The method of critical regions for two dimensional integrals and its application to a problem of antenna theory

Gewijzigd titelblad

THE METHOD OF CRITICAL REGIONS

FOR TWODIMENSIONAL INTEGRALS

AND ITS APPLICATION TO A PROBLEM

OF ANTENNA THEORY

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD V A N DOCTOR IN DE TECHNISCHE WETENSCHAP AAN DE TECHNISCHE HOGESCHOOL TE DELFT, KRACHTENS ARTIKEL 2 VAN HET KONINKLIJK BESLUIT VAN 16 SEPTEMBER 1927, STAATSBLAD NO 310, OP GEZAG VAN DE RECTOR MAGNIFICUS DR O. BOTTEMA,

HOOGLERAAR IN DE AFDELING DER ALGEMENE WETENSCHAPPEN. VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN

OP WOENSDAG 14 DECEMBER 1955 DES NAMIDDAGS TE 4 UUR

DOOR

JOH AN BERGHUIS

iininaBnanapmiHippp!ll^np>«iMW«iwii.iqiiii«<«w«miipHHP

THE METHOD OF CRITICAL REGIONS

FOR TWODIMENSIONAL INTEGRALS

AND ITS APPLICATION TO A PROBLEM

OF ANTENNA THEORY

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP AAN DE TECHNISCHE HOGESCHOOL TE DELFT, KRACHTENS ARTIKEL 2 VAN HET KONINKLIJK BESLUIT VAN 16 SEPTEMBER 1927, STAATSBLAD NO 310, OP GEZAG VAN DE RECTOR MAGNIFICUS DR O. BOTTEMA,

HOOGLERAAR IN DE AFDELING DER ALGEMENE WETENSCHAPPEN, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN

OP WOENSDAG 23 NOVEMBER 1955 DES NAMIDDAGS TE 4 UI

DOOR

JOHAN BERGHUIS

GEBOREN TE AMSTERDAM

Dit p r o e f s c h r i f t is goedgekeurd door de p r o m o t o r :

Prof. dr R. T i m m a n

Aan mijn ouders

Aan mijn vrouw

C O N T E N T S

1. Definitions of Prof, van der Oorput 2. A theorem on double Integrals 3. The transformation

4. Critical and non-critical regions 5. Exaaiple

6. Some theorems of Prof, van der Corput 7. Calculation of asymptotic residues 8. Generalisation to the complex case 9. Expandable and smooth curves Homogeneous Functions

Introduction

1. On the critical line of the second kind 2. On the critical line of the third kind 3. The critical point (homogeneous) 4. Problem 1

5. Problem 2

6. The critical line 7. A problem

8. Two critical lines

9. An example of an improper critical line Non homogeneous Functions

Introduction

1. Preponderance and ponderance; definitions 2. Simple properties

3. 3 theorems

4. Expansibility theorems 5. Theorem 5

6. Theorem 6

Examples for the non-homogeneous case 1. Problem 1

2. The critical line 3. The critical line II

7 8 10 10 11 13 15 17 18 19 24 25 27 27 28 29 31 35 35 39 40 42 43 45 45 45 46 47 SO SI S3 55 55 57 60

Cht^ter V. An old problem of antenna theory 66 Introduction 66 1. e = 71/2 69 2. e = 0 71 3. O < e < V 2 71 Appendix 76 References " 78

U I T T R E K S E L .

De methode van de kritieke gebieden voor tweevoudige integralen is een uitbreiding van de methode der stationaire fase. Behan-deld worden integralen van de vorm

ƒƒ g(w.x.y) e^^C^-^'^) dx dy.

D

waarin D een gebied van het reële (x,y) vlak is, w een grote positieve parameter is. De functie f(cü,x,y) heet „fasefunctie" en g(w,x.y) „amplitudefunctie". Aan deze beide functies worden nog voorwaarden opgelegd.

De methode der kritieke gebieden toont aan dat integralen van bovenstaande vorm voor grote positieve waarden van w asympto-tisch gelijk worden aan een som van „asymptoasympto-tische residuen". Asymptotische residuen zijn in feite Integralen over zgn. „kri-tieke gebieden", In hoofdstuk I wordt eerst het begrip „asymp-totisch analytisch" gedefinieerd en met behulp hiervan het kri-tieke gebied.

