The stationary-phase method applied to parametrically dependent integrals

THE STATIONARY-PHASEMETHOD APPLIED TO PARAMETRICALLYDEPENDENT INTEGRALS* A. G. PRUDKOVSKII Moscow (Received

THE application

27 December

1971;revised

of the stationary-phase

7 June 1972)

method to a single

integral

with analy-

tic, complex-values integrand dependent on a parameter a is discussed. theory is then extended to the multi-dimensional case, and in particular, concept

of the index of a complex

The present

paper examines

quadratic

The the

form is analyzed.

the asymptotic

behaviour

as h + + 0 of an integral

of the type I(CZ,h)=

j&l[S( A

5, a)h-‘]cD(z,

a)dx,

where A E RI,B E R',B > A; and a and h are real S(x, a> and @((x, a) are complex-valued, analytic with uous with respect to their arguments for x E U,, a E bourhood of the closed interval [A, ~1 in C’ and c0 is We shall

develop

a modification

parameters. The functions respect to x, and continE-c,, cJ (U, is a neigha positive

of the stationary

phase

number).

method and obtain

an asymptotic form for Z(a, h) which is uniform with respect to a and has accuracy O(h”), where M > 0 is arbitrary, in the neighbourhood of the point a = 0. The following assumptions are made: (1)

ReS(x,

a)<

(2)

the interval

S’*(X~, 0) = 0, ReS(Xi,

sional

We shall case.

*Zh. vychisl.

extend

0 for all x E [A,

[A, ~1 contains

~1, a E [-to, cl.,

only a finite

number of points

0) = 0; here S”22(3Ci, 0) + 0 for arbitrary our results

Mat. mat. Fiz.,

from the one-dimensional

13, 2, 275-293,

1973.

xi at which

i.

to the multi-dimen-

A. G. Pmdkovskii

2

Our main results are expressions (2.3) and (3.4) in the onedimensional and (4.2) and (4.3) in the multidimensional case. 1.

case.

case,

Notation and auxiliary lemmas

We shall prove the localization Consider the integral

principle,

2, a>h-~l~‘(“,

I,(a,h)=ieq[S(

first stated

in [l], for our present

a)&)(%

A

Bl.

where g (x) E Ctp bi,

For all x E supp g(x),

let either

S’ (x, 0) j 0 or Re S (x, 0) < 0. Then an

c, > 0 exists such that, when jai < tr, we have 1, (a, h) = 0 (hN), uniformly respect to a for any positive integer N. Proof. tinuous

Since S(X, a) and @(x, a) are analytic

with respect

with respect domain

to both their arguments,

to x are uniformly

U, x [-to, (,I of space

continuous

with respect

to x and con-

and their derivatives

in any closed

of all orders

sub-domain

C’ x R’, they are, in particular,

with

Y of the

continuous

in

Y, = [A, B1 x L-c,,~,l. We perform a decomposition

I=

el(s>

+ e,(s),

of the identity

e*(z)

E C”M

in [A, B] (see [2])

Bf,

et(s)

E C”EA,

BI.

such that \ &‘(x, 0) 1 > x for z E supp g(x) n supp e,(x) and Re --x for J: E suppgfa) [I supp e,(x), where x. > 0 is a number. Since S (x, a) and S’ (x, a) are uniformly

continuous

in Y 1 an 6, > 0 exists

such that, when la/ < cl, we have

(I.11

I&‘(z,

(l.2,

ReS(s,

a> 1 > x/2,

a) <---x-/2,

~1: E supp g(x)

II supp el

z E supp g(s) n supp

We write I, (a, h) as a sum of two integrals:

S(X, 0) <

(Z),

ez (L)

*

Parametrically

dependent

3

integrals

Integrating by parts, we obtain B

s

expl.S (2, a> VI

cD(.T, a) e, (z) g (5)

do

=

A

-h

jexp[S(x, a)h-ll--&tCD(x,

[9(x, a)]-‘} dx.

a)e,(x)g(x>

A

In view of (l.l), the integrand in the last integral is infinitely differentiable with respect to x and uniformly continuous in Y,; integrating by parts N times, the following asymptotic form, uniform with respect to a, is obtained: s

s

elip[S(z,a)h-‘]~(2,a)e,(z)g(s)dJ:=O(h”).

A.

Using (1.2), we obtain the following for the second integral in (1.3): B (1.4)

IS A

exp[S(s,

exp (This asymptotic

a)h-‘I@(

x/2h) (B - A)

~,ct)e,(rc)g(l-)~~l

(

max 4> (z, a) = 0 is.a)EY,

(h”) .

form is also uniform with respect to a.

The lemma is proved. Definition. if S’(x,,

We call x, E CA, ~1 a stationary

0) = 0, Re S(x,,

point of the function S(X, a)

0) = 0.

