Asymptotic expansions of certain integrals

ASYMPTOTIC EXPANSIONS OF CERTAIN INTEGRALS* A. A. TUZHILIN

Moscow (Received

1.

14 February

1968)

Statement of the Main Results

IN the present paper we obtain asymptotic expansions for the integrals m

s

O”&[z(chp +cht)'h]

exp(izcht)f(t)dt,

s.

0

as

IZ I-+ 00, 0 <

0

arg z <

,f(t)dt

(ch fi + ch t)v’2

.% where f(t) is a regular function in a half-strip

containing the positive half of the real axis and the origin (the point t = O), H’,1) (2) is Hankel’s function of the 1st kind of order V, v is a real number, v + 4, and p is an arbitrary real number. When arg z = 0 we take the upper limit in the integrals as M + ic, where c is an infinitely small positive number which has to be chosen so that the point DO + ic is in the strip in which f(t) is regular; when arg z = n the upper limit of the integrals is assumed equal to M - ir, with the same conditions on E. The function f(t) may be increasing as t + + M, but not excessively

fast.

These integrals are encountered e.g. in problems of wave diffraction and radiation in wedge-shaped regions. problems will be found in [l, 21.

Application of the present results to these

Notice that the integrals can be expanded asymptotically by using the stationary phase (or gradient) method, the resulting expansions being in the asymptotic power sequence

(z-‘n’2) mcN

(as Jz/ -+ W, N is the set sf positive

integers 0, 1, 2, . . .). But even for integrals of the first type, i.e.

s exp (iz ch t)

f(t) dt the stationary phase method is unreasonable, since it takzs no account of’the specific properties of the integral. For integrals of the second type, the *Zh.

v;chisl.

Mat. mat. Fiz.,9,5, 1024-1035, 1969.

54

Asymptotic

expansions

of certain

integrals

55

stationary phase method leads to extremely laborious working and an expansion (z-‘~@)~~N, in which the coefficients are in the asymptotic power sequence very difficult to visualize. A more effective method, that takes account of the properties of the integrals, is used in the present paper. For integrals of the first type we obtain an expansion in the asymptotic power sequence (z-“‘@),~N, but evaluation of the coefficients is simplified; for integrals of the second type, we obtain an expansion in the asymptotic sequence

the expansion coefficients differ only by a simple factor from the coefficients for integrals of the first type (see Theorems 1 and 2). If use is also made of the asymptotic expansion of the function H (t’ (2) in the power sequence (z-“2) mEiv, we get an expansion for integrals of the second type again in the asymptotic power sequence (z-rtLfi)‘nlEN. Notice that our method also gives a general indication of the way in which the d~ficulties of the stationary phase method may be overcome. Our main results are: Theorem 1 If the function f(t) is regular in a half-strip containing the positive half of the real axis and the origin, and

f(t) = O(exp PI (a + 11I21j

w

(n 30) uniformly in this strip as t, -f w (t = t, + it,), then, when k 2 ra and ]zf--+=~,

O
00

s 0

exp(iz ch t)/(t)&

=

expW

+ n/4)1 x -VP)

(I.21

56

A. A. Tuzhilin

where

GP =

c

2pg

=

$ Z) z2’c-gq -$‘2p+i] P”‘WL

2

(l-3)

s=.

(

;y:,

22’+l[

$3

&-)2p+2]

6=Of(21+w.

(1.4)

Theorem 2 If f(t) satisfies

k>n

the conditions of Theorem 1, then, when the real v >

and Izj +ob,

- 1/2,

O
1 if,“’[z(ch

p + ch t)‘h] (ch p + ch t)v!:!

.