De berekening van de asymptotische residuen is vervat in de hoofdstukken II (homogene fasefunctie) en IV (inhomogene fase-functie). Hoofdstuk III geeft de nodige beschouwingen om inho-mogene fasefuncties homogeen te kunnen maken. Vele voorbeelden worden behandeld.

Het laatste hoofdstuk geeft de toepassing op een reeds van Zen-neck en Sommerfeld daterend antenne-probleem: de berekening van de potentiaal van een vertikale dipool geplaatst in het hori-zontale grensvlak van 2 oneindige media van verschillende per-meabilitelt en in het bijzonder de asymptotische ontwikkeling daarvan.

I N T R O D U C T I O N

This t h e s i s handles on twodimensional i n t e g r a l s and i t gives in the f i r s t place an extension of the method of s t a t i o n a r y phase. Kelvin found the p r i n c i p l e of s t a t i o n a r y phase and A. Wlntner [ISÜ and E.T. Copson [17] gave a d e s c r i p t i o n of the method of s t a t i o n a r y phase for one-dimensional i n t e g r a l s .

Two y e a r s ago. Prof. J.G. van der Corput, wished to g e n e r a l i z e t h e method of s t a t i o n a r y phase, not only for one-dimensional i n t e g r a l s but a l s o for more dimensional ones. The author worked a l s o on t h i s p r o j e c t and he applied the new method, now c a l l « d t h e method of c r i t i c a l r e g i o n s , to a problem posed to him by Dr. C.J. Bouwkamp. This problem comes from the antenna theory • and i t was f i r s t t r e a t e d by Sommerfeld [ 3 ] ; t h e a s y m p t o t i c development i s given in t h i s t h e s i s .

So we deal with I n t e g r a l s of the form

ƒƒ g(ü).x,y) eifC"'*'-^) dx dy. (1)

D

where D is a certain region of the real (x,y) plane, to is a large positive parameter, and the functions f((o, x,y) and ] g(oa,x,y) have to satisfy some (very general) conditions. The

function f(to,x,y) may be called "fase function* and g(w,x,y) the "amplitude function".

The method of critical regions then shows that integrals of the fonn (1) can be written asymptotically for large values of 00 as a sum of the so-called "asymptotic residues". It appears that the region D can be subdivided into critical regions and non-critical regions. The contribution of a non-non-critical region is asymptotically equal to zero (if the integrand is premultiplied by a special function, called neutralisor). So the asymptotic behaviour of the integral (1) is the same as that of the inte-grals on the critical regions, the so-called asymptotic resi-dues.

In order to find the non-critical regions we define the notion of "asymptotically analytlclty" and we prove a general theorem for non-critical regions.

Then we only need to calculate the asymptotic contribution of the asymptotic residues. This problem is much more simpler in the case where f(üLi, x,y) is a homogeneous function in x and in y and therefore Chapter II treates the homogeneous case.

In order to reduce the general problem to the homogeneous one. we need tj»e notion of preponderance, given in Chapter I I I . "Hie following Chapter then gives some more difficult problems. In the l a s t Chapter V the attention turns to the problem of antenna theory,we find the potential of a vertical dipole plac-ed in the boundary layer of two i n f i n i t e mplac-ediums of different permeability, each f i l l i n g up a hal f-space.

C h a p t e r I

A S Y M P T O T I C R E S I D U E S

1. Definitions of Prof, van der Corput

Be Q an unbounded point set lying on the real axes, such that |w|^ 1 for each element w of Q. That the point set is unbounded means that it is possible to find for/ each given positive num-ber Y at least one element to of Q such that |oo| >

y-Any number and any function not depending on to is called 'fixed".

A number A is said to be at most of the sane order as B, in formula

A = 0(B). (1.1) if there exists a fixed positive number c such that

|A| g C | B | (1.2) for a l l elements to of S2. Formula (1.1) i s s a i d t o hold

uniform-ly i n one o r more p a r a m e t e r s , i f t h e number c, o c c u r r i n g in

( 1 . 2 ) can be chosen independent of t h e s e p a r a m e t e r s . But i f ( 1 . 1 ) i s s a i d t o hold for fixed v a l u e s of t h e s e p a r a m e t e r s . then c may depend on these paraüieters.

