibs

xν e−ax cos ϕ cos ψ sin (ax sin ψ) J ν (ax sin ϕ) dx 1 −ν− 12 ν νΓ ν + 2 √ a−ν−1 (sin ϕ) cos2 ψ + sin2 ψ cos2 ϕ sin ν + 12 β =2 π π π β a > 0, 0 < ϕ < , 0 < ψ < , Re ν > −1 ET II 34(12) tan = tan ψ cos ϕ 2 2 2 xν e−ax cos ϕ cos ψ cos (ax sin ψ) J ν (ax sin ϕ) dx 1 −ν− 12 ν νΓ ν+ 2 √ a−ν−1 (sin ϕ) cos2 ψ + sin2 ψ cos2 ϕ cos ν + 12 β =2 π π 1 β a > 0, 0 < ϕ, ψ < , Re ν > − tan = tan ψ cos ϕ ET II 38(37) 2 2 2

752

6.754 1. 2.

Bessel Functions

6.754

2 √ π − b2 b 8 I [b > 0] e ET I 108(9) 0 3/2 8 2 0 2

2

2

2 ∞ 2 2 a π a π a a 1 π −ax J0 cos − −Y0 cos + e cos x J 0 x dx = 4 2 16 16 4 16 16 4 0

∞

∞

3. 0

2 e−x sin(bx) I 0 x2 dx =

1 e−ax sin x2 J 0 x2 dx = 4

π J0 2

2

a 16

sin

MI 42 [a > 0]

a2 π a2 π a2 − −Y0 sin + 16 4 16 16 4

[a > 0] 6.755

∞

1. 0

2.

3.

4.

6.

7.

8. 9.

1 3 1 1 2 − 2 ν, 2 − 2 ν

[a > 0,

2a2

Re ν > 0]

ET II 366(14)

∞

√ 2 1 3 x−ν− 2 e−x cos 4a x I ν (x) dx = 2 2 ν−1 aν−1 e−a W − 32 ν, 12 ν 2a2 0 ET II 366(16) a > 0, Re ν > − 12 ∞

ν−1 Γ 3 − 2ν √ 2 ea W 3 ν− 1 , 1 − 1 ν 2a2 x−ν ex sin 4a x K ν (x) dx = 23/2 a π 21 2 2 2 2 Γ 2 +ν 0 a > 0, 0 < Re ν < 34 ET II 369(38) 1 ∞ √ Γ − 2ν a2 1 3 e W 3 ν,− 1 ν 2a2 x−ν− 2 ex cos 4a x K ν (x) dx = 2 2 ν−1 πaν−1 21 2 2 Γ 2+ 0 ν a > 0, − 12 < Re ν < 14 ET II 369(42)

√ √ πa Γ( + ν) Γ( − ν) 1 3 − 32 −x 2 x e sin 4a x K ν (x) dx = 2 F 2 + ν, − ν; , + ; −2a 2 2 2−2 Γ + 12 0 ET II 369(39) [Re > |Re ν|]

√ ∞ √ π Γ( + ν) Γ( − ν) 1 1 2 , + ; −2a + ν, − ν; x−1 e−x cos 4a x K ν (x) dx = F 2 2 2 2 2 Γ + 12 0 ET II 370(43) [Re > |Re ν|] ∞ √ 2 1 x−1/2 e−x cos 4a x I 0 (x) dx = √ e−a K 0 a2 2π 0 ET II 366(15) [a > 0] ∞ √ π a2 e K 0 a2 [a > 0] x−1/2 ex cos 4a x K 0 (x) dx = ET II 369(40) 2 0 ∞ √ 2 1 x−1/2 e−x cos 4a x K 0 (x) dx = √ π 3/2 e−a I 0 a2 ET II 369(41) 2 0

5.

ν−1 √ 2 x−ν e−x sin 4a x I ν (x) dx = 23/2 a e−a W

MI 42

∞

6.761

Bessel, trigonometric, and hyperbolic functions

6.756 1.

∞

1

√ x

√ sin a x J ν (bx) dx

ia i 1 a ia D −ν− 12 √ =√ D −ν− 12 √ − D −ν− 12 − √ Γ ν+ 2 2πb b b b [a > 0, b > 0, Re ν > −1] ET II 34(17)

1

√ x

√ cos a x J ν (bx) dx

a ia ia 1 1 D −ν− 12 √ =√ Γ ν+ D −ν− 12 √ + D −ν− 12 − √ 2 2πb b b b a > 0, b > 0, Re ν > − 21 ET II 39(42)

x− 2 e−a

0

∞

2.

x− 2 e−a

0

∞

3.

x−1/2 e−a

√ x

0

∞

4. 0

6.757 1.

∞

0

x−1/2 e−a

√ x

√ 1 sin a x J 0 (bx) dx = a I 14 2b

√ a I 1 cos a x J 0 (bx) dx = 2b − 4

2 a K 14 4b π |arg a| < , 4

2

2 a a K 14 4b 4b π |arg a| < , 4

a2 4b

b>0

ET II 11(40)

b>0

∞ " (−1)n Γ(ν − b + 2n + 1) Γ (ν + b) (ν + 2n − 1) J ν+2n+1 (a) Γ(ν − b + 1) Γ(ν + b + 2n + 2) n=0

[Re b > − Re ν]

∞

2. 0

∞

Γ(ν − b + 2n) Γ(ν + b) J ν (a) " + (ν + 2n) J ν+2n (a) 2(−1)n ν + b n=0 Γ(ν − b + 1) Γ(ν + b + 2n + 1) [Re b > − Re ν]

ET I 193(26)

e−bx cos a 1 − e−x J ν ae−x dx =

6.758

ET II 12(49)

e−bx sin a 1 − e−x J ν ae−x dx =2

753

π 2

π −2

ei(μ−ν)θ (cos θ)

ν+μ

(λz)−ν−μ J ν+μ (λz) dθ = π(2az)−μ (2bz)−ν J μ (az) J ν (bz) [Re(ν + μ) > −1] λ = 2 cos θ (a2 eiθ + b2 e−iθ )

ET I 193(27)

EH II 48(12)

6.76 Combinations of Bessel, trigonometric, and hyperbolic functions 6.761

0

∞

x

cosh x cos (2a sinh x) J ν (be ) J ν be

−x

√ J 2ν 2 b2 − a2 √ dx = 2 b 2 − a2 =0

[0 < a < b,

Re ν > −1]

[0 −1] ET II 359(10)

754

Bessel Functions

6.762

∞

0

cosh x sin (2a sinh x) J ν (bex ) Y ν be−x − Y ν (bex ) J ν be−x dx =0 =−

6.763

∞

−1/2 1/2 2 cos(νπ) a2 − b2 K 2ν 2 a2 − b2 π x

cosh x cos (2a sinh x) Y ν (be ) Y ν be

−x

0 < a < b,

|Re ν| <

1 2

0 < b < a,

|Re ν| <

1 2

ET II 360(12)

dx −1/2 1/2 1 J 2ν 2 b2 − a2 = − b 2 − a2 2 −1/2 1/2 2 = cos(νπ) a2 − b2 K 2ν 2 a2 − b2 π

0

6.762

[0 < a < b,

|Re ν| < 1]

[0 < b < a,

|Re ν| < 1] ET II 360(11)

6.77 Combinations of Bessel functions and the logarithm, or arctangent 6.771

∞

μ+ 12

x 0

1 3 2μ− 2 Γ μ+ν 3 a2 ν −μ 1 μ+ν 2 + 4 + +ψ + − ln ln x J ν (ax) dx = ν−μ 1 μ+ 3 ψ 2 4 2 4 4 Γ 2 +4 a 2 3 a > 0, − Re ν − 2 < Re μ < 0

