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Integrals of Bessel functions
Applied Mathematics Letters 26 (2013) 351–354
Contents lists available at SciVerse ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Integrals of Bessel functions D. Babusci a , G. Dattoli b , B. Germano c , M.R. Martinelli c , P.E. Ricci d,∗ a
INFN - Laboratori Nazionali di Frascati, via E. Fermi, 40, IT 00044 - Frascati (Roma), Italy
b
ENEA - Centro Ricerche Frascati, via E. Fermi, 45, IT 00044 - Frascati (Roma), Italy
c
Sapienza - Università di Roma, Dipartimento Me-Mo-Mat, Via A. Scarpa, 14 - 00161 Roma, Italy
d
Università Campus Bio-Medico di Roma, Via A. del Portillo, 21 - 00128 Roma, Italy
article
abstract
info
Article history: Received 5 July 2012 Received in revised form 4 October 2012 Accepted 5 October 2012
We use the operator method to evaluate a class of integrals involving Bessel or Bessel-type functions. The technique that we propose is based on the formal reduction of functions in this family to Gaussians. © 2012 Elsevier Ltd. All rights reserved.
Keywords: Integrals Generating function method Bessel-type functions
1. Introduction Integrals of the type In (a, b, α) =
∞
dx (ax + b)n e−α x
2
(1)
−∞
can be calculated using a general procedure based on the generating function (GF) method [1]. Multiplying both sides of Eq. (1) by t n /n! and summing up over n, we obtain G(a, b, α) =
∞ n t n =0
=
n!
In (a, b, α) = ebt
∞
2 dx e−α x +tax
−∞
π t 2 a2 exp + bt . α 4α
(2)
By exploiting the GF of the two-variable Hermite polynomials [2] ∞ n t n =0
n!
Hn (x, y) = ext +yt
2
Hn (x, y) = n!
[ n/2]
xn−2k yk
k=0
(n − 2k)!k!
,
from Eq. (2) we get G(a, b, α) =
∞ π α n=0
tn n!
Hn
b,
a2 4α
,
∗
Corresponding author. E-mail addresses: [email protected] (D. Babusci), [email protected] (G. Dattoli), [email protected] (B. Germano), [email protected] (M.R. Martinelli), [email protected] (P.E. Ricci). 0893-9659/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2012.10.003
(3)
352
D. Babusci et al. / Applied Mathematics Letters 26 (2013) 351–354
and, thus, In (a, b, α) =
a2 π H n b, . α 4α
(4)
2. The main results In a recent series of papers [1] it has been proved that the Bessel functions can be formally reduced to Gaussians according to the following representation: J ν ( x) =
xcˆ
ν
exp −ˆc
2
x 2
ϕ(0)
2
(5)
where cˆ µ ϕ(0) = ϕ(µ),
ϕ(µ) =
1
Γ (µ + 1)
.
(6)
The properties of the operator cˆ have been extensively discussed in [1]. By treating it as an ordinary constant, we find, for example, B0 (α) =
∞
√ 2 dx J0 ( α x) = √ ,
(7)
α
−∞
and, in analogy with Eq. (1), for ν > n (ν ∈ R),
√ 1 (ax + b)n dx Jν ( α x) √ = ν−1 ν 2 ( α x) −∞
π ν−1/2 a2 cˆ H n b, α αˆc 2 1 π a = ν−1 Bn b, ; ν 2 α α
∞
(8)
where we have introduced the polynomials Bn (x, y; ν) = n!
[ n/2] k=0
xn−2k yk
, (n − 2k)!k!Γ ν − k + 21
(9)
whose properties will be briefly described later. As a consequence of this result, for any function f (x) that can be written as f (x, m) =
m
(m < ν),
f k xk
(10)
k=0
the following identify holds:
√ f (ax + b) 1 = ν−1 dx Jν ( α x) √ ν 2 ( α x) −∞ ∞
m π a2 fk Bk b, ; ν . α k=0 α
(11)
It is interesting to note that the linear combination of Bessel functions Fn (x; a, b) =
n n k=0
k
an−k bk Jn−k (x)
(12)
can be written as Fn (x; a, b) =
ax 2
cˆ + b
n
exp −ˆc
x 2 2
ϕ(0)
(13)
and, therefore, according to Eq. (4), the following identity holds:
∞
√ −1/2 a2 dx Fn (x; a, b) = 2 π cˆ Hn b, cˆ ϕ(0) 4
−∞ [ n/2] √ = 2 π n! k=0
bn−2k a2k 4k (n − 2k)!k!Γ k +
1 2
.