Notes. 1. It follows from Lemma 1. I that the only contribution to the asymptotic form of I(cz, h) is from the integrals

over the neighbourhoods of

stationary points and the neighbourhoods of the ends A and B of the interval. view of assumption (Z), the stationary

points are isolated,

In

so that it is sufficient

4

A. G. Prudkovskii

to examine

the case when S(x, a) has just one stationary

2. When proving integration

BY

the lemma we made no use of the fact that the contour

is a segment

smooth contour

of the real axis,

in U, connecting

hypothesis, S,; (a,,

there exists

a neighbourhood

SXl (x, a) = 0 has a unique Definition. value

The symbol

of the square

point x,.

points

i.e. the lemma holds

of

for any piecewise

A and B.

0) f 0, so that,

Y,

by the implicit function theorem of the point (xc, 0) in which the equation

solution

x = z (a), the function

v“W, where W is a complex

root which satisfies

the condition

z(a)

being

number, arg fl

131,

continuous

refers

to the

E (-7r/2, + n/21.

Lemma 1.2 Let assumptions Then there exists

(1) and (2) be satisfied, a neighbourhood

and let 3c, be a stationary

point.

U, in C’ of X, and a number cg > 0 such that,

when \czI < tg, x E U,, the function

is analytic

with respect

to x and continuous

with respect

to both its arguments.

Proof. Put W (x, a) = IS (2 (al, a) - S (X, a>I Lz (4 - XP. Since z(a)

and S(x, a) are continuous

be proved provided

with respect

to !x, a), the lemma can

that it can be shown that v’[W (x, d)f is analytic

with respect

to x and bounded. Expanding

(1.6)

5(x, a) in a Taylor

series

W(X, a) f -1hS”(2(a),

a) -

in x at the point x = z(a),

we get

In view of assumption (I), Re W(x,, 0) >, 0, while \W (xc, 0) j = 1%S’k,, 0) / It 0 by virtue of assumption (2). Recalling respect

(1.6),

it can be asserted

to x, and continuous

that W(X, a) is bounded,

in Y,, so that,

given

arbitrary

analytic

numbers

%

with and %,

Parametrically 0 -=EXl -=C~1/2S”(Zh, 0) 1 < x2, there

point X, and an tl > 0 such that

xi <

dependent

exists

5

integrals

a neighbourhood

U, in C' of the

1W(Z, a) 1 C XZ, Re w(x,

CX)> --xi / 2

for x E Us, JCL/< 6,. Consequently,arg The function in modulus

W (x, a) C fn.

\/[W (x, a)] is therefore

analytic

with respect

to x and bounded

by the constant

The lemma is proved. We denote

by udi the image of the domain

Ui under the mapping

(1.5).

Corollary

A neighbourhood mapping

(1.5) defines

Y, = u, x L-c,,E,I of the point (x,, 0) exists such that the for every c E [-c4, c,l a unique analytic function x,(Y)

with y E Udi. Proof.

Consider

- dx

s”[z(a), al i-0(X-Z(d)

S,‘(x, cd

dr

=-

2Lz(a)--xldl[W(x, a>1 =-

2\l[W(x, a) 1

At the point x = x,, a = 0 we have

dY - %S” (x,, 0) k 0; dx= by the implicit every a E

function

theorem,

[-c4, c,l, the equation

there exist (1.5) defines

U, in C’ and c, > 0 such that, for a unique

in U,, (see [41). We introduce

(1.7)

the function

y (Y, a) = Q, [x, (y), al 2y [S’ (x,(y),

By what has been proved, (1.8)

Y((Y, a) E A(C’,J

n CO(Y,&),

a) I”.

analytic

function

xq(Y)

6

A. G. Prudkovskii

where Ya, = U,, x l-c,, ~~1. 2. Contribution Let x, E

of the identity

&(z)

E

c-t4

in [A,

~1

BIT

numbers.

from the stationary

I,(a, h)=

point

6,l and supp e,(x) E [x, - S,, X, t

= 1 for x E Lx, - S,, X, +

The contribution

Bl,

EGm[4

eo(s>

S,] E U,, where 6, < 6, are positive

(2.1)

stationary

(A, B), We perform a decomposition

1 = e,(z) + L%(x), such that e,(x)

from an interior

point

x, is described

by the integral

a) h-‘1 CD(x, a) e. (x) dx.

5, A

Theorem 2.1 Let Conditions

(1) and (2) be satisfied;

Ia\ < E, we have with any integer respect

(2.2)

then an 6 > 0 exists

I >, 0 the following

asymptotic

to a:

lo(a, h)-exp[S(z(a),

a)h-‘1

CY(zj)(O,

a) “‘(y)lljIVRhrli+1)1:=

j=o

where

Y(2j)(o,

a) =

PjY(y,

a>

&zj and the function The principal

(2.3)

such that, when form, uniform with

y=o

’

Y(y, a> is given by (1.5) and (1.7). term of the asymptotic

form is

Z,(a,h)=exp[S(z(u),cl)h~‘](1![-S”(z(a),a)1)-’(2nh)‘“X

(z(a),a)]-“X 1@(z(u),a)+;w

Parametrically

where z(a)

is the solution

of the equation

We carry out the change Io(a,h)=--

where y(a) mapping

of

(1.5) in the integral (2.1):

J exp[S(z(a),a)h-‘lexp(--zh-‘)Y((y,a)e”(y)dy, v(a) interval

lx, - a,, X, + &I under the

and g(y) = e,,(x,(y>).