ET II 32(25)

6.772

∞

1 ln x J 0 (ax) dx = − [ln(2a) + C] WA 430(4)a, a 0 ∞ 1 a

+C 2. ln x J 1 (ax) dx = − ln a 2 0 ∞ 2 3. ln a2 + x2 J 1 (bx) dx = [K 0 (ab) + ln a] b 0 ∞ 2 4. J 1 (tx) ln 1 + t4 dt = ker x x 0 √

∞ ln x + x2 + a2 ab ab 1 2 ab √ K0 + ln a I 0 K0 6.773 J 0 (bx) dx = 2 2 2 2 2 2 x +a 0 [a > 0, b > 0]

∞ √ 2 x + a2 + x ab dx [Re a > 0, b > 0] 6.774 ln √ = K 20 J 0 (bx) √ 2 + a2 − x 2 + a2 2 x x 0 ∞

1 6.77512 x ln a + a2 + x2 − ln x J 0 (bx) dx = 2 1 − e−ab b 0 [Re a > 0, b > 0]

∞ a2 2 1 − a K 1 (ab) 6.776 x ln 1 + 2 J 0 (bx) dx = [Re a > 0, b > 0] x b b 0 ∞ 2 6.777 J 1 (tx) arctan t2 dt = − kei x x 0 1.

ET II 10(27) ET II 19(11) ET II 19(12) MO 46

ET II 10(28) ET II 10(29)

ET II 12(55) ET II 10(30) MO 46

6.784

Combinations of Bessel and other special functions

755

6.78 Combinations of Bessel and other special functions

6.781

∞

0

1 si(ax) J 0 (bx) dx = − arcsin b

b a

[0 < b < a]

=0

[0 < a < b] ET II 13(6)

6.782 1. 2. 3. 4. 5. 6.12 7.12 6.783 1. 2. 3.

∞

√ e−z − 1 Ei(−x) J 0 2 zx dx = z 0 ∞ √ sin z si(x) J 0 2 zx dx = − z 0 ∞ √ cos z − 1 ci(x) J 0 2 zx dx = z 0 ∞ √ dx Ei(−z) − C − ln z √ Ei(−x) J 1 2 zx √ = x z 0 ∞ π √ dx − si(z) si(x) J 1 2 zx √ = − 2 √ x z 0 ∞ √ dx ci(x) − C − ln z √ ci(x) J 1 2 zx √ = x z 0 ∞ √ C + ln z − ez Ei(−z) Ei(−x) Y 0 2 zx dx = πz 0

NT 60(6) NT 60(5) NT 60(7) NT 60(9) NT 60(8) NT 63(5)

2 2 b [a > 0] x si a x J 0 (bx) dx = − 2 sin 2 b 4a 0

2 ∞ b 2 [a > 0] x ci a2 x2 J 0 (bx) dx = 2 1 − cos 2 b 4a 0 2

2 ∞ 2 2 b b 1 ci + ln + 2C ci a x J 0 (bx) dx = b 4a2 4a2 0

∞

∞

4. 0

6.784

NT 60(4)

∞

0

1 − si si a2 x2 J 1 (bx) dx = b

ν+1

x

1.

2 2

[1 − Φ(ax)] J ν (bx) dx = a

2

b 4a2

−ν

−

π 2

ET II 13(7)a ET II 13(8)a

[a > 0]

ET II 13(8)a

[a > 0]

ET II 20(25)a

2 Γ ν + 32 b2 b 1 1 1 1 exp − M 2 2 2 ν+ 2 , 2 ν+ 2 b2 Γ(ν + 2) 8a 4a π |arg a| < 4 , b > 0, Re ν > −1

ET II 92(22)

756

Bessel Functions

∞

2. 0

ν

x [1 − Φ(ax)] J ν (bx) dx =

6.785

2

1 b2 2 a 2 −ν Γ ν + 12 b 1 1 1 1 M exp − 2 ν− 4 , 2 ν+ 4 π b3/2 Γ ν + 32 8a2 4a2 |arg a| < π4 , Re ν > − 12 , b > 0

ET II 92(23)

6.785

∞

0

6.787

a2 2x

−x

x

0

6.786

∞ exp

1−Φ

a √ 2x

K ν (x) dx =

π 5/2 2 2 sec(νπ) [J ν (a)] + [Y ν (a)] 4 Re a > 0, |Re ν| < 12 ET II 370(46)

xν−2μ+2n+2 ex2 Γ μ, x2 Y ν (bx) dx 3 3

2

2 b b nΓ 2 − μ + ν + n Γ 2 − μ + n exp W μ− 12 ν−n−1, 12 ν = (−1) b Γ(1 − μ) 8 4 n is an integer, b > 0, Re(ν − μ + n) > − 32 , Re(−μ + n) > − 32 , Re ν < 12 − 2n

ET II 108(2)

∞

0

1

xν+2n− 2 J ν (bx) dx = 0 B(a + x, a − x)

π ≤ b < ∞,

−1 < Re ν < 2a − 2n −

7 2

ET II 92(21)

6.79 Integration of Bessel functions with respect to the order 6.791

∞

1. −∞ ∞

2. −∞ ∞

3. −∞

K ix+iy (a) K ix+iz (b) dx = π K iy−iz (a + b)

[|arg a| + |arg b| < π]

J ν−x (a) J μ+x (a) dx = J μ+ν (2a)

[Re(μ + ν) > 1]

ET II 382(21) ET II 379(1)

J κ+x (a) J λ−x (a) J μ+x (a) J ν−x (a) dx =

Γ(κ + λ + μ + ν + 1) Γ (κ + ⎛ λ + 1) Γ(λ + μ + 1) Γ(μ + ν + 1) Γ (ν + κ + 1) × 4F 5 ⎝

κ+λ+μ+ν +1 κ+λ+μ+ν +1 κ+λ+μ+ν κ+λ+μ+ν , , + 1, + 1; 2 2 2 2 ⎞

κ + λ + μ + ν + 1, κ + λ + 1, λ + μ + 1, μ + ν + 1, ν + κ + 1; −4a2⎠ [Re(κ + λ + μ + ν) > −1] 6.792

∞

1. −∞

ET II 379(3)

eπx K ix+iy (a) K ix+iz (b) dx = πe−πz K i(y−z) (a − b) [a > b > 0]

ET II 382(22)

6.794

2.

12

Integration of Bessel functions

∞

e −∞

iρx

K ν+ix (α) K ν−ix (β) dx = π

α + βeρ αeρ + β

ν K 2ν

757

α2 + β 2 + 2αβ cosh ρ

[|arg α| + |arg β| + |Im ρ| < π] ET II 382(23)

∞

3. −∞

e(π−γ)x K ix+iy (a) K ix+iz (b) dx = πe−βy−αz K iy−iz (c)

[0 < γ < π,

a > 0,

b > 0,

c > 0,

α, β, γ—the angles of the triangle with sides a, b, c] ET II 382(24), EH II 55(44)a

4.11

2ν ∞ h (2) (2) (2) e−cxi H ν−ix (a) H ν+ix (b) dx = 2i H 2ν (hk) k −∞

h=

∞

5. −∞

1

1

ae 2 c + be− 2 c ,

1

1

ae− 2 c + be 2 c

k=

[a, b > 0,

c is real]

ET II 380(11)

a−μ−x b−ν+x ecxi J μ+x (a) J ν−x (b) dx * 12 μ+ 12 ν ) # c

1/2 $ c 2 cos 2c 2 − 12 ci 2 12 ci a e exp (ν − μ)i J μ+ν 2 cos +b e = 1 1 2 2 a2 e− 2 ci + b2 e 2 ci [a > 0, b > 0, |c| < π, Re(μ + ν) > 1] =0 [a > 0,

|c| ≥ π,

b > 0,

Re(μ + ν) > 1]

EH II 54(41), ET II 379(2)

6.793 1.

