# proceedings, mathematical, physical and engineering sciences, vol. 452, no. 1955 (dec. 8, 1996),...

8/10/2019 # Proceedings, Mathematical, Physical and Engineering Sciences, Vol. 452, No. 1955 (Dec. 8, 1996), Pp. 2655-2690

1/37

Asymptotic Expansion of a General IntegralAuthor(s): A. SellierSource: Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 452, No. 1955 (Dec.8, 1996), pp. 2655-2690Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/52865

Accessed: 11/05/2010 15:01

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless

you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you

may use content in the JSTOR archive only for your personal, non-commercial use.

Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at

http://www.jstor.org/action/showPublisher?publisherCode=rsl.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed

page of such transmission.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of

content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms

of scholarship. For more information about JSTOR, please contact [email protected].

The Royal Societyis collaborating with JSTOR to digitize, preserve and extend access to Proceedings:

Mathematical, Physical and Engineering Sciences.

http://www.jstor.org
http://www.jstor.org/stable/52865?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/action/showPublisher?publisherCode=rslhttp://www.jstor.org/action/showPublisher?publisherCode=rslhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/stable/52865?origin=JSTOR-pdf


2/37

Asymptotic expansion

of

a

general integral

BY

A.

SELLIER

LADHYX,

Ecole

polytechnique,

91128 Palaiseau

Cedex,

France

For b

c

RI,

a real

function

h(x)

and also

complex pseudo-functions

f(x)

and

K(x,

u)

obeying

weak

assumptions,

an

asymptotic

expansion

of

the

general

integral

b

I(A) :- fp

f

(x)K[h(x), Ax]

dx,

is

proposed,

with

respect

to the real and

large

parameter A,

where the

notation

fp

designates

an

integration

in the finite

part

sense

of

Hadamard.

1. Introduction

For

>

0,

-oo

0 and

a

function

ha

with

h(x)

-

c

=

(x

-

a)Vha(x),

ha(x)

>

0 and

M

m~O

ha

(X)

=

E am log(x

-

a) + o(x

-a) in ]a,

a +rl],

m=0

where M

c

N and

aM

74

0.

2.

For

r'

:=-

y-r,

if

h(x)

-c

+

(x-

a)

(ha(x)

then

g

C

D

(c,

C),

else if

h(x)

c

-

(x

-

a)Yha(x)

then

g

E

Dr

(c,

C).

More

precisely,

there exist

real

values

r/

>

0

and

s'

>

r',

a

positive

integer I,

a

complex

function

Gr,

bounded in

[0, r],

a

family

of

positive

integers

(J(i)),

two

complex

families

(ai),

(gij)

such

that

Re(ao)

0,

i-y

r,

s'

>

r'

:=

y-r,

complex

families

(AZjq),

(Tp)

and

q

>

0 such that for

almost

any

x

E]a,

a

+

r1]

I

J(i)

Pi Qij(Pi)

w(x)

=

E

E E

giA

-pji

(x

-

a)'i

logi(x -a)

i=0

J=0pi=O

qi=O

I

J(i)

+

gij(x

-

a)sRii(x)

+

h(x)

-

cl

Gr,(lh(x)

-c),

(2.9)

i=0

j=0

where

each

quantity keeps

the

meaning

introduced

in

definition

5,

the

complex

func-

tions

Rii

are

bounded in an

adequate

neighbourhood

of a

and

Re(rpi)

r and Hr bounded in a

neighbour-

hood on

the

right

of

zero.

Case 1.

The case of

(h,g)

E

H+7l(a,C).

If

(gi)

=

(0),

then

w(x)

=

H(x)

C

Dr

(a, C).

If

there

exist

0

_q(O)

leads

to

J(O)

j

T,(x)

=

ha(x)?(x

-

a)

gOjE

C

q

loJ-q

[h,(a)]

log(x-

a)},

j=0 q=0

i.e. to

Td(x)

=

ha(x)a?(x

-

a)7?