An 6, > 0 exists

such that, when I a 1K t=, we have

Re (ya (xt’)) > 0, argy,(X) E (-n/2,n/2),

0.4)

S’ (x, a> = 0 in the neighbourhood

of variables

is the image of the closed

(LS),

integrals

a) 1 > 0.

the point x,, and Re\/[ - S” (z(a), Proof.

dependent

5E

[1co-62,ro-61],

Re (Y= (2)) < 0, arg Ya (z) E (n / 2, _3/2Tc), 5 E [lo + 6,, X0 + 621. This follows a = 0.

from the fact that the inequalities

In view of (2.4), by the assumption

(2.5)

in question

an x E LX,,- 6,, X, + a,1 exists

(I), Re kS(z(a),

ReS(z(a),a)
Let U be a ulosed

are satisfied

such that Reyd(x)

a) - Y,“(X)] = ReS(x,

when

< 0;

a) ,( 0, so that

for IaI
a E [-E,

~1. From (1.8),

(2.6)

y Q, a) - r,

of zero in C’ such that U z

U,, for any

given any h >/ 0, we have ycA) (0, a);

yk = Ye+’ fL+, (Y, a> 7

*=o where R n+l (y, a> is a function inY=Ux[-c,El. Then

(2.7)

IO((.& h) = 2 A=0

analytic

y(k) (0, +*(a,

with respect

to y and uniformly

IL)+ P+%A,

continua

a

A. G. Prudkovskii

where P(cr, h)=

J

exp[S(z(a),

a)h-‘lexp(-

y’h-‘)E(y)dy,

V(U)

J

~+l(o,h)=

exp[S(z(a),a)h-‘]exp(-_y2h-1)R,+,(y,.)~(y)dy.

v(a) We first obtain

J

~‘(o,h)= We divide

an asymptotic

exp[S(z(a),a)h-‘lexp(--zh-‘)e”(y)dy.

the contour

?3y> 5 1; y,(a)

form for the integral

y(a)

into two parts:

y1 (a) is the part on which

= y(a>\y,(a>,

J

~(a,h)=

exp[S(z(a),a)h-1]exp(y2h-‘)dyf

V,(U)

J

[S(z(a),a)h-‘]exp(-

y’h-‘)Z(y)dy.

V,(U) The integral over yI (a) is independent replaced by y,(a) ( see Fig. l), consisting

of its path, so that y1 (a> can be of two arcs of circles of radii

IYd(x,, - a,)( and jr,( x, + 8,)) and the segment VI9

IY&,

-

of the real axis [ - 1~~ (x, +

6,)(1.

Hence ~(a,h)=

J exp[S(z(a),a)h-~]exp(--~h-1)~(~)~y, Y,(U)

where y,(a)

= yz (a> u ys (a>, ;Cy) = 1 for y E ya (a).

On the contour

y,(a)

Re~S(z(a),

we have

a> - Y’) 6 0.

This follows from the fact that the inequality and from inequality (2.5). On y,(a)

we perform the decomposition

in question

of the identity1

is satisfied

on yz(a)

= gr (Y) + gz(y)y

such that supp gi (y) E I-X&, 3chl, gi (y) E C,” (73 (a) ) , fi2 (Y> = C” (r4 (a) ) where x1 is a positive number indepengl(y) s 1 for. YE [-x6/ 2, c/21. dent of a; this decomposition is possible by virtue of (2.4).

Parametrically

dependent

FIG.

9

integrals

I,

Then,

(X8}

x4

sexpCSb(a),

a)k-‘lexp(-y2h-‘)g,(y)dy.

-4

Applying Lemma 1.1. to the first integral method I31 to the second, we obtain

P(a, h) + expIS(z(a), uniformly

with respect

in (2.8) and applying

aP~‘~ltl(ffh)

to a, for any positive

= OQP). integer

N.

Integrating fk(a, h) by parts, we obtain the following, with respect to a, for any positive integer N: (2.9)

P(a,

Laplace’s

which holds

uniformly

h) +

k

Now consider

In+l (a, h). is bounded

l”+‘(a,h)“=C(a)lnfi(~,h)+

exp (-

y’h-‘)

=Q(hN).

odd

(k - i)!!2-k’2-f32; h

(Y, af, where C(a)

even i

From (2.6) we have R,+* (y, &I = ~(a) and Rncz (y, a> E

J exp[S(z(a),a)h-‘1X Y(a)

y n+2&+Z(Yt ~)~(~)~~.