2ν h e−cxi [J ν−ix (a) Y ν+ix (b) + Y ν−ix (a) J ν+ix (b)] dx = −2 J 2ν (hk) k −∞ 1 1 1 1 k = ae− 2 c + be 2 c [a, b > 0, Im c = 0] h = ae 2 c + be− 2 c ,

∞

∞

2.

e

−cxi

−∞

2ν h [J ν−ix (a) J ν+ix (b) − Y ν−ix (a) Y ν+ix (b)] dx = 2 Y 2ν (hk) k h=

3.10

∞

−∞

6.794

∞

1. 0

2.

1

1

ae 2 c + be− 2 c ,

1

1

ae− 2 c + be 2 c

k=

[a, b > 0,

Im c = 0]

ET II 380(10)

eiγx sech(πx) [J −ix (α) J ix (β) − J ix (α) J −ix (β)] dx = 2i H(σ) sign(β − α) J 0 σ 1/2

K ix (a) K ix (b) cosh[(π − ϕ)x] dx =

∞

cosh 0

ET II 380(9)

π

π x K ix (a) dx = 2 2

α, β, γ ∈ R,

α, β > 0,

σ = α2 + β 2 − 2αβ cosh γ

π K0 a2 + b2 − 2ab cos ϕ 2 [a > 0]

EH II 55(42) ET II 382(19)

758

Bessel Functions

∞

3. 0

5. 6.

π K 2ν 2a cos 2 2

[2|arg a| + |Re | < π] π

sech [a > 0] x J ix (a) dx = 2 sin a 2 −∞ ∞ π

x J ix (a) dx = −2i cos a cosech [a > 0] 2 −∞ ∞ sech(πx) [J ix (a)]2 + [Y ix (a)]2 dx = − Y 0 (2a) − E0 (2a)

4.

cosh(x) K ix+ν (a) K −ix+ν (a) dx =

6.794

ET II 383(28)

∞

ET II 380(6) ET II 380(7)

0

∞

7. 0

∞

8. 0

9.

10.

π

πa x K ix (a) dx = x sinh 2 2 x tanh(πx) K ix (b) K ix (a) dx =

[a > 0]

ET II 380(12)

[a > 0]

ET II 382(20)

π √ exp(−b − a) ab 2 a+b

ET II 175(4) [|arg b| < π, |arg a| < π]

2 π a a √ exp −b − x sinh(πx) K 2ix (a) K ix (b) dx = 5/2 8b 2 b 0 π b > 0, |arg a| < ET II 175(5) 4 ∞ x sinh(πx) π2 I n (b) K n (a) K ix (a) K ix (b) dx = [0 < b < a; n = 0, 1, 2, . . .] 2 2 x +n 2 0 2 π I n (a) K n (b) [0 < a < b; n = 0, 1, 2, . . .] = 2

∞

∞

11. 0

3/2

c a b ab π2 exp − + + 2 x sinh(πx) K ix (a) K ix (b) K ix (c) dx = 4 2 b a c

|arg a| + |arg b| <

π , 2

c>0

* √ ∞ π

π2 c (a + b) c2 + 4ab √ x K 12 ix (a) K 12 ix (b) K ix (c) dx = √ x sinh exp − 2 2 c2 + 4ab 2 ab 0 [|arg a| + |arg b| < π, c > 0] )

12.

ET II 176(8)

13. 0

ET II 176(9)

ET II 176(10) ∞

x sinh(πx) K 12 ix+λ (α) K 12 ix−λ (α) K ix (γ) dx = 0

2

π γ 22λ+1 α2λ z z = γ 2 − 4α2

=

[0 < γ < 2α] 2λ (γ + z) + (γ − z)2λ [0 < 2α < γ]

ET II 176(11)

6.797

6.795

Integration of Bessel functions

∞

cos(bx) K ix (a) dx =

1. 0

∞

3. 0

J x (ax) J −x (ax) cos(πx) dx = x sin(ax) K ix (b) dx =

−∞

4. −∞

x sin

0

1. 2. 3. 4.

∞

∞

5.

πb sinh a exp (−b cosh a) 2

∞

1. 0

0

ET II 380(4)

|Im a| <

π 2,

b>0

ET II 175(1)

[0 < a < b;

n = 0, 1, . . .]

[0 0

ET II 175(6)

1

2

2

π J0 4

(2)

b > 0]

ET II 380(8) EH II 55(47) EH II 55(48)

[a > 0,

b 2a sinh 2

b > 0]

ET II 383(27)

[a > 0,

b > 0]

ET II 382(26)

(2)

xeπx sinh(πx) Γ(ν + ix) Γ(ν − ix) H ix (a) H ix (b) dx √ = i2ν π Γ 12 + ν (ab)ν (a + b)−ν K ν (a + b) [a > 0,

2.

a>0

[|a| < 1]

b2 1 π 3/2 b πx K 12 ix (a) K ix (b) dx = √ exp −a − 2 8a 2a |arg a| <

sin(bx) sinh(πx) [K ix (a)] dx =

0

6.797

−1/2 1 1 − a2 4

e 2 πx cos(bx) J ix (a) dx = −i exp (ia cosh b) [a > 0, −∞ sinh(πx)

∞ 1 π πx K ix (a) dx = cos (a sinh b) cos(bx) cosh 2 2 0

∞ 1 π πx K ix (a) dx = sin (a sinh b) sin(bx) sinh 2 2 0

∞ b π2 2 cos(bx) cosh(πx) [K ix (a)] dx = − Y 0 2a sinh 4 2 0

π 2,

sin[(ν + ix)π] K ν+ix (a) K ν−ix (b) dx = π 2 I n (a) K n+2ν (b) n + ν + ix = π 2 K n+2ν (a) I n (b)

∞

5.

6.796

|Im b| <

π −a cosh b e 2

EH II 55(46), ET II 175(2) ∞

2. 0

759

∞

b > 0,

Re ν > 0]

ET II 381(14)

iπ 3/2 2ν (2) (2) (b−a)−ν H (2) xeπx sinh(πx) cosh(πx) Γ(ν +ix) Γ(ν −ix) H ix (a) H ix (b) dx = 1 ν (b−a) Γ − ν 2 0 < a < b, 0 < Re ν < 12 ET II 381(15)

760

Functions Generated by Bessel Functions

∞

3.

xe

πx

sinh(πx) Γ

0

ν + ix 2

6.811

ν − ix (2) (2) Γ H ix (a) H ix (b) dx 2

− 1 ν a2 + b 2 = iπ22−ν (ab)ν a2 + b2 2 H (2) ν [a > 0,

4.

