Asymptotic expansion of a class of definite integrals

ASYWTOTIC

EXPANSION OF A CLASS OF DEFINITE INTEGRALS* N. V.

SLONOVSKII Yharkov 14

(“eceiued

1. TYE integral

1956)

Jurre

transforlls

b

1 Jn w(t)1

fn (5, b) =

f(f)df

(2)

0

(J”(t) is the n-th order problems of mathematical (2)

is

sin2t

obtained exp(ig

An asymptotic imaginary,

in

cos

t)

[d

Bessel function) are often encountered in physics. An asymptotic expansion of integral for

with

paper

particular h = s/2.

of

for

expansion in

was obtained

The present

the

n = 1,

(2)

[21 as

considers

fn (z,k, 00) =

the

case

)1(t)

q = px,

= sin

p constant.

n = 0 and f(t)

t,

f(t)

=

r 4.a.

= t exn(-t2)and

II

1x 1 + W asY!iwtotic exPa:lSion Of

1 expp(fkt)J, Izt)elrp(-rt)f(t)dt

(3)

0

as constant y >O; k, x are parameters. k-too, x+00; n=O, 1, 2,. . . ; grals of this type are encountered in problems of potential theory occurring 2.

l

Zh.

in electron

Bessel’s

vychisl.

opt its.

function

.&fat. mat.

of

integral

Fiz.

7.

order

4, 226

885

R can be written

- 888,

1967.

lnte-

Asymptotic

expansion

l

(4)

in

definite

227

integrals

(+a)

s-” --! exp [_Ti(s-f)]ds.

J, (2) = --i\

Substituting

of

(3)

and changing

the

(4)

order

of

integration,

we get

(6) The order inside it

the

of

integration

can be changed

provided

and fourth Isl = 1 in the first and third, and f(t) E L (0, a).

circle

in the

second

Formal

integration

by parts

the

contour

quadrants.

r is

and outside

gives N

2

fn(s, k, 00) =

(-l)W-‘)(O)Ej

(7)

+ RN,

j-1

where

(8) (9) r

The disposition by the condition

of

the

poles

of

the

ajsn+f = This

equation

integrand

in

(9)

is

determined

0.

(10)

has a root .si.= 0

of

multiplicity

each

of

n f

1 -

s* -

2(a -

multiplicity

j.

.j and roots

ifl)s-1

= 0,

s2,

(ii) s3,

o=ylx,

by

given

B=

(12)

k/x,

From (12), sm = a-$*

)l(W (c

-

B” + I)2 + 4aq32)+ (a2 2

‘12 -

fi”-1) 3

(13)

228

N.V.

-_i

c

v’((a2-

-

Slonovskii

~2+~)z+4u2~2)-(az-~z+~)

% -

2

1

m = 2, 3;

I>

the root sg is assumed conventionally to correspond to the minus sign in (13). Qualitatively, the disposition of the roots s2 and s3 in the complex plane is as follows. Let urn= Re(s,),

u,,, = Im(sm);

we have um = a -(-l)m

v* = _ @+(-_l)m

l’((02 - B” + 1)2 $4aqP)-f- u2 - 82 + 1 2

f

I

l((a” -BZ+~)2+4a2~2)-(u2-g~+1) 2 [

‘k t

3

$12 .

($4) (15)

From these, (urn+ 6) 2 = a2 - gz + 1, (am -a)*-(a, - of (G. + B) = --a&

(W (171

whence UmZ- 2X&,-vvm2-2~vm-1

=o

(W

and

Substituting

(19)

in (13) gives 2_ v7n -

1--umz+2aatn

2 [(urn -a)/uJ-i

*

whence

wh.ere

Obviously, :$, is defined as a function of u,,, outside the interval (0, 2a). The graph of the function is given in Fig. 1. Wow consider u, as a function of 13, p ez ( -co, Co); we naturally assume that a << 1 (for x >> 1).

Asymptotic

FIG.

By differentiating the the

expansion

of

definite

PIG.

3.

the

right-hand

side

extrema of u,~(P) are easily shown to corresponding extremal values are

From

these

properties,

a qualitative

of

223

integrals

(14)

occw

picture

with

at the

of

the

4.

respect point:;

u,(a)

to Q, - x), 0, A;

curves

can

be obtained; they are illustrated in Figs. 2 and 3. Com!?arison of Figs. 1 - 3 shows that the loci of the roots ~2, s 3 are the vurves 1,~ and t3 of Pig. 4 (up to higher powers of ,~f. From Fig. 4, the roots s 1 and 52 r by the contours rl of (IO) must lie inside the contour i-. Replacing and Tz circuiting

the

points

s1 and ~2~ we find

that

‘L.Y.

230

IJsing laborious,

Slonovskii

the residue theorem and performing calculations, we find that, for

essentially simple, n + 1 - j GO.

though

W) and for

n + 1 -

i > 0, (-1)’ I(j -

1 *-’ -2 1) II2 8r-j I_o

Putting n >o:

(j-j)!(n

j = 1 in

-j)!

(_-l)iss2i+

j-1

(n-j+pP)!(2j-2-P)!

~

d

(ss*+i)2j-i

b

get

2 Ei = ----. z

1

and (26).

of

l(SSZ + 1) pj- -

p!(i - 1 -P)!

p-0

(25)

(26)

il(n-j-i)!

s:+’

(-i)n+fi

+

(n-i-i)!(j--++i)l

to

a first

,

approximation

for

1

ss2+i

(27)

s,“-’

an integer-valued

argument

is

given

by

P-l

T(j~n)=~(i---i-v)

for

j>n+p,

v-o e

for

To find

the

conditions

under

which

i,

j&n++.

series

(7)

is

asY!!!PtOtiCas x + a,

h + ,a, let

fl’ 0;

be a constant theorem we

find

to

the

that

inner

(7)

is

then

integral asymptotic

lsx-s~l in

(8)

up to

(m > )‘(u* +

A). Applying

and assuming its

(,b -

1)-th

that

Let

f(t)

satisfy

the f@‘(r)

conditions eL(O,

001,

p =

0,1,2,..., N,

residue

PN)(t) EL(O,

term.

Theorem

the !rence

oo),

Asymptotic

expamion

then, when (28) is satisfied, series (n - l)-th term as p + co, x -‘a. The question treatment.

of

the

An asymptotic

necessity expansion

1x1 -+;u and k + 33 can be obtained

1.

2.

PODDUSNYI, G.V. Construction functions, Izv. I/(.‘?ou. ,Cer.

definite

of

of

(7)

- 431,

1963.

asymptotic

condition of

(26)

up to

requires

f,,(x, b, a,) for

it5

furhter

ilaqinary

T a.;

similarly.

of the mnterr!.

NIL SllGV, L. N. and ?lIZNETSOV, P. I.

419

is

231

inteorals

asyogtotic series of 6, 130 - 134. 1365. nn

the

evaluation

of

a class

t!le

of

integral