“ttjdi

=

(14

I)

exp( - inv) = ----_ (ch P + W2 >; f

’ im+lrr @ + %&,fi~ (2 m! In=O

(ch p + 1) ‘/2 (m+w ) +o( z

\

H$+2,,2_,[z(ch

(;1J,2- ,,[z (ch fj + 1) ‘121x 711

/3+1)‘“](

(ch s ,’ ‘)“’

)(h+2”2)}_

where the C, are given by (1.3) and (1.4). If we also use the asymptotic expansion of the Hankel function [3] as ( --JT <

I+~

arg 2 <

2~) namely,

and substitute it in (1.5) (see Lemma 1 in [41), we get

Theorem 3 If f (t) satisfies

k > IZ, IzI+

T II:,” (z(ch .\

(ch

the conditions of Theorem 1, then, when the real v >

CO,0 < arg z G nt, p + ch t)““)

p

+

- ‘12,

ch

t)vi2

(1.6)

‘f(t)dt=

0

- 1)“2-a~~/2)} (cl1 /?J+ 1) (v+‘)D

k

exp {i(z(chp+

(I;

m=O

v+‘i

>

x

Asymptotic

expansions

(

of certain

57

integrals

(ch 6 + 1) ‘12 mlz’-i x

--

I m/21

2 ,,=O

>

2

I-[(m+ I)/2 -sn-2p (In - 2P)! P! r (4

+o(

- v-2p+1)(2(chp+l))jJ

J-)}, ~@+3)/2

where the Cm are given by (1.3) and (1.4). To prove Theorems 1 and 2, we require a theorem on the expansion of an f(t) satisfying the conditions of Theorem 1 in powers of the function ch t - 1 (Theorem 5). The ideas used for proving this theorem can provide the basis for a generalization of the Darboux, BUrmann and Teixeira theorems (see [5l, pp 176, 181-185). The theorem itself provides an example of the direction in which these latter theorems may be generalized. 2. Proof of the Theorems To prove Theorems 1 and 2, the following two preliminary theorems are needed. Theorem 4 Given

p 2 0, z # 0

and

0 < arg z < n

we have

00

s

exP(izcht)(cht--)~ch~2dt=

exp[i(~+il/4)llii~:p~~~~~),

(2.4)

0

and whatever the complex V,

O”&’ (Z(chB+cht)‘h),(cht_ll)Pchtdt_ s 0

(ch 6 + ch t) VP

exp[in(p

2

-

(2.2)

+ ‘/2 - 2, )Ppr(p+ '/2) &t;j+ x (ch p + l)v’2 (z(ch fi + I)%) ( (ch pZ+,,,,>,+“.

Note 1. When arg z = 0 we take the upper limit in the integrals in (2.1) and (2.2) as m + ic, and when arg z = R, as m - ic, where t is a positive number as small as desired (see the similar note in Section 1).

A. A. TuzhiEin

58

Proof of Theorem 4. Equation (2.1) is easily proved by the change of variables ch t - 1 = u, ch(t / 2)dt = 2-“~~u-‘~~duetc. To prove (2.21, notice that its left- and right-handsides are regular in z when z f 0, 0 6 arg z 4 C, and II is any complex number,so that we only need to prove (2.2) e.g. when z = ix , 3~> Q and v > 0. Using the equation

(2.3) where K, (z) is MacDonald’s function, and, when larg z) < n/4 (see C61,Section 6.22, (1511, K;(z) = +(

;,‘3

exp [- ‘t - z2/42] $,

we obtain (see the similar derivation in 1’71,Lemma3)

(ch2-

t 1)Pch--&. 2

The change in the order of the integrationis easily justified by the usual criteria. If we apply (2.1) for evaluat~g the inner integral (we have to put z = ix2 ,/ 4t), we get

Asymptotic

expansions

of certain

integrals

59

If we again use (2.4), we get L

=

exP[--~(~+1)/212p+'r(~+i/2)

K”_,_,(,(chp+l)‘,~)x

n(ch fi + 4)‘+‘2

Using the relationship K_, (z) = K, (z) , again putting (2.3), we obtain (2.2).

ix = z

and using

Theorem 5 Let f = {t : -3t < Im t < n), and let f(t) be regular in a half-strip R, containing the positive half of the real axis and the origin, while

f(t) = O(exp

(tin / 2)).