I f i t i s possible to find a fixed real number p such t h a t A = 0 ( | t ü | ' ' ) . then A i s c a l l e d asyn^tótically finite. I f for each fixed real number q

A - B = odtt)!""»),

then A and B are called asymptotically equal, in formula A ~ B.

Two numbers, which are asymptotically equal, possess the same

asymptotic behaviour.

Ctae says that ah tends asymptotically to an asymptotic limit s

if it is possible to find a fixed integer k and furthermore for each fixed integer h ^ k a fixed real number qh such.that qh tends to Infinity with h and that for each fixed integer h ^ k

^h - S = 0(|a)|"**''). 2. A theorem on double i n t e g r a l s

In the following pages the boundaries of the twodimensional regions of integration are assumed to consist of a f i n i t e num-ber of analytical arcs. An analytical arc i s an arc given by a parametric representation

u = u(t,co) V = v(t,w)

for values of t in the interval 0 < t < 1, each function being indefinitely many times differentlable with respect to t for values of t in 0 < t < 1.

For shortness the points of intersection of these arcs are called vertices.

First we prove:

Itieorem A. Let A be a region of the real twodimensional (u,v) plane bounded by a closed continuous curve r, given by the pa-rametric representation u = u(t), v = v(t).

Let A(a), u, v) be a function of' ou, u and v such that

1°. on A and its boundary the derivative — ^ A(<o, u, v) exists for each non-negative integer k; '^

2°. the integrals

-ik

Jk = IS {^^ A(to,u,v)}.ei" du dv A ^

exist and tend asymptotically to zero if k - » for large posi-tive values of to;

3°. the integrals

L k = > e ^ ^ A ( ( o . u , v ) } - ^ d t

exist and are asymptotically equal to zero for each non-nega-tive integer k.

AssertioB: the integral

I = ƒ ƒ A(ü),u.v)e'" du dv A

is asymptotically equal to zero.

Renark: The same assertion holds if the factor i of the expo-nent is replaced by +1 or -1.

Proof: B y . p a r t i a l i n t e g r a t i o n with r e s p e c t t o u we find I = - i ƒƒ A(u).u.v)de*"dv = A = - i ƒ A ( t o . u . v ) e i " 4 r d t + 1 / / e ^ - l ^ d u dv = r dt ^ du = - 2 ' ( • l ) " * ' ƒ e'" ^ ^ dt f ( f i ) " " ƒƒ e ^ - l ^ d u dv ii»» T> ^u** dt ^ "da^

~ J k'

Since t h i s relation holds for each p o s i t i v e integer k we obtain I ~ 0.

Reaarii:

The integrals L^ can be equal to zero if e.g. -^dv eQuals zero; dt

then the theorem A holds k fortiori.

It may be that C contains vertices or points where the deriva-tive of V with respect to t does not exist, but if the ampli-tude-function A(u), u, V) and all its derivatives with respect to u are equal to zero .also Lk can be asymptotically equal to zero. Consider now the more general integral

I = ƒƒ e'"('^'*''y)h(a),x,y)dx dy. . D

Where D is a region of the real twodimensional (•x.y)plane. bounded by a closed continuous curve C. The phase function u(<«^. x.y) is a real function of x, y and <o, such that its parti-al derivatives u, and u^ exist everywhere in D, that they are continuous on that region and that they are not both equal to zero. The amplitude function h(to,x,y) may be complex.

We transform the integral I into an integral of the form

J J e'" A(od. u,v)du dv o

considered in theorem A by the transformation u = u((o.x,y)

V = v(io.x.y)

and we apply theorem A t o t h i s i n t e g r a l . The new amplitude function i s now defined by

A(co.u.v) ..h(!^llll^.

U V -U V

» y y X

The second variable V(Ü}, x,y) can be chosen freely, with the restriction that the transformation has to give a (one,one) correspondence between D and its image G of the (u,v) plane, this means that the jacobian A = u^v - u^v^^ may not vanish in D.

3. The transfonation

In order to determine the second coordinate v we assume a con-tinuous function A(x,y), which is > 5 > 0 everywhere in D and determine v by the condition

U,Vy - UyV, = A .