11

∞

0

b > 0,

Re ν > 0]

ET II 381(16)

x sinh(πx) Γ(λ + ix) Γ(λ − ix) K ix (a) K ix (b) dx = 2λ−1 π 3/2 (ab)λ (a + b)−λ Γ λ + 12 K λ (a + b) [|arg a| < π,

Re λ > 0,

b > 0] ET II 176(12)

∞

5. 0

∞

6. 0

∞

7. 0

x sinh(2πx) Γ(λ + ix) Γ(λ − ix) K ix (a) K ix (b) dx =

λ

5 2

ab 2λ π 1 K λ (|b − a|) |b − a| Γ 2 − λ a > 0, 0 < Re λ < 12 , b > 0

ET II 176(13)

ab √ K 2λ x sinh(πx) Γ λ + 12 ix Γ λ − 12 ix K ix (a) K ix (b) dx = 2π 2 a2 + b 2 2 a2 + b 2 |arg a| < π2 , Re λ > 0, b > 0 x tanh(πx) K ix (a) K ix (b) 1 3 1 3 1 dx = 2 Γ 4 + 2 ix Γ 4 − 2 ix

ET II 177(14)

πab 2 + b2 exp − a a2 + b 2 |arg a| < π2 ,

b>0 ,

ET II 177(15)

6.8 Functions Generated by Bessel Functions 6.81 Struve functions 6.811

∞

1. 0

∞

2. 0

∞

3. 0

Hν (bx) dx = −

Hν

a2 x

Hν−1

cot

νπ 2

b

Hν (bx) dx = − 2

a x

√

J 2ν 2a b b

[−2 < Re ν < 0,

b > 0]

a > 0,

Re ν > − 32

b > 0,

ET II 158(1)

ET II 170(37)

√

1 dx Hν (bx) = − √ J 2ν−1 2a b x a b

a > 0,

b > 0,

Re ν > − 12

ET II 170(38)

6.812

1. 0

∞

H1 (bx) dx π [I 1 (ab) − L1 (ab)] = x2 + a2 2a

[Re a > 0,

b > 0]

ET II 158(6)

6.821

Combinations of Struve functions, exponentials, and powers

∞

2. 0

761

b cot νπ Hν (bx) 3 − ν 3 + ν a2 b 2 π 2

L ; ; 1; dx = − (ab) + F ν 2 1 νπ x2 + a2 1 − ν2 2 2 2 2a sin 2 [Re a > 0, b > 0, |Re ν| < 2] ET II 159(7)

6.813

∞

s−1

x

1. 0

∞

2. 0

∞

3. 0

x−ν−1 Hν (x) dx =

2−ν−1 π Γ(ν + 1)

x−μ−ν Hμ (x) Hν (x) dx =

1

0 1

5. 0

WA 429(2), ET I 335(52)

Re ν > − 32

ET II 383(2)

√ 2−μ−ν π Γ(μ + ν) Γ μ + 12 Γ ν + 12 Γ μ + ν + 12 [Re(μ + ν) > 0]

4.

2s−1 Γ s+ν s+ν 2 π tan Hν (ax) dx = s 1 1 2 a Γ 2ν − 2s + 1

3 , 1 − Re ν a > 0, −1 − Re ν < Re s < min 2

xν+1 Hν (ax) dx =

1 Hν+1 (a) a

x1−ν Hν (ax) dx =

1 aν−1 √ 1 − a Hν−1 (a) ν−1 2 πΓ ν + 2

a > 0,

WA 435(2), ET II 384(8)

Re ν > − 32

[a > 0] 6.814 1.

∞

(x2 +

0

6.815

1. 0

2.

xν+1 Hν (bx)

1

ET II 158(3)a

2μ−1 πaμ+ν b−μ [I −μ−ν (ab) − Lμ+ν (ab)] Γ(1 − μ) cos[(μ + ν)π] Re a > 0, b > 0, Re ν > − 32 , Re(μ + ν) < 12 , Re(2μ + ν) < 32

a2 )1−μ

ET II 158(2)a

dx =

ET II 159(8)

√ 1 x 2 ν (1 − x)μ−1 Hν a x dx = 2μ a−μ Γ(μ) Hμ+ν (a)

Re ν > − 32 , Re μ > 0 ET II 199(88)a

1 √ 1 3 3 a2 3 B(λ, μ)aν+1 , ν + , λ + μ; − 1, λ; xλ− 2 ν− 2 (1 − x)μ−1 Hν a x dx = ν √ 2F 3 2 2 4 2 π Γ ν + 32 0 [Re λ > 0,

Re μ > 0]

ET II 199(89)a

6.82 Combinations of Struve functions, exponentials, and powers 6.821 1.6 0

∞

−n− 12 1 1 e−αx H−n− 12 (βx) dx = (−1)n β n+ 2 α + α2 + β 2 2 α + β2 [Re α > |Im β|]

ET I 206(6)

762

2.

6

Functions Generated by Bessel Functions

∞

0

−n− 12 1 1 e−αx L−n− 12 (βx) dx = β n+ 2 α + α2 − β 2 2 α − β2

√

∞

6.822

e−αx H0 (βx) dx =

2 π

α2 +β 2 +β α

ln

[Re α > |Re β|]

ET I 208(26)

[Re α > |Im β|] ET II 205(1) α2 + β 2

β ∞ arcsin α 2 −αx 4. e L0 (βx) dx = [Re α > |Re β|] ET II 207(18) π α2 + β 2 0 ∞ a

a

a

a π (ν+1)x cosec(νπ) sinh I ν+ 12 − cosh I −ν− 12 6.822 e Hν (a sinh x) dx = a 2 2 2 2 0 [Re a > 0, −2 < Re ν < 0] 3.

0

ET II 385(11)

6.823

∞

λ −αx

x e

1. 0

∞

2. 0

λ+ν +3 3 3 b2 λ+ν bν+1 Γ(λ + ν + 2) 3 F 2 1,

+ 1, ; ,ν + ;− 2 Hν (bx) dx = √ 3 2 2 2 2 a 2ν aλ+ν+2 π Γ ν + 2 [Re a > 0, b > 0, Re(λ + ν) > −2] ET II 161(19)

ν

xν e−αx Lν (βx) dx =

Γ(2ν + 1) αβ (2β)ν Γ ν + 12 β −ν− 12 − P

2ν+1 −ν− 12 √ 2 1 1 α π ν+ π α − β2 α β 2 − α2 2 4 2 Re α > |Re β|, Re ν > − 12

ET I 209(35)a

6.824 1.

∞

0

6.825

1

a

1

ea Φ 2ν+1

1 √ a

MI 51

1 1 1 1 a − 2ν, t e e γ MI 51 2 a Γ 2 − 2ν a2ν+1 0

∞ β ν+1 Γ 12 + 2s + ν2 ν+s+1 3 3 β2 s−1 −α2 x2 1, x e Hν (βx) dx = ν+1 √ ν+s+1 ; , ν + ; − F 2 2 2 2 2 4α2 2 πα Γ ν + 32 0 Re s > − Re ν − 1, |arg α| < π4

2.

√

tν e−at L2ν 2 t dt =

∞

ν −at

√

L−2ν t dt =

1

ET I 335(51)a, ET II 162(20)

6.83 Combinations of Struve and trigonometric functions

6.831

0

∞

x−ν sin(ax) Hν (bx) dx = 0 =

√

π2

−ν −ν

b

ν− 12 b 2 − a2 Γ ν + 12

0 − 12

0 < a < b,

Re ν > − 12

ET II 162(21)

6.847

Combinations of Struve and Bessel functions

6.832

∞√

2 2

x sin(ax) H 14 b x

0

dx = −2

−3/2 √

π

√

a

b2

Y

1 4

763

a2 4b2 [a > 0]

ET I 109(14)

6.84–6.85 Combinations of Struve and Bessel functions

6.841

∞

0

0 < b < a, 0 < a < b,

Hν−1 (ax) Y ν (bx) dx = −aν−1 b−ν =0

6.842 6.843

∞

0

∞

1. 0

∞

2. 0

4 K [H0 (ax) − Y 0 (ax)] J 0 (bx) dx = π(a + b)

√ 1 J 2ν a x Hν (bx) dx = − Y ν b

a2 4b

1 2

|Re ν| <

1 2

b > 0]

a > 0,

b > 0,

2

a b

ET II 114(36)

|a − b| a+b [a > 0,

√ 2 Γ(ν + 1) S −ν−1,ν K 2ν 2a x Hν (bx) dx = πb ν

|Re ν| <

ET II 15(22)

−1 < Re ν <

5 4

ET II 164(10)

[Re a > 0,

b > 0,

Re ν > −1] ET II 168(27)

∞ √ √ √ μ−ν μ−ν π J μ a x − sin π Y μ a x K μ a x Hν (bx) dx cos 2 2 0

2

2 a a 1 W − 12 ν, 12 μ = 2 W 12 ν, 12 μ a 2b 2b |arg a| < π4 , b > 0, Re ν > |Re μ| − 2 ET II 169(35)

6.844

6.845 1.