(14)
D. Babusci et al. / Applied Mathematics Letters 26 (2013) 351–354
353
The method illustrated is more general than it may appear and can indeed be extended to other families of Bessel-like functions. For example, the spherical Bessel functions [3], j n ( x) =
π 2x
Jn+1/2 (x),
(15)
within the present formalism can be written as j n ( x) =
√ π
x 2
xn cˆ n+1/2 exp −ˆc 2n+1
2
ϕ(0).
(16)
By using this expression, the integral
∞
dx jn (x),
bn =
(17)
−∞
can easily be calculated by exploiting the GF method, getting b2n =
π (2n)! 22n (n!)2
b2n+1 = 0.
,
(18)
The Struve functions are defined by the series [3] H ν ( x) =
∞
x 2k+ν+1 (−1)k . 3 2 Γ k+ Γ k+ν+ 2
k=0
(19)
3 2
Even in this case we can apply the operational method, albeit in a slightly different form, and these functions can be written as 1/2 ν+1/2 c2
Hν (x) = cˆ1
ˆ
x ν+1 2
1 1 + cˆ1 cˆ2
x 2 ϕ1 (0)ϕ2 (0)
(20)
2
where the operator cˆi (i = 1, 2) acts only on ϕi and verifies the identity (6). Moreover, the use of the Laplace transform allows us to write 1/2 ν+1/2 c1 c2
H ν ( x) = ˆ
ˆ
x ν+1 2
∞
x 2 ds exp −s 1 + cˆ1 cˆ2 ϕ1 (0)ϕ2 (0) 2
0
(21)
which can be used to prove that, for µ + ν not an even integer, the following identity holds: ∞
dx xµ Hν (x) = (−1)µ+ν
0
2µ π 1 . π sin (µ + ν) 2 Γ 1−µ−ν Γ 1−µ+ν 2
(22)
2
As a final example, we consider the Wright–Bessel functions [4] Wα,β (x) =
∞
xk
k=0
k!Γ (kα + β)
,
(23)
which can formally be defined as
Wα,β (x) = cˆ β−1 exp cˆ α x ϕ(0).
(24)
Using this expression, it is easy to show that ∞
dx Wα,β (−x2 ) =
√ π
−∞
1
Γ (β − α/2)
,
(25)
and (d > 0) ∞
dx Wα,β (−x)e−dx = 0
1 d
Eα,β
−
1
d
(26)
where Eα,β (x) =
∞
xk
k=0
Γ (α k + β)
is the modified Mittag-Leffler function [4,5].
(27)
354
D. Babusci et al. / Applied Mathematics Letters 26 (2013) 351–354
3. Concluding remarks Now, let us come back to the polynomials Bn (x, y; ν), introduced in Eq. (9). These are auxiliary Hermite-like polynomials, and their properties can be deduced from those of Hermite polynomials written in umbral form. For example, by using the first of Eqs. (3) and (24), it can be shown that the generating function is given by ∞ n t
n! n =0
Bn (x, y; ν) = ext W−1,ν+1/2 (yt 2 ).
(28)
It is also interesting to note that these polynomials satisfy the differential equation
ˆ y Bn (x, y; ν) = ∂x2 Bn (x, y; ν) D
Bn (x, 0; ν) = xn ϕ
ν−
1
2
(29)
ˆ y = cˆ ∂y has been introduced. Therefore, we get where the derivative operator D 1 −1 2 n ˆ Bn (x, y; ν) = exp c y∂x x ϕ ν − ,
2
(30)
which, taking into account the operational definition of the Hermite polynomials Hn (x, y) = exp y∂x2 xn ,
(31)
can easily be shown to coincide with the expression given in Eq. (9). These polynomials can be framed within the Appell family. However, they deserve to be studied carefully and this will be done in a future paper. References [1] [2] [3] [4] [5]
D. Babusci, G. Dattoli, G.H.E. Duchamp, K. Górska, K.A. Penson, arXiv:1105.5967v1 [math.CA]. P. Appell, J. Kampé de Fériét, Fonctions Hypergéometriqués et Polynômes d’Hermite, Gauthier-Villars, Paris, 1926. L.C. Andrews, Special functions for Engineers and Applied Mathematicians, MacMillan, New York, 1985. H.M. Srivastava, H.L. Manocha, A Treatise on Generating Functions, Wiley, New York, 1984. D. Babusci, G. Dattoli, M. Del Franco, Lectures on mathematical methods for physics, Internal Report ENEA RT/2010/5837.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Integrals of Bessel functions D. Babusci a , G. Dattoli b , B. Germano c , M.R. Martinelli c , P.E. Ricci d,∗ a
INFN - Laboratori Nazionali di Frascati, via E. Fermi, 40, IT 00044 - Frascati (Roma), Italy
b
ENEA - Centro Ricerche Frascati, via E. Fermi, 45, IT 00044 - Frascati (Roma), Italy
c
Sapienza - Università di Roma, Dipartimento Me-Mo-Mat, Via A. Scarpa, 14 - 00161 Roma, Italy
d
Università Campus Bio-Medico di Roma, Via A. del Portillo, 21 - 00128 Roma, Italy
article
abstract
info
Article history: Received 5 July 2012 Received in revised form 4 October 2012 Accepted 5 October 2012
We use the operator method to evaluate a class of integrals involving Bessel or Bessel-type functions. The technique that we propose is based on the formal reduction of functions in this family to Gaussians. © 2012 Elsevier Ltd. All rights reserved.