(2.5)

uniformly in this strip as tr -+ 00 ft = tl + it2) . Then, whatever the t in the half-strip R,, containing the positive half of the real axis and the origin, and such that its closure is contained in i’ fl R, and whatever the k 2 n, we have

r(t)=

’ 2mi2a, 2 m, m=o *

(sh+)m

+

where

(2.7) and the contour I’ embraces R, and lies in f ll R G&e Fig. I). A neigh~~h~ U of the origin exists (U c RI), such that, for all t E U m 2m~2a,

f(t)= 2

n1=0

m,

( sh+)7n =5 m=o

>

(cht-

I)@.

(2.3)

.

Proof. Elementaryworking (see Fig. 2) shows that the function ZJ= 72 sh (t / 2) = (ch t - 1) ‘h maps one-to-one and confo~ally the strip I into the complex y plane with cuts along the imaginaryaxis 12 4 1Im ZJI< CG and that the real t axis thereby maps into the real y axis, while any half-strip

60

A. A. T~zhil~n

ReZ

FIG.

1.

containing the positive (or negative) part of the real axis and the origin and included in I, maps into a star-shaped region with respect to y = 0. Let x(y) be the inverse to the function t t (ch t - 1) “2== y (t E 1). I fl H in the mapping t .--f (~11t -. 1) v2~ Then, whatever the IJ E B the function f o x (the composition of f and x) is regular, and since B is a star-shaped region with respect to y = 0, we obtain for all y E B, in accordance with a particular case of Darboux’s formula (see [5l, p. 176, and 181, (4.5)), Let B be the image of the half-strip

In addition, if y is a closed contour lying in B and ~ontain~g the point z, we have

(2.10)

Asymptotic

expansions

of certain

integrals

61

FIG. 2.

Now let I1 be the contour on the t plane, consisting of a finite part of the contour I’ and the vertical line II’, to the right of the origin (see Fig. 1; the “finite part of the contour I” is the part to the left of r,‘). Let T, be the region bounded by rl, I31 the image of T,, and I!‘, the image of I1 under the mapping t --f (ch t - 1) ‘12(see F ig. 2). It is easily seen that B, is a star-shaped region with respect to the origin. Hence, if t E TI, we have ty2 sh (t / 2) E B, for all r such that 0 < 7 < 1. Consequently, for all t E T1

62

A. A. Tuzhilin

Let the point s be the preimage of the point fox[y2sh (t/2)] =f(t), wehave

5 (5 = y2 sh (S / 2) ) . Then, since

(2.11)

Now consider,the contribution from the contour lYl’ to the integral round rl in (2.11). Since f(s) = O(exp (s,n / 2)), where s = sI + is2, uniformly with respect to s, as s 1 -+m, we obtain for suffkiently large s 1,

f(W(42)ds r,,

[sh\(s/2)-Tsh(t/2)]R+2

I

=-exp[sf(n-k---1)/21-

Hence, if k 2 n, we have f(s)ch(s/2)ds lim 1~ q-Z r,, [sh(s/2)-mh(t/2)]d+2

=”

(2.12)

Thus, when k sn, we can replace the contour rl by I- in (2.11), and the latter holds for all t E RI. Now consider

(f 0 x)(“)(O).

From (2.10),

(fox)‘“‘(O) = g-j

T”)

(2.13)

where the contour y lies in B I and embraces the origin. On changing the variable: in (2.13), and denoting by ; the preimage of the 6=j’2sh (s/2) contour y containing the point t = 0, we get (fcx)(rn)(3)=

m! 21-w/2

zJ4T_ P+* [sh (s/2) /@+l Y

Expressions

(2.6) and (2.7) are thus proved.

We now show that (2.8) holds.

When t = 0 we have f(0) = (f o x) (0),

(2.14)

Now let t & 0; then, when k > II,

i.e. (2.8) holds.

-_

w-1

-j

lini sh (t/2)

o

(1 -T’GfkdT

X

f(s)ch (s/2)ds ’ i fsh(s~2)/sh(t/2)-~]“+2[~h(s/2)/sh(t/2)-~C]k-’-~ If t is sufficiently

for all s on r.