∞

0

0

|Re ν| <

1 2

2

2 √

√ a a 1 2 + sin(νπ) Hν Hν (bx) dx = K 2ν 2a b − Y 2ν 2a b J −ν x x b π a > 0, b > 0, − 32 < Re ν < 0 ∞

0

6.847

b > 0,

ET II 73(7)

∞

2.

6.846

a

a √

4 − Y −ν J ν (bx) dx = cos(νπ) K 2ν 2 ab H−ν x x πb |arg a| < π,

0

∞

ET II 170(39)

√ √ 2 a 1 K 2ν 2a x + Y 2ν 2a x Hν (bx) dx = J ν π b b a > 0, 2

b > 0,

|Re ν| <

dx νπ νπ π J ν (ax) + sin Hν (ax) 2 [I ν (ak) − Lν (ak)] cos = 2 2 x + k2 2k a > 0, Re k > 0,

1 2

ET II 169(30)

− 12 < Re ν < 2

ET II 384(5)a, WA 467(8)

764

6.848

Functions Generated by Bessel Functions

∞

1. 0

0

∞

1. 0

∞

2. 0

6.851

1.

2 a ν−1 1 cos(νπ) 2 π b a + b2 Re a > 0,

x [H−ν (ax) − Y −ν (ax)] J ν (bx) dx = 2

x K ν (ax) Hν (bx) dx = a−ν−1 bν+1

#

2 J 12 ν (ax) − Y

Re a > 0,

1 + b2

x [K μ (ax)]2 H0 (bx) dx = −2−μ−1 πa−2μ z = 4a2 + b2

∞

0

a2

cos(νπ) ν−1 1 b aν π a + b |arg a| < π,

x

2μ

2μ

−1 < Re ν < − 21

− 21 < Re ν,

b > 0,

2 $ 1 (ax) Hν (bx) dx = 0 2ν

ET II 73(5)

ET II 164(12)

0 0

Re ν > − 23

+ (z − b) sec(μπ), bz Re a > 0, b > 0, |Re μ| < 32

(z + b)

=

b > 0,

ET II 74(12)

∞

2.

6.849

x [I ν (ax) − L−ν (ax)] J ν (bx) dx =

6.848

ET II 166(18)

− 32 < Re ν < 0 − 32 < Re ν < 0

ET II 164(7) ∞

2. 0

xν+1 [J ν (ax)]2 − [Y ν (ax)]2 Hν (bx) dx

=0 −ν− 12 23ν+2 a2ν b−ν−1 2 1 b − 4a2 = √ πΓ 2 −ν

0 12 ,

Re(μ + ν) > 1

ET II 383(4)

xμ−ν+1 Y μ (ax) Hν (bx) dx =0 =

ν−μ−1 21+μ−ν aμ b−ν 2 b − a2 Γ(ν − μ)

0 0,

− 32 < Re μ <

1 2

0 < a < b,

Re(ν − μ) > 0,

− 32 < Re μ <

1 2

ET II 163(3)

6.854

Combinations of Struve and Bessel functions

∞

3.

3 3 b2 2μ+ν+1 bν+1 3 F 1, μ + ν + ; ; − 2 K μ (ax) Hν (bx) dx = √ μ+2ν+3 Γ μ + ν + πa 2 2 2 a Re a > 0, b > 0, Re ν > − 23 , Re(μ + ν) > − 32 ET II 165(13)

μ+ν+1

x 0

6.853 1.

∞

0

x1−μ [sin (μπ) J μ+ν (ax) + cos(μπ) Y μ+ν (ax)] Hν (bx) dx

=0

μ−1 b ν b 2 − a2 = μ−1 μ+ν 2 a Γ(μ)

0 − 23 ,

Re(ν − μ) <

1 2

0 < a < b,

1 < Re μ < 32 ,

Re ν > − 23 ,

Re(ν − μ) <

1 2

ET II 163(4) ∞

2. 0

1

xλ+ 2 [I μ (ax) − L−μ (ax)] J ν (bx) dx

765

∞

3. 0

Re a > 0,

! ⎞ !1+μ μ μ ! , 1 − , 1 + 2 2 1 cos(μπ) 3 ⎜ b2 ! 2 ⎟ b−λ− 2 G 22 = 2λ+ 2 ⎠ 33 ⎝ 2 ! 3 π a ! + λ+ν , 1+μ , 3 + λ−ν ! 2 2 2 4 4 Re(μ + ν + λ) > − 32 , − Re ν − 52 < Re(λ − μ) < 1 ET II 76(21)

b > 0,

⎛

1

xλ+ 2 [Hμ (ax) − Y μ (ax)] J ν (bx) dx

! ⎞ !1−μ μ μ ! , 1 − , 1 + 2 2 1 cos(μπ) 3 ⎜b ! 2 ⎟ b−λ− 2 G 23 = 2λ+ 2 ⎠ 33 ⎝ 2 ! 3 π2 a ! + λ+ν , 1−μ , 3 + λ−ν ! 2 2 4 2 4 |arg a| < π, Re(λ + μ) < 1, Re(λ + ν) + 32 > |Re μ| ET II 73(6) ⎛

2

4.

2 ν− 1 −ν 1 a 2b √ x I ν− 12 (ax) − Lν− 12 (ax) J ν (bx) dx = 2 π a + b2 Re a > 0, b > 0,

∞√

0

0

∞

6.

1. 0

a b 1 b2 1 2 μ−ν+1 F 1, ; ν − μ + ; − 2 x [I μ (ax) − Lμ (ax)] J ν (bx) dx = √ 2 2 a π Γ ν − μ + 12 1 −1 < 2 Re μ + 1 < Re ν + 2 , Re a > 0, b > 0 ET II 74(13)

1 b2 1 2μ−ν+1 a−μ−1 bν−1 F 1, + μ; + ν; − 2 [I μ (ax) − L−μ (ax)] J ν (bx) dx = 1 2 2 a Γ 2 − μ Γ 12 + ν 1 Re a > 0, Re ν > − 2 , Re μ > −1, b > 0 ET II 75(18)

μ−ν+1

x

1 2

μ−ν+1 μ−1 ν−2μ−1

0

6.854

|Re ν| <

ET II 74(11)

∞

5.

b > 0,

∞

xH

1 2ν

2

ax

K ν (bx) dx =

Γ

1

2ν + 1 1 1− 2 2 ν aπ

S

− 12 ν−1, 12 ν

b2 4a [a > 0,

Re b > 0,

Re ν > −2] ET II 150(75)

766

Functions Generated by Bessel Functions

∞

2. 0

2

x H 12 ν ax

1 J ν (bx) dx = − Y 2a

1 2ν

b2 4a

a > 0,

6.855

b > 0,

−2 < Re ν <

3 2

ET II 73(3)

6.855

∞

2ν+ 12

x

1. 0

ET II 76(22)

a

a √

dx 4 − Y −ν−1 J ν (bx) H−ν−1 = − √ cos(νπ) K −2ν−1 2 ab x x x π ab |arg a| < π, b > 0, |Re ν| < 12

∞

2. 0

ET II 74(8)

∞

3.