This is a first order partial differential equation for v with characteristic equations

i i = ^ = dv ^ ^^

-Uy u , A

The projections of the characteristics in the (x,y,v) space on the (x,y) plane are given by

u,dx + Uydy = 0 , they appear to be the equlphase lines.

Since Uj, and u nowhere in D vanish together and they are con-tinuous, the equations have no singular point in D.

If the values v are given on a certain arc C' represented by X = x(t), y = y(t), V = v(t),

it is a well-known result from the theory of differential equa-tions, that V is determined in a certain neighbourhood of that arc. if the determinant

du ,

-This means that the arc C' does not contact an equi-phase line. Since C is a closed contour, the values of v may not be given along the complete curve, in order to determine v uniquely in D. Hence we divide C into a finite number of arcs C^ by the verti-ces and contact points. We state without proof the following topological properties of the equlphase lines:

Since nowhere in D. u, = Uy = 0, the equlphase lines are either closed curves inside D or they intersect the boundary C twice. (Since u is only defined inside D, a segment of an equlphase line, which is the analytic continuation of a segment which lies inside D is counted as a new segment). The equlphase lines starting from a segment between two successive vertices or con-tact points sweep a region of D, where v is defined uniquely by its values on the segment of the contour. They intersect the contour a second time in another segment between two vertices or contact points.

Hence the arcs, in which C is divided can be connected pairwise. Considering the integration along a curve u = const

we have

where s denotes the arclength.

That means that v is an increasing function of s. Since we can choose V at one arc of the boundary (e.g. v = 0), it follows now that at the corresponding segment of the boundary v can also be chosen freely (e.g. v = 1): we have to choose the func-tion A(t,oo) for each value of t such that holds

dv A dx - " y (dv)» . A» . dy . " x ds» • - — u»+u^ dX dv = A(u» + u») •'^ds ^ X y '

ipijm-u.- .•• — — • I • !•• •..ipi-i.iii . ^ . . ^ i . . , , , ,

o (U»+Up'*

where L denotes the length of the are of the equlphase line contained between the two points of intersection with the boun-dary.

Hence choosing on the first of two corresponding arcs v = 0 and dv

on the second v = 1, the derivative -jr occurring in the Inte-dt

grals Lk of theorem A vanish along the boundary.

4. Critical and non-critical regions

If it is possible to transform an integral by means of the the-ory of the preceding sections into an integral satisfying the co;iditions of theorem A, we call the region of integration non-critical and h(tó,x.y)e^"(*">*"y) is called asymptotically analy-tic on D. All other regions are called crianaly-tical regions. These regions are the neighbourhoods of the points where the amplitude function A(u).u,v) shows some irregularity: e.g. if the function A(Ü), u, v) is not differentlable indefinitely with respect to u. or if u, = U y = 0 , then A((«>, u, v) becomes infinite, or if the derivative of u along the boundary C, in formula-rr,

dt equals zero, or if A(to, u, v) does not satisfy toe integral con-ditions of theorem A.

In order to separate the n o n - c r i t i c a l part from the c r i t i c a l parts of an integrationreglon we use a neutralisor:

This i s a function N(u) defined by for 0 g u g 1 holds N(u) = 1, for l+e g u holds N(u) = 0,

where e < l denotes a fixed small p o s i t i v e number: moreover N(u) i s Indefinitely differentlable and a l l derivatives of N(u) equal zero for u = 1 and u = l+e e.g. for 1 g u g l+e we take

N(u) = {1 + e - ( " - l ) " - ( " - l - ^ ) " ' } " .

Consider now the Integral

I = II g(a),x.y)ei"ï'(".x.J^)dx dy

D

where D is a twodimensional region of the real (x,y) plane and

cp(uy,x,y) Is a real function of to, x. and y.

Be T a set formed by points or boundary points of D. Let U(<«), x.y) be a real function of «o, x. and y such that 0 < U g 1 for points (x,y) belonging to T and Ü > 1 for all points

belong-ing to D - T = S: moreover, let U(a).x,y) be indefinitely diffe-rentlable with respect to x and to y for all values of ü > 0 . Suppose that

{l-N(U)}g(co.x.y)eiV("'.*.y) is asymptotically analytic on S and

I ~ ƒƒ N(ü)g(a),x.y)e^*«'(<*'*'J^)dx dy = I,

D ' is not asymptotically analytic. Then T is called a critical set

and Ij is called the asymptotic residue of g(üd, x,y)e^'^("''*"^^ on D on the critical set T.