+ y~~+~

A @.I) fiC"(Y),so that

10

A. G. Przdkovskii

We put n = 2r + 1 in (2.10); r + 2 times finally

and noting

that,

then,

from (2.9),

evaluating

the integral

in (2. IO) by parts

we have I*‘+* (a, h) = 0(h(*rf3)/*),

we

get l*‘+*((r,

(2.11)

/r) =

O(jp+3)‘*).

The form (2.2) follows The theorem

from (2.7), (2.9),

and (2.11) whe*r \a( < F.

is proved.

Since the factor expLS(z(a), a)h-‘I appears in all the terms of the series for I(a, h), we can strengthen the form (2.2), and write

Note. asymptotic

7

Zo(a, h)-

Y(zj) (0, a) w - I)!! JFn h(zJ+l)/z = (Xj) ! 2j c j=O c~)h-~]0(h(Z’+3)/2)+ O(JLM),

exp[S(z(a),

exp[S(z(a),

a)h-‘1

where M is arbitrary. 3. We now consider point A.

Contribution

the contribution

If A is not a stationary

all x E [A, A +

point,

from the boundary to the asymptotic positive

form from a boundary

E and 6 exist

ReS(x,

XE

The contribution

of the identity:

e,(x) E C”[A,

1 = e,(x) + e,(x), suchthate,(x)cland

Bl,

e,(x) E C”[A, Bl,

[A,A+6/2I,e,(x)=OforxE

from the point A will then be described

jA = jexp[S( 2, a)h-%I+, a)q(x)dx. A If Re S(x, a> < 0, we have j, = o(/& uniformly with respect Parts,

N by virtue

We can expand

jA=-

inequalities:

S’ (X, a) f 0.

a) < 0,

We perform a decomposition

tive integer

such that, for

61 and a E L-c, cl, we have one of the following

of (1.4);

otherwise,

jA into the asymptotic

[A+&BI. by the integral

to a for any posi-

S’ (x, a f 0, and integrating series

ha64 a) exp[S(A, a)h-‘IS'(A,a)

,,~‘(A,~)S’(A,~)-~(A,~)S”(A,~) S’3 (A, cc)

exp[S(A,a)h-‘I-...+O(hN),

by

Parametrically

where the remainder

is uniform

Now let A be the unique

position

of the identity

1=e,(x)

dependent

with respect

to a.

point of Sk,

stationary

11

integrals

a>. We perform the decom-

in !A, Bl +-e(x),

en (x)

81,

rtY[A,

E

e

(x)

E

CObfAt Bl,

such that 6?A(x) z 1 for x E [A, A f &I, supp eA (x) = [A, A + &I E u4, where 6, and 6, are positive numbers. The contribution

point A is eiven bv

from the stationarv

IA(

jexpD(

a)eA(x)dx,

5, a)h-*]@(5,

A We put T(a) by means

= Et (a) - Ald IES(z(a), a> - SCA, cdlk (a) - Al -‘h

of recurrence

For T(a)

the quantities D, (a) and J,‘(a,

relations

f 0,

L)o(a)=Y(O,a),

Y(Tbh+-~~~,~)

D*(a)=

T(a)

(3.1)

U”(0,+-wd

(a)=

I)

We introduce II).

&(a>=

2 -T(a)

’

Y’(T(a),a)-D,(a)-L22(a)T(a) .

'

[T(a)

I”

For s >, 2 even, i=s--1 &

(a) =

Y(*) (0, a) - s!

D,&)C:-~

[- T(u) lzi--s +

C{

i-8/2 lIei+, (a)

]zi-s+‘}]{s!‘[-

c:-‘-’ [-- T(a)

D2a+i(a)=

E

[Y'"I(T(CX),a)-S!

T(a) Is>-‘,

{L)2i(U)C18-i[T(a)]2i-“+

i=s/z

For s >, 3 odd, i=s- t

&(a)=

[y(“(O,a)-s!L),(a)-s!

II,,+, (a) ci”-i-i [-T(a)

JzW]

z {D,i(a)C,s-i i=(s+1)/2 fs![-

T(~x)]~)-“,

[-T(a)]2i-“+

A. G. ~~kovski~

12

s! D2s(a)

T”(a)

1~s!~T~u)Is++.

When ~(a) = 0, we have D,(a)

D, (a) are continuous

= (l/k!)

Y’““@,a).

in a neighbourhood

It will be shown that these

of the point a = 0: -‘I*

1,” (a, h) = -h”exp[S(z(a),a)h-‘1 r:+

CJ.2)

r:

7

exp(- t’)dt (1

I

(a, h) =

I

exp[SfA,

a)h-’

Jo0(a, FL)

with k even,

Theorem 3.1 Let coad~t~oas (1) and (2) be satisfied; then an E > 0 exists such that, when 1a I,< 6, we have the following asymptotic form, uniform with respect to a, for any integer M >, @: (3.3)

I,(a.+~

(Dzll(a>J89(a,k)+D,,,,(a)J,“,,(a,k)} (1-O

(MCZ)/Z 0th f O(h (MW)/2)

if

M

is even,

ifMisodd.