1

x2ν+ 2

0

6.857

∞

0

a

a − Y ν+ 12 J ν (bx) dx Hν+ 12 x x

√

√

1 1 1 2abe 4 πi K 2ν+1 2abe− 4 πi = −25/2 π −3/2 aν+ 2 b−ν−1 sin(νπ) K 2ν+1 |arg a| < π, b > 0, −1 < Re ν < − 61 ET II 74(9)

6.856

1 a

a √

√

3 aν+ 2 2 √ − Lν+ 12 J ν (bx) dx = 2 I ν+ 12 J 2ab K 2ab 2ν+1 2ν+1 x x πbν+1 Re a > 0, b > 0, −1 < Re ν < 12

2 √ √ 1 a x Y ν a x K ν a x Hν (bx) dx = 2 exp − 2b 2b b > 0,

∞

1.

x exp 0

a2 x2 8

K 12 ν

k = 14 ν, 2.

∞

σ−2

x 0

a2 x2 8

m=

1 2

|arg a| <

π 4,

Re ν > − 32

ET II 169(32)

Hν (bx) dx

2

2 νπ 1 b b 2 − ν −1 ν −1 2 2 Γ − ν exp W k,m b cos =√ a 2 π 2 2 2a a2 1 3 3 + 4ν |arg a| < 4 π, b > 0, − 2 < Re ν < 0 ET II 167(24)

1 2 2 1 2 2 a x Hν (bx) dx exp − a x K μ 2 2 √ + μ Γ ν+σ −μ π −ν−σ ν+1 Γ ν+σ 2 2 b = ν+2 a 2 Γ 32 Γ ν + 32 Γ ν+σ 2

ν+σ 3 3 ν+σ b2 ν +σ + μ, − μ; , ν + , ;− 2 × 3 F 3 1, 2 2 2 2 2 4a b > 0, |arg a| < π4 , Re(σ + ν) > 2|Re μ| ET II 167(23)

6.866

Lommel functions

767

6.86 Lommel functions 6.861

∞

λ−1

x

1. 0

6.862 1.

12

u

0

Γ

S μ,ν (x) dx =

1

2 (1

+ λ + μ) Γ 12 (1 − λ − μ) Γ 12 (1 + μ + ν) Γ 12 (1 + μ − ν) 22−λ−μ Γ 12 (ν − λ) + 1 Γ 1 − 12 (λ + ν) − Re μ < Re λ + 1 < 52 ET II 385(17)

√ 1 1 xλ− 2 μ− 2 (u − x)σ−1 s μ,ν a x dx

aμ uλ+σ Γ(λ + 1) (μ − ν + 1) (μ + ν + 1) Γ(λ + σ + 1) a2 u μ−ν+3 μ+ν +3 , , λ + σ + 1; − × 2 F 3 1, 1 + λ; 2 2 4 [Re λ > −1, Re σ > 0] ET II 199(92) 1 1 ∞ √ √ B μ, 12 (1 − λ − ν) − μ u 2 μ+ 2 ν 1 ν μ−1 2 x (x − u) Sλ,ν a x dx = S λ+μ,μ+ν a u μ a u ! √ ! !arg a u ! < π, 0 < 2 Re μ < 1 − Re(λ + ν) ET II 211(71) = Γ(σ)

2.

6.863

0

6.864

∞√

0

6.865

xe−αx s μ, 14

x2 2

∞

2

√ α 3 dx = 2−2μ−1 α Γ 2μ + S −μ−1, 14 2 2 Re α > 0, Re μ > − 34

ET I 209(38)

exp[(μ + 1)x] s μ,ν (a sinh x) dx = 2μ−2 π cosec(μπ) Γ() Γ(σ) a

a a a

Iσ − I − I −σ × I 2 2 2 2 2 = μ + ν + 1, 2σ = μ − ν + 1 [a > 0, −2 < Re μ < 0] ET II 386(22) ∞√ μ−ν 1 B 14 − μ+ν 2 , 4 − 2 sinh x cosh(νx) S μ, 12 (a cosh x) dx = S μ+ 12 ,ν (a) √ μ+ 3 a2 2 0 |arg a| < π, Re μ + |Re ν| < 12 ET II 388(31)

6.866 1.

12

∞

0

x−μ−1 cos(ax) s μ,ν (x) dx =0 =2

2. 0

μ− 12

√

πΓ

μ+ν +1 2

1 μ+ 1 −μ− 1 μ−ν+1 Γ 1 − a2 2 4 P ν− 1 2 (a) 2 2

[a > 1] [0 < a < 1] ET II 386(18)

∞

1 μ− 1 μ− 1 √ μ−ν 2 μ+ν −μ −μ− 12 Γ 1− a − 1 2 4 P ν− 12 (a) x sin(ax) S μ,ν (x) dx = 2 πΓ 1− 2 2 2 [a > 1, Re μ < 1 − |Re ν|]

ET II 387(23)

768

6.867

Functions Generated by Bessel Functions

π/2

6.867

cos(2μx) S 2μ−1,2ν (a cos x) dx

1. 0

a

a

a

a π22μ−3 a2μ cosec(2νπ) J μ+ν Y μ−ν − J μ−ν Y μ+ν Γ(1 − μ − ν) Γ (1 − μ + ν) 2 2 2 2 [Re μ > −2, |Re ν| < 1] ET II 388(29) π/2 a

a

Jσ cos [(μ + 1) x] s μ,ν (a cos x) dx = 2μ−2 π Γ() Γ(σ) J 2 2 0 2 = μ + ν + 1, 2σ = μ − ν + 1 [Re μ > −2] ET II 386(21) =

2.

6.868 6.869

π/2

0

∞

1.

1−μ−ν

x 0

π

π

cos(2μx) π22μ−1 S 2μ,2ν (a sec x) dx = W μ,ν aei 2 W μ,ν ae−i 2 cos x a [|arg a| < π, Re μ < 1]

√ ν−1 12 (μ+ν−1) μ+ν−1 πa Γ(1 − μ − ν) 2 a J ν (ax) S μ,−μ−2ν (x) dx = − 1 P μ+ν (a) 1 2μ+2ν Γ ν + 2 a > 1, Re ν > − 12 , Re(μ + ν) < 1

ET II 388(28)

∞

2. 0

x−μ J ν (ax) s ν+μ,−ν+μ+1 (x) dx μ = 2ν−1 Γ(ν)a−ν 1 − a2 =0

ET II 388(30)

0 < a < 1, 1 < a,

Re μ > −1,

Re μ > −1,

−1 < Re ν < −1 < Re ν < 32

3 2

ET II 388(28)

∞

3. 0

2 2 1 1 b 1 Γ μ + ν + 1 Γ μ − ν + 1 S −μ−1, 12 ν x K ν (bx) s μ, 12 ν ax dx = 4a 2 2 4a Re μ > 12 |Re ν| − 2, a > 0, Re b > 0 ET II 151(78)

6.87 Thomson functions 6.871

1/2 β4 + 1 + β2 e−βx ber x dx = 2 (β 4 + 1) 0

1/2 ∞ β4 + 1 − β2 e−βx bei x dx = 2 (β 4 + 1) 0

1.

2.

∞

ME 40

ME 40

6.874

6.872

Thomson functions

∞

1. 0

⎡

π⎣ 1 3νπ 1 cos + J 12 (ν−1) β 2β 2β 4 ⎤

3ν + 6 ⎦ 1 1 cos + π − J 12 (ν+1) 2β 2β 4

√ 1 e−βx berν 2 x dx = 2β

∞

2.

e

−βx

0

3. 4. 5. 6. 7.

2.

3. 4.