Keywords: Integrals Generating function method Bessel-type functions
1. Introduction Integrals of the type In (a, b, α) =
∞
dx (ax + b)n e−α x
2
(1)
−∞
can be calculated using a general procedure based on the generating function (GF) method [1]. Multiplying both sides of Eq. (1) by t n /n! and summing up over n, we obtain G(a, b, α) =
∞ n t n =0
=
n!
In (a, b, α) = ebt
∞
2 dx e−α x +tax
−∞
π t 2 a2 exp + bt . α 4α
(2)
By exploiting the GF of the two-variable Hermite polynomials [2] ∞ n t n =0
n!
Hn (x, y) = ext +yt
2
Hn (x, y) = n!
[ n/2]
xn−2k yk
k=0
(n − 2k)!k!
,
from Eq. (2) we get G(a, b, α) =
∞ π α n=0
tn n!
Hn
b,
a2 4α
,
∗
Corresponding author. E-mail addresses: [email protected] (D. Babusci), [email protected] (G. Dattoli), [email protected] (B. Germano), [email protected] (M.R. Martinelli), [email protected] (P.E. Ricci). 0893-9659/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2012.10.003
(3)
352
D. Babusci et al. / Applied Mathematics Letters 26 (2013) 351–354
and, thus, In (a, b, α) =
a2 π H n b, . α 4α
(4)
2. The main results In a recent series of papers [1] it has been proved that the Bessel functions can be formally reduced to Gaussians according to the following representation: J ν ( x) =
xcˆ
ν
exp −ˆc
2
x 2
ϕ(0)
2
(5)
where cˆ µ ϕ(0) = ϕ(µ),
ϕ(µ) =
1
Γ (µ + 1)
.
(6)
The properties of the operator cˆ have been extensively discussed in [1]. By treating it as an ordinary constant, we find, for example, B0 (α) =
∞
√ 2 dx J0 ( α x) = √ ,
(7)
α
−∞
and, in analogy with Eq. (1), for ν > n (ν ∈ R),
√ 1 (ax + b)n dx Jν ( α x) √ = ν−1 ν 2 ( α x) −∞
π ν−1/2 a2 cˆ H n b, α αˆc 2 1 π a = ν−1 Bn b, ; ν 2 α α
∞
(8)
where we have introduced the polynomials Bn (x, y; ν) = n!
[ n/2] k=0
xn−2k yk
, (n − 2k)!k!Γ ν − k + 21
(9)
whose properties will be briefly described later. As a consequence of this result, for any function f (x) that can be written as f (x, m) =
m
(m < ν),
f k xk
(10)
k=0
the following identify holds:
√ f (ax + b) 1 = ν−1 dx Jν ( α x) √ ν 2 ( α x) −∞ ∞
m π a2 fk Bk b, ; ν . α k=0 α
(11)
It is interesting to note that the linear combination of Bessel functions Fn (x; a, b) =
n n k=0
k
an−k bk Jn−k (x)
(12)
can be written as Fn (x; a, b) =
ax 2
cˆ + b
n
exp −ˆc
x 2 2
ϕ(0)
(13)
and, therefore, according to Eq. (4), the following identity holds:
∞
√ −1/2 a2 dx Fn (x; a, b) = 2 π cˆ Hn b, cˆ ϕ(0) 4
−∞ [ n/2] √ = 2 π n! k=0
bn−2k a2k 4k (n − 2k)!k!Γ k +
1 2
.