=

Parametrically Writing out the principal

term of this form, we have

r,(a,h)-_((z(a),a)exp[S(z(a),

(3.4)

13

dependent integrals

a>h-‘1~~[-5”(2(a),

a)lPX

T(a)lfh ~(23th)

J

++n-‘”

(

exp(-

t”)dt )

0

+ exp[S(A,

cz)h-‘lD(a)h

+ O(h3’2),

where T(a)

= (z(a)

D(a)-=

-s(k~)lM~)

-A)?I{IS(z(a),a)

-A]-“$,

- @‘(A,a) -t- @((A,a)S”‘(A, a) As”’ @;a) 3(S” (A, CL))~~ -

@(A, a) @(z(a),a) S’(A,a) + 1I2T(a)lI[--SM (z(a),a)

3

if

~(ct)=A,

if

z(a)+Ay

z(a) is the solution of the equation S’ (x, a) = 0 in the neighbourhood of the point A, and the symbol \/ denotes the branch of the square root with positive real part. The functions T(a), z(a), and D(a) are continuous in the neighbourhood of a =O. Note.

If the point z(a)

explS(A,

a)h-‘ID(a)h;

{,(a,

S [A,

if z(a)

h) = @(z(a),

~1, the

principal

term in the form (3.4) is

=A, then T(a) =O and

a) exp [S(z(cz), o)h-‘Iol[--S”(z(u),

a)ll-‘Y(~hz-l) +

D(b) exp [S(A, a)h-‘]iL + O(W2). Proof of the theorem.

We change

the variables

in the integral

in accordance

with (1.5):

where y(a) &)

is the image of the interval

[A, A + S,] under the mapping (1.5), and

= eA (x~ (y>>. An 6 > 0 exists

such that, when Ial < c, we have ReY,(x) < 0, argy,(x) E This follows beca:.t?, the inequality in

(m/2, “/, 7: for x E [A + a,, A + a,]. question

holds when a = 0.

Let U be a closed neighbourhood of zero in C’ such that U E Uo, for arbitrary a E C-r, cl. From (1.8) the functions

A. G, Prudkovskii

14

are analytic hence,

with respect

given

to y and uniformly

any positive

continuous

integer

M, we have

- T(a))““‘RE$

(y, a),

in Y = U x F-6, ~1;

(3.5) y”+l(y

whereD,(a)

E], Kifjt(y,

GE C’[-E,

a) E A(U)

The function y M+l (y - T (a)j”+’ if T(a) f 0, the function itself,

fi

Co(Y).

RiI: (y, a) is analytic with respect to y and its first M derivatives with respect

in U;

to y, vanish at the points y = 0, y = T(a). If T(a) = 0 the function itself and its 2M + 1 derivatives with respect to y vanish at y = 0. The recurrence relations (3.1) follow from these Using (3.9,

conditions.

we write I, (a, h) as the sum of integrals M

IA(a,h)=

-

c

rDzs(a)d,‘(a,h)+.L),,,,(a)J,“,,(a,h)~

+JTC(a,h),

s= 0

where

K’z shall

(3.6)

tirst

obtain

Jo”(a,h)=

the asymptotic

form of the integral

f expfS(z(a),a)h-‘Jexp(-y?h-‘)e”(y)dy, r(a)

We denote

by yr the contour

the Points T(a), the focalization J,“(a,h)=

- / Y,(A

principle,

(Fig.

2) consisting

of any smooth contour

joining

+ 6,) /, and the ray f - j y,(A + 8,) /, - ml. Using we can assert

that

J exp[S(z(a),a)h-‘]exp(-y2h-‘)dy=O(hN). r*(a)

Parametrically

dependent

FIG. uniformly

withrespect

We change

2.

to a for any positive

the variable

loO(a, h) = llh

15

integrals

integer

in the integral

(3.6):

N. t = y/\ih=

exp[S(z(a),Gl)h-i]exp(-tZ)dt+O(hN).

7 T(a)/Jh

It will now be shown that JoO(a, h) = show that the integral -co j= expP(z(a), s

O(dh). For this it is sufficient

a)h-‘lexp(--t’)dt

T(a)l/h

is bounded

by a constant,

If Re T(a) $0,

I

we

have ReS(z(a),

exp(-

J

of h and a.

independent

a>< 0 by virtue

t”>exp[S(z(a),

a)h-I]&

< I

lT(a)l/Jh

-_

T(a)/0

Ti2T+ If 1 T(a)

of (2.6% and

I

J iT(a)l/ih

exp~~(z(~),

.

a)h-*I& I

1< Cdh, where C is an arbitrary

positive

constant,

T(a)lJh

(3.7)

1

J

If I T(a) I > C~h,

exp(-

t’z)exp[S(z(a),

a)h-‘]dt

1-CCC.

to

A. G. hudkouskii

16

Expressions (3.7) and (3.8) show that the integral If ReT(a) C 0, then iii ~ i -‘zYy expfS(z(af,