This asymptotic residue is asymptotically independent of the choice of the neutralisor, of the choice of the function U and independent of the choice of the number e.

Let namely N(u) and N*(u) be two neutralisors and let U and U* be two functions having the mentioned properties.with respect to the critical set T.

Moreover, let

{l-N(U)}g(u>.x,y)eiV(«".*.y) and {l-N*(U*)}g(to. x,y)i^('^.^.y) be asymptotically analytic on S. Then it follows that

//N(ü)g((o.x,y)el^<'*».x..y)dx dy ~//N*(ü*)g(co.x,y)ei^(<^. ^.•y)dx dy D D The asymptotic behaviour of the asymptotic residue is indepen-dent of the choice of the fixed number e. If N(ü) is a neutra-lisor and 5 a fixed positiv-e number <e, then N ( ^ —) is again

1+6'

a neutralisor and since the asymptotic behaviour of the residue is independent of the choice of the neutralisor, it is also in-dependent of the choice of the fixed number e.

It has to be shown that at least one function ü((o, u, v) exist but this will be done for every special problem.

It may be remarked that it is possible that the whole set D is a critical set; then one cannot find a subset T of D and a function U(to, x,y) such that

{l-N(U)}g(oo,x,y)ei^('^'*.-y) is asymptotically analytic on D.

It is possible (and in general this will h ^ p e n ) that D con-tains two or more critical sets Tj. Then one uses the method

described above for each critical set separately, with another function U(<o.x,y).

Let us have n critical sets Tx T„ and, therefore, n func-tions ü|(«,x,y),...,ü„(w,x.y).

Then the function [ U {l-N(üi)}]g(w.x.y)e*'P<**>*'y^ is

asymp-totically analytic on D and

I ~ 2 ƒƒ N(üi)g(<o.x.y)e^^("'*'y)dx dy.

i = l

If for w -. 00 a given critical set tends to one point we call this a critical point and if the given critical set tends to a curve we call this a critical curve.

5. Example

Let D be the region of the real (x,y) plane defined by O ^ y g x . O ^ x g l . Moreover, let be g(w,x.y) = (l+oD*'xy^) •*

and cp(w,x,y) = ay(x+y). We use as new coordinates

u = uy(x+y). (5.1) P = (x-l)/(y-l). (5.1) The equlphase lines are the hyperbolae

y» + xy = u.(o'*

and t h e s e g n e n t 0 < x < 1 of t h e x - a x i s c o r r e s p o n d s t o t h e equlphase l i n e u = 0. The p o i n t x = y = 1 i s a v e r t e x .

The second v a r i a b l e p i s equal to zero for x = l and unity for X = y, the two segments of the boundary.

The c r i t i c a l s e t s are the x - a x i s and the v e r t e x x = y = l . Hence we choose t h e functions

Ui = u.to"^ (5.2) Uj = (o">^+K3(l-y) + ( l - X ) } (5.2)

Let Tj be the set along the x-axis where upon Uj < 1 and let Tj be the set with the vertex (1.1) where upon Uj g 1.

le now solve for y and x

p-l+{(l-p)»+4—(P+1)}^

p{(l-P)»+4 — (P+l)}*'-P-p»+2

X = ^ . (5.3)

2(p+l)

so that fay differentiating these functions with respect to u we get a factor to"*.

Moreover

^ {x+3y-2y»-2xj}

" ' ^ ' - "'^" = ( , . I ) . ^

being only equal to zero for y = x =

Let S be defined by

X+d 0.

S = D - Ti - Tj.

By means of the transformation (5.1), S is transfomed into a

region G: 0 < p < 1, u)'^ < u g 2m-u)^.

The function

[l-N(üi)][l-N(Ua)](y-l)» w(x+3y-2y»-2xy) (l+ü)^y»x)

satisfies the conditions of theorem A and so we have /ƒ [l-N(U,)] [l-N(Uj)] - ^ — j ^ ^ ^ dx dy ~ 0.