⎡

MI 49

1 3ν sin + π 2β 4 ⎤

3ν + 6 ⎦ 1 1 sin + π − J 12 (ν+1) 2β 2β 4

√ 1 beiν 2 x dx = 2β

π⎣ J 12 (ν−1) β

1 2β

MI 49

√ 1 1 e−βx ber 2 x dx = cos β β 0 ∞ √ 1 1 e−βx bei 2 x dx = sin β β 0 ∞ √ 1 1 1 1 1 cos ci + sin si e−βx ker 2 x dx = − 2β β β β β 0 ∞ √ 1 1 1 1 1 −βx sin ci − cos si e kei 2 x dx = − 2β β β β β 0

∞ √ √ 3νπ 2 2 1 Jν sin + e−βx berν 2 x beiν 2 x dx = 2β β β 2 0

∞ √ √ 1 2 ber2ν 2 x + bei2ν 2 x e−βx dx = I ν β β 0

6.873

1.

∞

6.874

769

ME 40 MI 50 MI 50

[Re ν > −1]

MI 49

[Re ν > −1]

ME 40

√

e π 1 3π 3νπ 1 √ ber2ν 2 2x dx = Jν cos − + x β β β 4 2 0 Re ν > − 12

∞ −βx √

e π 1 3π 3νπ 1 √ bei2ν 2 2x dx = Jν sin − + β β β 4 2 x 0 Re ν > − 12

∞ √ −βx ν 3νπ 2−ν 1 + [Re ν > −1] x 2 berν x e dx = 1+ν cos β 4β 4 0

∞ √ −βx ν 3νπ 2−ν 1 2 + [Re ν > −1] x beiν x e dx = 1+ν sin β 4β 4 0

ME 40

∞ −βx

MI 49

MI 49 ME 40 ME 40

770

6.875 1. 2. 6.876 1. 2.

Mathieu Functions

6.875

√ √ 1 1 π 1 1 ln β cos + sin ker 2 x − ln x ber 2 x dx = e 2 β β 4 β 0 ∞ √ √ 1 1 π 1 1 ln β sin − cos e−βx kei 2 x − ln x bei 2 x dx = 2 β β 4 β 0

∞

∞

−βx

1 arctan a2 2a 0 ∞ 1 ln (1 + a4 ) x ker x J 1 (ax) dx = 2a 0 x kei x J 1 (ax) dx = −

MI 50 MI 50

[a > 0]

ET II 21(32)

[a > 0]

ET II 21(33)

6.9 Mathieu Functions (m)

Notation: k 2 = q. For deﬁnition of the coeﬃcients Ap

(m)

and Bp

see section 8.6.

6.91 Mathieu functions 6.911

2π 0

2π

2. 0

2π

2

∞ 2 " (2n+1) A2r+1 =π

2π

MA

[m = p]

sem (z, q) sep (z, q) dz = 0

0 2π

5.

2

[se2n+1 (z, q)] dz = π

0

MA

∞ 2 " (2n+1) B2r+1 =π

MA

∞ 2 " (2n+2) B2r+2 =π

MA

r=0

2π

6.

2

[se2n+2 (z, q)] dz = π

0

MA

r=0

4.

∞ 2 2 " (2n) (2n) 2 A2r [ce2n (z, q)] dz = 2π A0 +π =π

[ce2n+1 (z, q)] dz = π

0

MA

r=1

3.

[m = p]

cem (z, q) cep (z, q) dz = 0

1.

r=0

2π

7.

sem (z, q) cep (z, q) dz = 0

0

[m = 1, 2, . . . ;

p = 1, 2, . . .]

MA

6.92 Combinations of Mathieu, hyperbolic, and trigonometric functions 6.921

1. 0

(2n)

π

cosh (2k cos u sinh z) ce2n (u, q) du =

πA0 (−1)n Ce2n (z, −q) ce2n π2 , q [q > 0]

MA

6.922

Mathieu, hyperbolic, and trigonometric functions

(2n)

π

2.

cosh (2k sin u cosh z) ce2n (u, q) du =

0

771

πA0 (−1)n Ce2n (z, −q) ce2n (0, q) [q > 0]

(2n+1)

π

3.

sinh (2k sin u cosh z) se2n+1 (u, q) du =

0

MA

πkB1 (−1)n Ce2n+1 (z, −q) se2n+1 (0, q) [q > 0]

(2n+1)

π

4.

sinh (2k cos u sinh z) ce2n+1 (u, q) du =

0

MA

πkA1 (−1)n+1 Se2n+1 (z, −q) ce2n+1 π2 , q [q > 0]

(2n+1)

π

5.

sinh (2k sin u sin z) se2n+1 (u, q) du =

0

MA

πkB1 se2n+1 (z, q) se2n+1 (0, q) [q > 0]

6.922

[q > 0] π

2.

sin u sinh z cos (2k cos u cosh z) se2n+1 (u, q) du =

0

πA1 Ce2n+1 (z, q) 2 ce2n+1 (0, q)

cos u cosh z cos (2k sin u sinh z) ce2n+1 (u, q) du =

0

(2n+1)

π

1.

MA

π

3.

sin u sinh z sin (2k cos u cosh z) se2n+2 (u, q) du =

0

MA

πB1 (2n+1) π Se2n+1 (z, q) ,q 2 se2n+1 2 [q > 0] (2n+2) πkB2 − 2 se2n+2 π2 , q

Se2n+2 (z, q)

[q > 0]

cos u cosh z sin (2k sin u sinh z) se2n+2 (u, q) du =

0

MA

(2n+2)

π

4.

πkB2 Se2n+2 (z, q) 2 se2n+2 (0, q) [q > 0]

π

5.

sin u cosh z cosh (2k cos u sinh z) se2n+1 (u, q) du =

0

πB1

MA

(2n+1)

2 se2n+1

π 2

,q

(−1)n Ce2n+1 (z, −q)

[q > 0]

π

6.

cos u sinh z cosh (2k sin u cosh z) ce2n+1 (u, q) du =

0

MA

(2n+1) πA1

2 ce2n+1 (0, q)

(−1)n Se2n+1 (z, −q)

[q > 0] 7. 0

MA

π

sin u cosh z sinh (2k cos u sinh z) se2n+2 (u, q) du =

MA

(2n+2) πkB2 π (−1)n+1 ,q 2 se2n+2

Se2n+2 (z, −q)

2

[q > 0]

MA

772

Mathieu Functions

(2n+2)

π

8.

cos u sinh z sinh (2k sin u cosh z) se2n+2 (u, q) du =

0

6.923

πkB2 (−1)n Se2n+2 (z, −q) 2 se2n+2 (0, q) [q > 0]

6.923

∞

1. 0

2. 0

cos (2k cosh z cosh u) sinh z sinh u Se2n+1 (u, q) du = −

0

∞

0

cos (2k cosh z cosh u) sinh z sinh u Se2n+2 (u, q) du = −

0

sin (2k cosh z cosh u) Ce2n (u, q) du =

πA0 Ce2n (z, q) 2 ce2n 12 π, q

cos (2k cosh z cosh u) Ce2n (u, q) du =

(2n) πA0 1 − 2 ce2n 2 π, q

[q > 0]

∞

6. 0

sin (2k cosh z cosh u) Ce2n+1 (u, q) du =

kπA1 Fey2n+1 (z, q) 2 ce2n+1 12 π, q

cos (2k cosh z cosh u) Ce2n+1 (u, q) du =

(2n+1) kπA1 2 ce2n+1 12 π, q

[q > 0]

∞

8. 0

(2n)

π

1.

cos (2k cos u cos z) ce2n (u, q) du =

0

πA0 1

ce2n

2 π, q

0

π

sin (2k cos u cos z) ce2n+1 (u, q) du = −

MA

ce2n (z, q) [q > 0]

2.

MA

Ce2n+1 (z, q)

[q > 0] 6.924

MA

(2n+1)

∞

0

MA

Fey2n (z, q)

[q > 0] 7.

MA

(2n)

∞

5.