[gojdJ(x

-

a)j

j=o

if

d(x- a)

:=

log[ha(a)(x

-

a)7].

Because

(goj)

5

(0),

it

is

easy

to

prove

that

T

(x)

is

not

the zero

function

in

a

neighbourhood

on

the

right

of

a.

Consequently,

if

(h,g)

e

7r,2

(a,

C)

then

S+

(g

o

h)

-

Re(oao).

The reader

may

easily

introduce

the

definitions of

sets

H_'1

a,

C),

H7'2

a,

C)

and

show that

(h,g)

c

H17'm(a,

C)

for m

c

{1,2}

implies

w(x)

:=

g[h(x)]

c

DL

(a,

C).

Definition

9. For r >

0,

the

complex

function h is of the

second kind

on

the

set

]0, r[

if

and

only

if

there exist

a

complex

function

H,

a

family

of

positive

integers

(M(n)),

and

two

complex

families

(/3n)

and

(hr,)

such

that

N

M(n)

Vc

]0, r[,

h(c)=E E

hnmeO

logm(c)

+

H(e),

(2.14)

n=O

m=K(n)

Re(/N)

+oo

j,Re(cai )?r m,Re(yn)

r

(2.18)

mean that

there

exist a real s

>

r,

a

complex

function

Fr

bounded

in

a

neighbourhood

respectively

on the

right

of

zero and on the

left of

infinity

in which

f(x)

-=

aixi

logi

x

+

xsFr(x)

j,Re(ai)


13/37

Finally

for

real

values r1

and

r2,

2(]0,

b[, C)

is the set of

complex

functions

such

that

lim

f(x)=

aijx'

logi

x-+O

j,Re(ai )

0

and

0

ri

-

So(f).

For

t

:

[s

-

rl

+

So(f)]/2

>

0,

aij(Ax)f(x)xsRi(x)

-

xrl+taij(Ax)f(x)Fij(x)

where

Fij(x)

:=

xt-S(f)f(x)Rii(x)

is bounded

on the

right

of

zero. The

change

of variable u

= Ax

yields

A/A

Iij

(A)

aij

(Ax)f)R

x ) (x)

dx

=

rl

+taiJ(u)Fij(au/A)du

A-(rl+l+t)

(3.24)

Proc. R. Soc.

Lond.

A

(1996)

2671


19/37

Because

So(ai3)

>

-1-

r1

and

Fij

is

bounded

near zero

the last

integration

exists

and

is

bounded

for

large

enough

A.

Finally,

t

>

0

ensures that

Iij(A)

=

o[A-(rl+)]

and

thereby

that

A/A

I/f

S"(x,

Ax)

dx

o[A-(r+l)].

Jo

If m

C

N

and

f

C

C

with

Re(/)

>

ri,

the choice of

Rij(x)

=

1 or Ri

(x)

=

F(x),

f(x)

'=

logm(x)

with

So(f)

-

0

shows

that

A/A

Jo,m(A)

:=

'

iA

(x)x3

log

xdx

= o[A-(l+l)]

and

AA/X

JO3,m(X)

=

ai(Xz)F(x)x3

logx

z

dx

=

o[X-(r+1)].

As a finite

sum

of such

integrals

J3,m(A)

or

J'

,(A)

the

contributions

A/A

A/A

A/

S(xr,

x)dx

and

S'(x, Ax)

dx

Jo

Jo

are also

equal

to

o[A-(~l+l)].

Thus,

relation

(3.24)

is

proved.

(c)

If

(aij)

-

(0)

then

the second

sum

on the

right-hand

side of

(3.9)

is

zero.

Else

for this

sum

So(f)

+

yS

A

ensures

that

[T

+

T'

+

T"] [x, (Oi

(u))]

=

xs1

R(x,

u)

with

R(x,

u)

bounded for

(x,

u)

c

[0,

r71

[A,

+oo[.

Moreover,

one

gets

xv~;

ho(x)";

f(x)

=

x1G(x)

with

G

bounded for x

E

[0,

r7].