Therefore the integral i s equal to the sum of the asymptotic residue of (l+ti#*xy2)-iei"'y(x+y) on D along the c r i t i c a l x-axis and the asymptotic residue of that function on D at the vertex

( l . l ) .

6. Some theorems of Prof, van der Oorput

Theorem C: Be s a fixed number with Re s > 0 and suppose

Re T ^ 0 and |T:| i ceo®, where c and 9 denote suitably chosen fixed positive numbers.

Then / ^ x » - i e - « d x - r ( s ) 0

if on the Integration path (0,i) holds - u < arg x < TI. For each fixed integer m ^ 0 holds:

f x»-i(log x)"e-'dx ~ r<») (s), 0

where ?<-) (s) = { ^ r(x)}

This theorem has been proved by J.G.van der Corput in "On t h e method of c r i t i c a l p o i n t s " , Indag.Math.Vol .X, F a s c . 3 , 1948. Furthermore Prof, van der Corput has proved:

Theorem D: Be s a fixed number Re s > Q and T a r e a l number > l+e, e being a s u i t a b l y chosen fixed p o s i t i v e number.

T; •" i s Then ƒ N(x)x''-'e'''^Mx - r ( s ) .to"'e 2 .

0

Por each fixed i n t e g e r m g 0 holds:

71 I s

r N(x)x»-i (log X)- e''^ dx - {I(s)e_2_^C> ^

0 OJ* 7t i s 71 I x

Where {mi^} . {^ ^W^ ^ }

I . At f i r s t the asymptotic r e s i d u e of (l+oj^^xy^)-le-'-"'-^^^''''^) on D along t h e c r i t i c a l x - a x i s i s c a l c u l a t e d . Put 9 = 1 + e and

introduce in the f i r s t place the new v a r i a b l e s P = y . x - \ z = y(x+y), (7.1) U = üji-'^z = ooi-V(x+y). Prom (7.1) X = {p-i(p+l) " ^ z P (7.2) y = { p ( P + i ) ' ^ z } « .

If E is the subset of D whereupon 0 ^ Uj g 9 holds then E con-tains the X-axis as boundary line. The boundary of E is recti-fiable:

i. this assumption is obvious for the part of the x-axis be-tween zero and one;

ii. be Xg a given value of x in the interval 0 g x ^ 1. Then the upper side of E is given by

y = Vi {-x„ + (x„2 + 4910^-^)"}:

the function y occurring in this representation is differentla-ble in the interval 0 < Xo ^ 1 and by means of theorem 18 of

[2]; one sees that this part of the boundary is also

rectifia-ble, or it is possible that the upperside of E is given by

X = y > 0 and 0 g y» ^ 2"'&*)^'^ and, therefore, it is

ill. the last boundary line of E is given by

X = 1 0 ^ y ^ 2'H-l+(l+4eu)'''*)''} which is also rectlfiable.

Prom the relations (7.2) it follows that x and y are continu-ously differentlable with respect to p and to z and, therefore, also with respect to p and U, if p > 0 and U > 0.

Be F the image of E lying in the (p.U) plane. On the interior of F the functions x and y are continuously differentlable with respect to p and U. According to theorem 17 of [2] there holds

ƒƒ N(U)(l+o^xy2)-iei'^J^(''+J') dx dy =

E

= oü'^-»2-' II N(U)(l+co«xy2)-iei'^p-i(p+l)-i dp dU (7.3)

F

In the last integral U runs through the interval 0 < U < 9; for a fixed positive value of U the variable p runs through the interval (Po.l), where p^ is the continuous solution of the equation

z = Uoo'^-i = p(p+l)

tending to zero as 00 increases Indefinitely. According to theo-rem 8 of [2] there exists one and only one such solution and this solution can be expanded for small positive values of z into the form

p^ = z(l - z + 2z2 . . . ) .

We now show, that by a suitable choice of X the expansion {l+COÏ^Xyï}-! = { l + ( J ' z 3 / 2 p l / 2 ( p + l ) - 3 / 2 } - l =

= 1 Cjjl) t o " / V ' » z 3 » / » ( p + l ) - 3 " / 2 (7,4)

n = 0

is valid throughout the region F and, moreover, that this ex-pansion is an asymptotic exex-pansion for sufficiently large po-sitive values of w for each point of P.