MA

kπB2 (2n+2) Se2n+2 (z, q) 4 se2n+2 12 π, q

[q > 0]

MA

kπB2 (2n+2) Gey2n+2 (z, q) 4 se2n+2 12 π, q

sin (2k cosh z cosh u) sinh z sinh u Se2n+2 (u, q) du = −

[q > 0]

4.

MA

πB1 (2n+1) Gey2n+1 (z, q) 4 se2n+1 12 π, q

[q > 0] ∞

3.

πB1 (2n+1) Se2n+1 (z, q) 4 se2n+1 12 π, q

sin (2k cosh z cosh u) sinh z sinh u Se2n+1 (u, q) du = −

[q > 0] ∞

MA

MA

(2n+1)

πkA1 ce2n+1 (z, q) ce2n+1 12 π, q [q > 0]

MA

6.926

Mathieu, hyperbolic, and trigonometric functions

(2n)

π

3.

cos (2k cos u cosh z) ce2n (u, q) du =

0

πA0 1

ce2n

2 π, q

773

Ce2n (z, q) [q > 0]

(2n)

π

4.

cos (2k sin u sinh z) ce2n (u, q) du =

0

MA

πA0 Ce2n (z, q) ce2n (0, q) [q > 0]

(2n+1) πkA1 − ce2n+1 12 π, q

π

5.

sin (2k cos u cosh z) ce2n+1 (u, q) du =

0

MA

Ce2n+1 (z, q)

[q > 0]

(2n+1)

π

6.

sin (2k sin u sinh z) se2n+1 (u, q) du =

0

MA

πkB1 Se2n+1 (z, q) se2n+1 (0, q) [q > 0]

6.925 1.

MA

Notation: z1 = 2k cosh2 ξ − sin2 η, and tan α = tanh ξ tan η 2π sin [z1 cos(θ − α)] ce2n (θ, q) dθ = 0.

MA

0

2π

2. 0

2π

3. 0

cos [z1 cos(θ − α)] ce2n (θ, q) dθ =

(2n)

2πA0 Ce2n (ξ, q) ce2n (η, q) ce2n (0, q) ce2n 12 π, q

MA

(2n+1)

sin [z1 cos(θ − α)] ce2n+1 (θ, q) dθ = −

2πkA1 Ce2n+1 (ξ, q) ce2n+1 (η, q) ce2n+1 (0, q) ce2n+1 12 π, q MA

2π

4. 0

2π

5. 0

cos [z1 cos(θ − α)] ce2n+1 (θ, q) dθ = 0 sin [z1 cos(θ − α)] se2n+1 (θ, q) dθ =

MA (2n+1)

2πkB1 Se2n+1 (ξ, q) se2n+1 (η, q) se2n+1 (0, q) se2n+1 12 π, q MA

2π

6. 0

2π

7. 0

2π

8. 0

cos [z1 cos(θ − α)] se2n+1 (θ, q) dθ = 0

MA

sin [z1 cos(θ − α)] se2n+2 (θ, q) dθ = 0

MA

cos [z1 cos(θ − α)] se2n+2 (θ, q) dθ =

2πk 2 B2 (2n+2) se2n+2 (0, q) se2n+2 12 π, q

Se2n+2 (ξ, q) se2n+2 (η, q) MA

6.926

0

π

sin u sin z sin (2k cos u cos z) se2n+2 (u, q) du = −

(2n+2)

πkB2 se2n+2 (z, q) 2 se2n+2 π2 , q [q > 0]

MA

774

Mathieu Functions

6.931

6.93 Combinations of Mathieu and Bessel functions 6.931

2 (2n) π A0 π ce2n (z, q) 0 ,q ce2n (0, q) ce2n 2 2 (2n) 2π 2π A0 1/2 π Fey2n (z, q) ce2n (u, q) du = Y 0 k [2 (cos 2u + cosh 2z)] 0 ce2n (0, q) ce2n ,q 2

1.

2.

π

J 0 k [2 (cos 2u + cos 2z)]1/2 ce2n (u, q) du =

MA

MA

6.94 Relationships between eigenfunctions of the Helmholtz equation in diﬀerent coordinate systems Notation: Particular solutions of the Helmholtz equation in three-dimensional inﬁnite space ∇2 Ψ + k 2 Ψ = 0 in Cartesian (x, y, z), spherical (r, θ, φ), and cylindrical (ρ, z, φ) coordinates are Ψkx ky kz (x, y, z) ∝ ei(kx x+ky y+kz z) with k 2 = kx2 + ky2 + kz2 k imφ Z l+1/2 (kr) P m Ψlm (r, θ, φ) ∝ e l (cos θ) r

Ψmkz (ρ, z, φ) ∝ ei(mφ+kz z) Z l+1/2 ρ k 2 − kz2 with P m l (cos θ) the associated Legendre function, Z is any Bessel function, m = 0, 1, . . . , l; l ∈ N, r2 = ρ2 + z 2 , ρ = r sin θ, z = r cos θ, φ = arccot(x/y), and kt2 = k 2 − kz2 . 6.941 k

z

2πk m p iρz l−m 2 2 dp = i J l+1/2 (kr) P m e J m ρ k − ρ Pl 1. l k r r −k

∞

2. −∞

3. 0

∞

e−iρz J l+1/2 (kr) P m l

z

r

dz = im−l

[ρ > 0, l ≥ m ≥ 0]

ρ

2πr J m ρ k 2 − ρ2 P m l k k

[ρ > 0, l ≥ m ≥ 0]

x dx J m (ρkt ) cos kx x + m arcsin ρ

(−1)m kx 2 2 = cos y kt − kx + m arccos 2 2 kt kt − kx

=0

2 kx < kt2 2 kx > kt2

6.941

Eigenfunctions of Helmholtz equation

x dx Y m (ρkt ) cos kx x + m arcsin 0 ρ

m (−1) kx 2 2 = sin y kt − kx + m arccos 2 2 kt kt − kx

(−1)m |kx | 2 2 = exp −y kx − kt − m sign (kx ) arccosh kt kx2 − kt2

∞ z

2πr (j) 2 kz (j) −ikz z m−l 2 Pm e H ρ H l+1/2 (kr) P m dz = i k − k z l m l r k k −∞

4.

5.

6.

7.

8. 9.

10.

∞

775

2 kx < kt2 2 kx > kt2

[ρ > 0] The result is true for j = 1 if π > arg k 2 − kz2 ≥ 0, for j = 2 if −π < arg k 2 − kz2 ≤ 0.

∞

z

2πk (j) kz (j) m ikz z l−m 2 2 e H l+1/2 (kr) P m H m ρ k − kz P l dkz = i l k r r −∞ The result is true for j = 1 if π > arg k 2 − kz2 ≥ 0, for j = 2 if −π < arg k 2 − kz2 ≤ 0.

∞

2 2πr kz m z −ikz z m−l e J m ρ k 2 − kz2 P m kz < k 2 J l+1/2 (kr) P l dz = i l r k k −∞ 2 =0 kz > k 2

k

z

2πk kz m ikz z l−m 2 2 e J l+1/2 (kr) P m J m ρ k − kz P l dkz = i l k r r −k

∞

2 2πr kz m z m −ikz z m−l 2 2 e Y m ρ k − kz P l kz < k 2 Y l+1/2 (kr) P l dz = i r k k −∞

2 2r kz K m ρ kz2 − k 2 P m kz > k 2 = −2im−l l kπ k

k

kz eikz z dkz il−m Y m ρ k 2 − kz2 P m l k −k

m kz 4 ∞ 1 K m ρ kz2 − k 2 eikz z dkz − cos kz z + 2 π(m − l) P l π k k z

2πk Y l+1/2 (kr) P m = l r r

Definite Integrals of Special Functions

Definite Integrals of Special Functions

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