(3.9)

j is bounded

a)h”]exp(-&‘)&

-k I

T(C%)/Jh -m I

exp[S(z(a), J --fT(afl/Jh

a)&‘]exp(-

.

t”)ds I

The first integral is bounded by virtue of (3.7) and (3.8), the variables, 0 = t + 1T(a) / /I,&, in the second: j-

when Re T(n) > 0.

exp[S(z(G1),a)h-“lexy,(-t2)dt

while we change

) =-

-ITpllifh

In the proof we used the obvious Re

a>-

LS(z (a),

In short, the integral tive integer N, (3.10)

JoP(a, h)-t

uniformly

with respect

1T(a) 1“I < 0.

j is also bounded

I(&)

o.d-In [

inequality

{exp[S(z(a),

in this case

Hence,

for any posi-

a)h-‘]X

x(a)/&

J 0

exp (- t”) dt

I)

= Ufh”),

to a, where the expression

in braces

is bounded.

Parametrically Integrating

J, 1 (a, h) by parts,

dependent

we obtain

17

integrals

the recurrence

relations

(3.2),

which hold up to I. To prove our theorem, J““+i(a,h)= M+i

it remains

to consider

the integral

S exp[S(z(a),a)h-‘IX Y(a)

The following

recurrence

Using (3. IO) and (3. II),

7::: The uniformity integrands

relations

can be proved by integration

we get

(a, h) = { ~;~L,, with respect

in Y. Hence

M+2)‘2), if M is odd, if M is even.

to a follows

from the uniform continuity

we have (3.3) when Ial < t. The

4.

multi-dimensional

The theorem

h)=

fexp[S( 0

where 52 is an N-dimensional N the parallelepiped a(~,

rI i-i

of the is proved.

case

The above method also enables us to determine of multi-dimensional integrals of the type I(a,

by parts:

the asymptotic

behaviour

~,@)h-‘]cf,(x,a)d~,

open domain

[Ai,B
a) are analytic with revs3

in Cw with boundary

the complex-valued

I’, lying inside

functions

to every Xi and continuous

S(x, a> and

with respect

to all

their arguments for xi E Ui, a E [--fo, <,I, i = 1, . . . , N (Ui is the neighbourhood in C’ of the interval EAi, BJ, and Ed is a positive number). Definition.

We call x, E Q a stationary

point of the first kind of the

function S(x, a> if gradS(x,, 0) = 0, ReS(x,, 0) = 0. We call X, E I- a stationary point of the second kind of S(x, a> if ReS(x,, 0) = 0, and the vector gradS(x,,

0) is non-zero

and normal to the boundary r at the point x,.

A. G. Prudkouskii

18

We call x, E det {aij(q

a a non-degenerate

0) 1 = det

the analytic

point of the first kind if

i, j, = 1, . . . , N.

- ~(““‘O))zO. 1 3

Now let the stationary Consider

stationary

point of the first

or second

kind x0 lie on r.

diffeomorphism

where 21 is a neighbourhood

of the point x0, V is an open set in RN, the point

x, corresponds

to y = 0, and r is given by the equation

let the yN axis

be directed

We call

a stationary

inwards

into the domain a).

point of the first

- &o,

det {aij(O, 0) > = det

kind x0 E

0) }+ 1

i,j=I,...,

0,

I= det

We shall

assume

I

(2)

the domain fi contains

Notice

it asserts

comes

solely

r is called

kinds,

E-c,,

Theorem

,...,

if

N-l.

only a finite

EJ,

number of stationary

principle

that the contribution

from stationary

for integration

points

of the

all of which are non-degenerate.

points

holds here as in the one-dimensional

to the asymptotic of the first

by parts is replaced

behaviour

and second

is proved in the same way as in the one-dimensional expression

non-degenerate

i,j=l

a) < 0 for all x: E a, a E

that the localization

case;

N,

that:

ReS(x,

and second

kind x, E

- -&(o’O)}+O, z ,

(1)

first

if

i,i=f....,N-1;.

point of the second

det{&(O,O)

r non-degenerate

3

det(d.l(O,O~)=dot{-~j(O,O)}#O.

a stationary

YN = 0 (for clarity,

case

by Green’s

kinds. except

of I(a,

Iz)

The principle that the

formula.

4.1

Let conditions

(1) and (2) be satisfied

of the first

kind;

the following

asymptotic

then holds

in the neighbourhood

of a = 0:

and let X, E

fi be a stionary

form, uniform with respect

point

to a,

Parametrically

(4.2)

dependent

19

integrals

CL,(~(ct),a)exp~~(z(a),a)h-‘-_(il2)Indfa,,~](2nh)~~~

ko(%h)-

=

I’ I det (a<,) I

O(h’

Iv+2>/2

),

where z(a) is the solution of the point x,,

ai.i=--

of the equation

gradS(x,

a) = 0 in the neighbourhood

i, j= 1, . . . . N,

(z(a), a),

dxidXj

and Ind @aiiI is given by (5.2) and (5.4). When a = 0, the form (4.2) becomes

Note.

the same as (3.12) of [6].