Since on F

Uj = u.oo"^ = z.u)!"^ < 9 we have [zj < 9to^-* and

0 < p„ < - ^ < z < 900^^"* p+1

Consequently

| t ^ / 2 p n / 2 2 3 n / 3 ( p + i ) - 3 n / 2 | ^ 0(U^"/2p.n/2(i+p) - 3n/2^X3n/2-n)

For X < 2/3-Ti, where T) > 0, the series is seen to be (asympto-tically) convergent, and also uniformly convergent.

Substitution of the relation (7.4) into the integral occurring in the right hand side of equation (7.3) gives for the wanted asymptotic residue

N 9 7k.

2 ('1) w^/z ƒ N(U)e''« "j(z)dU + 0(oo-3NV2)

n=0 0

Where J(Z) = ƒ % ( " - 2 ) / 2 ( p + l ) - ( 3 n + 2)/223n/2(jp

From this result it follows that we only need to expand asymp-totically for large positive values of oo the integral

ƒ N(U) e*"»^ J(z)du (7.5) 0

For this we consider: 1: the integral J(z).

Since the function p("-2)/2(p+i)-(3n+2)/2 g^jj jjg expanded for small positive values of p and since p ^ can be expanded for small positive values of z, as is shown above, the conditions of theorem 5 of [2] are satisfied. From this theorem we deduce that J(z) can be expanded for small positive values of z with a logarithmical terra (that means can be expanded for large posi-tive values of oo).

11: Since the integrals of the form

J® N(ü)ei<^''"Ü*dU and /® N(U)ei'^'^"lf log U dU with a > -1 0 0

exist and can be majorized by fixed numerical constants (see theorem D) it appears that substitution of the expansion (with the exception of logarithms) for J(z) into (7.5) gives an asymp-totic expansion of the double Integral.

Summation of all the partial asymptotic sums gives the wanted asymptotic sum for the asymptotic residue of (l+oi^xy^)-^e^'^'^**''^ on D along the critical x-axis.

First one has to think on the parameter X, we have introduced during the treatment of the problem of asymptotic convergence. It is obvious that the asymptotic sum is independent of X; this parameter is only needed to prove the asymptotic convergence of the asymptotic sum. This sum is of the form

n>

N -q_-Xr IJ -q'-Xr' -Q». log w 2 a„ü> ^ " + 2 b„w " " + 0(0) ^ )

(1=0 n = 0

*n. ^n' %• *ïn« ^n' ^'n ""d ft„ denoting fixed real numbers, q qj,, r„, r^, Q,, tending to infinity with n. This sum has to be independent of X, so that all terms depending on X have to

va-nish. During the calculation of a few terms we may choose X, and since the problem of the convergence of the asymptotic sum

plays no role during this calculation, we can choose even values of X > -| - c. So we can choose e.g. X = l.

Another reason for this choice can be the homogenity in X; the true variable is z instead of U and so we find in each term of the sum after each factor oo^ a factor Ü. ^plying theorem D we see that the result of the calculation for each term does not depend on X, so that we may choose X = i.

The wanted terms are readily computed: Abbreviating

T

e i * ^ <p(ü)dU

0

instead of f N(U)ei"*^ cp(U)dU

0

one has by choosing X^ i:

H T dz ƒ dp ei'"Z{l+a^p'«(p+l)-'/»z'/»}-*p-»(P+l)-' 0+ 1 = %ƒ e^^ dz I p-»(p+l)-» dp 0 P„ ._ ff^ ƒ* e^^z z^/» dz ƒ p-«(p+l)-5''» dp + . . . 2 0 Po 0+

= 14 ƒ e^^ dz {-log z - log 2 • 2z - 3z» . . . }

0

_ W^ ƒ* giwZ 23/2 dz {5-2»' - 22" + 4- z'^» . . .}

2 0

IE: Ttl 7d ^ = ^ ^ ^ 00-1 _ ^(i).e^.oo-i+e2.iogto.a)-i-14 1og 2 . e ^ . w - i

57d + e'"^ w-2 - 5 2*'l2-^ e 4 r(5/2) co'^ 37d + 2 e 2^ (O-*/2 37d - 3 e 2 u)-^ + 7 00' 7/2