Consequently,

the

complex

function

Ws,S2

satisfies

Ws1,s2(x,

)

=

R(x, u)

+

G(x)wt,s

[h(x)

-

c,

u]

and is

bounded for

(x,

u)

E

[0,

r7]

x

[A, +oo[.

Gathering

all

the

properties

satisfied

by pseudo-function

K(x,

u),

it

appears

that

K

c

rF2

(]0, b[,

C)

where this set is defined

in

Sellier

(1994).

Application

of theorem 1

of this latter

paper

allows us to

expand

b

I(A)

-

fp

K(x, Ax)

dx

Jo

and thereafter ensures the stated result.

A

As outlined

right

after definition

5

the case of a

function h which is

constant

in

a

neighbourhood

on the

right

of zero is

not taken into account

by

definition

11

and

previous

theorem

12.

Assume that

(f,

h,

K)

obeys

the next

modified

properties

of

definition

11:

1.

Property

1 is

unchanged.

2.

Properties

2a and 2b are

unchanged

with

this time

Vr2(c,u)

bounded near

infinity, property

2c

replaced by:

gnm(x)

:=

f()Knm[h(x)]

c

Loc(],

b],

C),

h(x)

=

c

for x

E

[0, r/]

where

rT

remains the real number introduced

by

properties

2a

and 2b

and

(3.5)

is

replaced

by

Knm[c]

=

Knm[c],

i.e. i

=

0

-

J(0),

ao

=

0, Kno

=

Knm[c]

and also

Vnm

=

0.

Note

that

these features

agree

with

expansion

(2.7)

introduced

in

definition 5

if

p-

Qo(0)

-

0,

A

=

1, A?

-

0.

3.

Expansion

(3.6)

and

assumption

(3.7)

hold with

a?(u)

=

K[c,

u]

and

Ht

=

0.

Finally,

one assumes that

lim

K[c,

u]

=

ApqUpP

logq

u

q,Re(3p)

R

and

(3.9)

is deduced

by applying

(3.3)

for X

=

c. Thus

Oo(u)

=

V2(c, u),

which is

bounded in a

neighbourhood

of

infinity.

4.

Naturally property

4 is true with

Wt,2

=

0.

Proc. R. Soc.

Lond.

A

(1996)

2673


21/37

Under

this set

of

assumptions,

one

gets

for

r

0 and

Uc

a

neighbourhood

of

point

c such that

02C+L(x,

y)

exists and is

bounded

for

(x,

y)

E

[0, A]

x

Uc.

3. Vi

E

{0

...

, I},

the functions

O0O1

L(u,

c)

for 0

0. In

such a

case,

b

=

r

and

inequality

(3.3)

holds. If

7

b

b,

h

bounded

in

[0,

b]

ensures

that

th(x)/(Ax)

-

0 as

A

->

+oo

for

any

(x,

t)

c

[r],

b]

x

[0,

1].

Consequently,

HA(x)

=

(N )-1

(I

-

t)N02N+lL[1,th(x)/(Xx)]dt

Jo

is

bounded

for A

large

enough

and x c

[ri,

b].

Moreover,

x-(N+l-~)[h(x)]n

turns

out

to be

bounded too on

[I, b].

Therafter

and since

f L

oc(]0,

b],

C)

there

exists a

real

B

such that

0

r2= r. More precisely,

two

cases

occur:

Proc.

R.

Soc.

Lond.

A

(1996)

2677


25/37


26/37

Asymptotic expansion

of

a

general

integral

With

the

new index n

=

I

+

1

+

j,

(4.14)

may

be

rewritten

as

N

X-c

cl

Ht

[X

-

c,

u]

=

W

W

(X

-

c)IX

-

clt

u-(n"-)

+u-~2

IX-cltl

wtl,s,

(X-c,

u)

n=O

with

Wn(X

-

C)

=

0 =

V(X

-

C)

for

0

?

n

# proceedings, mathematical, physical and engineering sciences, vol. 452, no. 1955 (dec. 8, 1996),...

Documents