(14)
D. Babusci et al. / Applied Mathematics Letters 26 (2013) 351–354
353
The method illustrated is more general than it may appear and can indeed be extended to other families of Bessel-like functions. For example, the spherical Bessel functions [3], j n ( x) =
π 2x
Jn+1/2 (x),
(15)
within the present formalism can be written as j n ( x) =
√ π
x 2
xn cˆ n+1/2 exp −ˆc 2n+1
2
ϕ(0).
(16)
By using this expression, the integral
∞
dx jn (x),
bn =
(17)
−∞
can easily be calculated by exploiting the GF method, getting b2n =
π (2n)! 22n (n!)2
b2n+1 = 0.
,
(18)
The Struve functions are defined by the series [3] H ν ( x) =
∞
x 2k+ν+1 (−1)k . 3 2 Γ k+ Γ k+ν+ 2
k=0
(19)
3 2
Even in this case we can apply the operational method, albeit in a slightly different form, and these functions can be written as 1/2 ν+1/2 c2
Hν (x) = cˆ1
ˆ
x ν+1 2
1 1 + cˆ1 cˆ2
x 2 ϕ1 (0)ϕ2 (0)
(20)
2
where the operator cˆi (i = 1, 2) acts only on ϕi and verifies the identity (6). Moreover, the use of the Laplace transform allows us to write 1/2 ν+1/2 c1 c2
H ν ( x) = ˆ
ˆ
x ν+1 2
∞
x 2 ds exp −s 1 + cˆ1 cˆ2 ϕ1 (0)ϕ2 (0) 2
0
(21)
which can be used to prove that, for µ + ν not an even integer, the following identity holds: ∞
dx xµ Hν (x) = (−1)µ+ν
0
2µ π 1 . π sin (µ + ν) 2 Γ 1−µ−ν Γ 1−µ+ν 2
(22)
2
As a final example, we consider the Wright–Bessel functions [4] Wα,β (x) =
∞
xk
k=0
k!Γ (kα + β)
,
(23)
which can formally be defined as
Wα,β (x) = cˆ β−1 exp cˆ α x ϕ(0).
(24)
Using this expression, it is easy to show that ∞
dx Wα,β (−x2 ) =
√ π
−∞
1
Γ (β − α/2)
,
(25)
and (d > 0) ∞
dx Wα,β (−x)e−dx = 0
1 d
Eα,β
−
1
d
(26)
where Eα,β (x) =
∞
xk
k=0
Γ (α k + β)
is the modified Mittag-Leffler function [4,5].
(27)
354
D. Babusci et al. / Applied Mathematics Letters 26 (2013) 351–354
3. Concluding remarks Now, let us come back to the polynomials Bn (x, y; ν), introduced in Eq. (9). These are auxiliary Hermite-like polynomials, and their properties can be deduced from those of Hermite polynomials written in umbral form. For example, by using the first of Eqs. (3) and (24), it can be shown that the generating function is given by ∞ n t
n! n =0
Bn (x, y; ν) = ext W−1,ν+1/2 (yt 2 ).
(28)
It is also interesting to note that these polynomials satisfy the differential equation
ˆ y Bn (x, y; ν) = ∂x2 Bn (x, y; ν) D
Bn (x, 0; ν) = xn ϕ
ν−
1
2
(29)
ˆ y = cˆ ∂y has been introduced. Therefore, we get where the derivative operator D 1 −1 2 n ˆ Bn (x, y; ν) = exp c y∂x x ϕ ν − ,
2
(30)
which, taking into account the operational definition of the Hermite polynomials Hn (x, y) = exp y∂x2 xn ,
(31)
can easily be shown to coincide with the expression given in Eq. (9). These polynomials can be framed within the Appell family. However, they deserve to be studied carefully and this will be done in a future paper. References [1] [2] [3] [4] [5]
D. Babusci, G. Dattoli, G.H.E. Duchamp, K. Górska, K.A. Penson, arXiv:1105.5967v1 [math.CA]. P. Appell, J. Kampé de Fériét, Fonctions Hypergéometriqués et Polynômes d’Hermite, Gauthier-Villars, Paris, 1926. L.C. Andrews, Special functions for Engineers and Applied Mathematicians, MacMillan, New York, 1985. H.M. Srivastava, H.L. Manocha, A Treatise on Generating Functions, Wiley, New York, 1984. D. Babusci, G. Dattoli, M. Del Franco, Lectures on mathematical methods for physics, Internal Report ENEA RT/2010/5837.