Proof of Theorem 4.1. We rotate the coordinate system the principal minors of the matrix k~,j are non-&generate: (A,=detj~ijl,i, The contribution integral

whereei(xi)

A,fO,k=f,...,N).

j=l,...,k;

form from the point x0 is given by the

to the asymptotic

I,,(a,h)=

j exp ES (2, a) -6

je,(s,)dz,... -b

E-‘C,Oa[-a,

h-’ ] Qt (2,

81, ei(xi) = 1 for xi E [-a/2,

Since d2S/c3x,’ = - A, f 0, we can appfy Theorem while regarding Then

all the remaining

(2~~)

exp (- W&3

a) ei 6%) d%

S/21. 2.1 to the last integral,

x,, . . . , x+, and a as parameters.

arguments

j e,(rz)exp[9(z,a)h-Lj~(Tra)X

je,(r,)h... -b

IIO(~,h)=

in such a way that

-a

dT + 2

. ..f

lltc-ht where~(~,a)=S(X,(~,, “N, a), x,,

. . . , xN,

from the equations

*-*jxN,a),x2, a).

The function

. . ..xN.a),~(x,a)=cp(XI(x,,

X,(x1,

. . . , xN, a) is defined

. ....

implicity

A. G. Prudkouskdi

20

pi =

Ipi lexp (iYi> = - --g(X& i2

It may easily

be seen that the matrix

)...)

ZN,cz),& ,...,

of second

derivatives

&%a). of the function

Rxz, *--t $, a) is connected with the matrix ~d*S/d~$c~~ by Eqs. (5.3); by Lemma 5.2, the principal minors of the matrix !gijjr = { -~z~~~~~~~~ are respectively equal to the minors of the matrix la. .) divided by a,,, and hence are nonative to the variables x,, . . . , xN, zero. On further applying Theorem 2.1 re“r’ we obtain

the required

result.

Now let the stationary

The theorem

point X, E ‘r;

is proved. we perform the change

of variables

(4.11, in which case X, + y0 = 0, I’ -) yn! = 0. If X, E I’ is a stationary point of the second kind, we obtain from Theorem 4.1 the following form for IXO(a, &): I

a)h-’ - (i/2)Ind{&$] (an) (N-i”zh(N+i”z+ (-dS(~(‘~),a)/dy,)1’Idet{dij}I

(a h) _

CD(~(,a), a)exp[S(z(a),

20 T @(h where y = z(a)

(N-i-3)/2

is the solution

&Yay,

d,=-

NOW let x, es rbe

of the system

= 0, . . ., t%/dy,_,

in the neighbourhood

Theorem

), of equations yiv = 0,

= 0,

of the point y = 0, and CJ2S ‘YidYj

bb>,

a stationary

a>,

i, j = 1, . . . , N - 1.

point of the first kind.

4.2.

Let conditions (1) and (2) be satisfied of the first kind; we then have, uniformly hood of a = 0:

and let x0 E rbe a stationary point with respect to a, in the neighbour-

~(~(a),a)exp[S(z(~),~)h-‘-(~/2)I~d{~i~}l(2~~)”’~

(4.3)

Lo(a, h) =

I’ Idet&)

exp(-

t’)dt

1 +

exp[S(E (a), a)h-’ ]D(cc)h(N”‘)‘2 + 0 where f(a)

= U(O, a),

1

(h’N+2)/‘2),

Parametrically

T(a) =~NbHw@w,

-

f'

dependent

21

integrals

a) --SE(a), 43bN(e-%

(0, a) [s” (0, q-1 + l/Qf (0, a) SW(0, a) [SN(0, a)]-2 nif

D (o) =

- f P,4

fs’(O,

~)r1-t112f

f

(YN,a> =

=;!,(a),

bv (4, a)tT (4 x

X f{--li2S” S(YN, a) =“S(U($N,

z(a)

@)I > II if 2 (a) # E (a)7

(3~ (a),

a>, Yivt a),

~(~(Y~,~),~~,U)eXp[(-i/2)Ind{d;j~](Z~)’~-*”~ 9

I’ Idet{dij) I

is the solution of the system of equations dS/dy, = 0, i = 1, . . . , N - 1 in the neighbourhood of the point y = 0, a = 0, while z(u) is the solution of the

U(Y,, a) equation

gradS(y,

a) = 0 in the neighbourhood

of the point y = 0, a = 0; z(a) =

lz, (a), * - - , .zN (a) 1; while d2S d;j

=

dyi dyj ’ 82s

a*j =

i,j=l

ayi dy;’

and (Y1, ***, yN) are connected Proof.

i,i=I*...