*

small, we can put

Hence

as k -$ 01). Theorem 5 is proved, Note 2. It is clear from the above proof that, if f(t) is regular in a strip conning the real axis, and if we understand by I? a contour embracing this entire strip {stretching to infinity leftwards in Fig. l), then (2.6) and CL71 will hald in any strip R, embracing the real axis and having a closure lying inside the region bounded by l?. We now turn to the proof of Theorems 1 and 2. Roof of Theorem 1. The function conditions of Theorem 5, so that

co

s

exp(izch

U

CP(t) = f(t) / ch (t / 2)

satisfies

the

a0

t)i(t)dt =

s exp(iz ch t)ch+)dt 0

=

(2.15)

64

s

r fsh (s/2) -

t sh(t/2))kz-

’

where, in accordance with CZ.71, am =

~~{f(~~(

~h~~~~~

&; 2

lL p/2

But

sish

is

(1”>[

j@

( ‘I”)

)“3,,,

=

g(

2’[

sh;,/~,,m+i”js~,

-!&(

q&y’]

an even fun&ion, so that its odd derivatives

P”e’=

s=.

Pvo).

vanish, i.e. from (1.3)

and (1.41, I Um =

-

2mi2

cm*

(2.16)

Consider the expression

When arg z = 0, thti upper limit of the integral with respect to t in (2.17) is cx + i6, or N - ri when arg z = F. Put (2.18)

I.& A=minlsh(s/2) ---zsh(t12)1, wheretheminisoverallsonl?and all t on the contour in f2.171, while A f 0. Since k >:“, we have

If we divide the contour !J of the integral over s in f2.13f into a finite part and a

Asymptotic

expansions

of certain

integrals

65

part in the neigh~~ho~ of the infinitely remote point s, the double integral over the finite part of r will be bounded with respect to t, since when t is bounded (on the contour in t2.17)) it defines a continuous function of t, while it tends to zero when t is large. Thus, to assess the behaviour of (2.19) as a function of t, we need to consider the integral over the part of r in the neighbourhood of the infinitely remote point s (denote this part by I’ ). When t is bounded, the integral defines a continuous function of t, i.e. is also bounded. Now let s and t be so large that sh(s/2) and sh(t/2) can be replaced by their asymptotic values sh$=

shsh-=

q+iImI‘*

t

2

-

2 sh-

ti =tie

2

sh%os-2 -

Im r* 2

sh

1+

Im I?+ i&---2---

>

t (2.2Qf

(2.21)

In (2.201, Im l?* are the points of intersection of I with the imaginary axis (see Fig. 1); in (2.211, f > 0 when arg z = 0 or arg z = R, and c = 0 otherwise. We find from (2.20) and (2.21) that, given large s and t, 1sh (S / 2) - 1; ~1: (t / 2) 1represents the distance between two points on two straight lines which intersect at the origin. If LX,y) is a point on one straight line (y = az) and (x,, y,,) a point on the other (yO = bx,), the minimum distance between these points, whatever the fixed x,, is

Hence, given large t and large s (s on I ’ ; we assume c < Im I’+ and E < IIm r-i),

Hence, from (1.11, given large t on the contour in (2.17), 1

s -r)kdz~ (1

0

If bwl

r, (sh(s/2)-zsh(t/2)I”+‘d

Hence, for all t on the contour in (2.17),

A. A. Tuzhilin

66

(2.22)

Rk(t) = O(1). Thus,

Qk = f exp (iz ch t)( sh :-)“’

(2.23)

ch;O(l)dt.

i

0

We change the variable:2 (ch t - 1) = s, in (2.23); then, on the new contour, 0 (1) in (2.23) will remain O(1) for all s. We obtain as a result, as 121+ 00, OD

Qk

exp (iz) 21+k/2++k/Z

=

exp (i arg 2)

exp(is)sknO(l)ds

s

= 0 ( Z).

(2.24).

0

When arg z = 0, the upper limit in (2.24) is ~0exp tic), and when arg z = 77, boexp (-ic). If we now substitute

(2.24) in (2.15) and use (2.1), we obtain (1.2).