We apply Theorem

N-

1,

,*-a, N;

with (x1, . . . , xN) by the mapping 2. I with respect

to the variables

(4.1). yr, . . . , yN_l ;

then

@(U(~Y, a), ye, o)exp[S(U(y~,

a),

YN,

a)h-’

+(~/2)Ind{di:}]

(2nh)cN-‘)‘2

y’ldatfdij)! We apply Theorem The theorem

3.1 to the integral

(4.4) and obtain

the required

+...

result

.

(4.3).

is proved. 5.

We shall

of complex quadratic forms in real space RN

Algebraic

consider

and (5.4) below)

properties

in the present

of a non-degenerate

satisfying

the condition

(5.1)

Re

N

cc

section quadratic

the concept

N

a$& 2 0 for arbitrary

,=Ij=i

of an index (see (5.2)

form {aijj (aij = oji, detcij

Xi E

RN.

+ 0),

22

A. G. Prudkovskii

We shall state without proof some elementary lemmas of linear algebra, showing that (5.2) and (5.4) are equivalent and that the index is invariant under a nondegenerate

linear

The notation

mapping

in RN.

of the basis

11). . . , ik is the minor of the matrix {aij) at A lll.. #il,

is as follows:

l

of rows i,, . . . , i, and columns j,, . . . , j,;

the intersection intersection principal

i.e. A *s * ’ * s kk, will be called

of the first h rows and columns, minors

of the matrix jcij)

the minors at the

l,...,

and denoted

by A,;

Xi are the eigenvalues

of the matrix lU;jl, and argW is the value of the argument of the complex number W which satisfies the condition argW e [ -7~, ~1.

Lemma 5.1 An orthogonal

mapping

in RN exists

of the basis

minors of the matrix ~aij~ are non-zero i-2 , *a’, N.

in RN exists,

By Lemma 5.1, a basis

such that the principal

and such that Reari

in which the principal

In this basis, of the matrix (aij] are non-zero. be introduced in the following way. Let pI = A, = alI, pk = A, (A,,,)

- 0 for arbitrary

minors A,

an index of the matrix (aijf can

-’ for k >,2; then

iv IIldIUiJ =

(5.2)

We now consider “ij~

a*g 6% c A=,

argpkE(---,nl.

the matrix lbij),

i, j = 1, . . . , N - 1, connected

with

by

bij ,s (a,,) “A ;I 2,.

(5.3)

,

i,....,ik

We denote by Bi,s,.,,j, II, . . . , ik and columns

’

the minor of the matrix ibij] at the intersection

of rows

j,, . . . , jk.

Lemma 5.2 iaijI and {bij) be connected is _i N - 1, j, ,( N - 1, we have

Let a,, Y 0 and let the matrices for arbitrary

by (5.3);

then,

Parametrically

dependent

23

integrals

Lemma5.3 Let a,, f 0 and let kriji satisfy the inequality (5.1); connected with krijj by (5.3), also satisfies

then the matrix ibijl

condition (5.1).

Corollary If faij] satisfies

(5.1) and its principal minors are non-zero, then

argpk = arg(Ak (A,_,) “1 E [ -7712, 77/21.

Lemma5.4 Let the symmetric matrix @,I Re 9,

F, Cijz,Zj,

O

satisfy the condition for arbitrary 5~ * 0,

x8 E RN,

i=l j=* then the matrix k2 ij 1 is non-degenerate. Corollary If the matrix la,1 sa t is f ies (SJ),

its eigenvalues must lie in the right-hand

half-plane:

ar&E

[ -n/2;n/21.

Lemma 5.5 Let the symmetric non-degenerate matrix iaij 1 satisfy

(5.1);

then the index

of the matrix (5.2) is invariant under linear mappings of the basis in RN.

Lemma5.6 Let the symmetric non-degenerate matrix {aij] satisfy (5.1); the matrix (5.2) then satisfies

the equation

where Xi are the eigenvalues

of the matrix.

the index of

24

A. G. Prudkovskii

(To prove the last lemma, we use Lemmas In conclusion general

I thank V. P. Maslov

guidance,

and V. L. Dubnov

5.1, 5.3, and 5.5.)

for suggesting

the problem

and G. A. Voropaev

and for

for valuable

Translated

criticisms.

by D. E. Brown

REFERENCES

1.

VAN DER CORPUT, 1, 15-38,

J. G.

Zur Methode der station&en

Phase,

Compositio

Math.,

1934.

2.

GEL’FAND, I, M. and SHILOV, G. E. Generalized Functions and Operations on Them (Obobshchennye funktsii i deistviya nad nimi), 1, Fizmatgiz, Moscow, 1958.

3.

GOURSAT,

4.

HORMANDER,

5.

ERDELY,

6.

FEDORYUK, difference

510-540,

E.

A.

Cows L.

d’analyse

Introduction

Asymptotic

mathematique, to Complex

Expansions,

M. V. On the stability and partial differential 1967.

Gauthier-Villars,

Analysis,

Dover,

Paris,

Van Nostrand,

1943.

1966.

1956.

in C of the Cauchy problem for finiteequations, Zh. vj%hisl. Mat. mat. Fiz.,

7, 3,