Theorem 1 is proved. F’roof of Theorem 2. We find in the same way as when proving Theorem 1:

OD d”[z F 9. G

(ch p + ch t) ‘“1

(ch p + ch +‘1

* F+T [(m + 1)/Z]

2 m=o

ml

exp(--inv)

‘(‘)d’

(2.25)

= (ch 6 + 1)“r-l’

C,nH&~y~_v [z (ch fi + 1) ‘ia]( (ch

k + 1 O”H;“[z (ch 6 + ch t) ‘/=I 4ni0 s (ch~+cht$--

p‘fl’“)cw”a+

hi-i ch+dt

r

1 (1 -qck D

s[W/2)

x

f(s)& - zsh(t/2)]k+2

Asymptotic

s

r

expansions

f(s)&

of certain

T H?[z(ch

- z sh (t/2)]k+2 = o

[sh(s/2)

67

integrals

fi + ch t) ‘/iJ

(ch fi + ch t)“/z-

’

k+i

ch [z / (ch l3 + 1)‘h] (ch t -

W”t;,:ange the variable:

2-h/2--i

Qk=

(ch p+j)‘” s

(chfi+I)v/Z’

1) = s

in (2.26), and

Wr+‘X >

pa-1

or Qk =

(ch p + I)?“:!

,J&2M2_,,[z(ch

6 + i)‘l’I(

(ch “,+ I)”

)‘,’

If we now let 121+ 00, use the asymptotic form

i(k+2)/2 exp ( -in

v)

X

z (ch 6 + 1) ‘12 ( (1

H$;2j,2_v Iz ( ch p + 1) ‘h]( (ch ’ ;

I) “’

)@+‘) ’ x

(ch6+1)v/2

(2.28)

0~exp(i X

(2.27)

sWql)&.

of the Hankel function, then change the variable in (2.27): +s/[z(ch6+1)‘~]}‘~--1)=z, weget

QR=

$0(l)&.

s

arg z)

exp(iz)zk!

I +

X

--v+1/2

X X

2z(ch 6 + 1)“2

kl2

>

O(1)dx.

But, given a real v > 4, the integral in (2.28) is uniformly bounded with respect to z when jz j > 0, 0 \( arg z & 1~(when arg z = 0 or R, the contour in the integral of (2.28) is displaced in the complex x region so that the integral is absolutely convergent). Thus,

68

A. A. Tuxhilin

1

Qk = (ch p +

l)v/2

which completes the proof of Theorem 2. Translated

by D. E. Brown

REFERENCES

1.

TUZHILIN, A. A. New representations regions with ideal faces, Akustich.

2.

TUZHILIN, A. A. Short-wave asymptotic expansions for the diffracted electromagnetic fields generated by arbitrarily oriented dipoles in a wedge-shaped region with ideally conducting faces, III Vses. simpozium po difraktsii uoln, Ref. kokl., Nauka, Moscow, 93-95, 1964.

3.

GRADSHTEIN, I. S. and RYZHIK, I. M. Tables products (Tablitsy integralov, summ. Ryadou 1962.

4.

TUZHILIN, A. A. Theory of MacDonald (Teoriya integralov Makdonal’da. II. ur-niya, 3, 10, 1751-1765, 1967.

5.

WHITTAKER, E. T. and WATSON, Univ. Press, 1940.

6.

WATSON, Press,

7.

TUZHILIN, A. A. Representation of the electromagnetic fields generated by dipoles in the presence of ideally conducting half-planes by MacDonald integrals, Differ. ur-niya, 3, 11. 1971-1989, 1967.

8.

TUZHILIN, A. A. Theory of MacDonald integrals, I, Recurrence convergent series, Differ. ur-niya, 3, 7, 1195-1212, 1967.

G. N. 1944.

A Treatise

for the diffraction Zh. 9, 2, 209-214,

fields in wedge-shaped 1963.

of integrals, sums, series and i proizuedenii), Fizmatgiz, Moscow,

integrals, II, Asymptotic expansions) Asimptoticheskie razlozheniya), Differ.

G. N.

on the Theory

A course

of Bessel

in Modem

Analysis,

Functions,

Cambridge

Cambridge

relations,

Univ.

Uniformly