II. The asymptotic residue of the function (l+ü^xy2)-ie^'**"5'(*+J^) on D at the point (1.1) can be expanded in a similar manner. Putting

y = 1-vl v = 1-yj 0 ^ x ^ y ^ 1, X = l-u) u = 1-x ) 0 $ u ^ v ^ 1,

it has been proved that the asymptotic residue mentioned is equal to:

e2ito ƒƒ e-i'^CSv+u) ei'^(v»+vu){i^.^^.2v.u+2vu+v2-v2u)}-idu dv

E'

Now the function UJ(3V+U) is used as cp((o, x,y) and ei'^(v^+vu){Uw«(l-2v-u+2vu+v2-uv2)}-i as g(u),x.y).

As before it can be shown that for each point of E' this last function can be expanded asymptotically if oo - oo.

As under I one can apply theorem 17 of [2] and if 2 = Uoo^'*, this yields .^2X-2 e2iw ƒ e-i'^"J(U)U dU 0 where 1 gloj(v2+vu) ^p "^^"^ " °^(3+P)»{l+to^+to^(--g|^+ ^P^'^f - ^ ^ 3 ) } ^ ^ 3+P (3+p)2 (3+p)3' Expanding e^"'^^ '''^^^ in a power series and expanding the last

factor of

(Uco«)-»{l + ^ ^ (_2z+p_z^ 2 p z 2 . z 2 _ _ p z 3 _ . ^ - x (l+to«) 3+p (3+P) 2 (3+P) 3

by means of the binomial theorem, one gets the desired asympto-tic expansion; the proof can be given in a manner similar to that of I. The computation gives

12-l,e2"*.u)-»(l+«>»)-» + 0(a)-''/2).

This asymptotic residue is apparently of much smaller order than the other one.

8. Generalisation to the complex case

We have mentioned already that the factor i of the exponent e'"

can be replaced by +1; the determination of the critical and the non-critical regions is made In the same manner.

We wish however, to generalise the problem of the determination of critical regions for the case of a complex function u(ü:kx,y).

In this case we transform a part of the real (x,y) plane into a region A of a surface in the twodimensional complex (u,v) space. Since theorem A does not use properties connected with the ana-lytlclty of the functions u, v and A(oi), u, v), but only the prin-ciple of partial Integration, this theorem holds also in this case.

The transformation is a more difficult problem. In fact we transform a complex (x,y) space into a complex (u,v) space by means of

u = u(w,x,y) V = v(a),x,y).

We have chosen this transformation in section 3 such that it is one-valued and that the Jacobian is unequal to zero for real x

and y. According to [l8], this transformation can be inverted,

if u and v are analytic functions of x and y and the Jacobian *

0-Then we have

X = X(t«). u.v) y = y(w.u.v)

also an analytic and one-valued mapping. Therefore the mapping i s topological, so that the number of intersections of two cur-ves i s invariant.

If we write for a moment

,x = Xj + i x j ; u = Ui + iuj; y = yj + i y j ; v = Vj + i v j ;

we hare transformed the plane x, = y, = 0 into a surface

u - u(o),Xi,yi), V = v((ü,X2.yj).

The definition of the integral

I = II h(oo,x.y)ei"('*''^'y> dx dy,

D

can be given in the ordinary way (see e.g. [l9] and [20]) but also as a surface integral in the 4-dimensional (Xj,X2,yi,y2) space.

Let A be the image of D in the surface of the (u,v) space and let

Ii = II A(a),u, v)e'" du dv A

be defined in the same way. Choosing

A(oo.u,v) = h(oü.x,y) | i j ^

we can prove that Ij = Ij; this proof is given in the same way as in [20].

Returning to our non-critical regions, we define these regions in the same way as In the case of real u function. Since we found"it not necessary to generalise the notion of neutralisor for complex arguments (we can always choose a positive function on the critical regions) the determination of critical regions for the case of a complex phase function follows the same lines as above. Section 4 can be used without any change in the com-plex case.

9. Expandable and smooth curves

A curve of the real twodimensional (x,y) plane starting from the initial point (a,6), where a and 6 denote finite fixed real numbers, is called "an expandable curve in the neighbourhood of

(a,b)" if one of the following two conditions is satisfied:

i: it can be represented by y = cp(x) with cp(a) = 6; moreover, if in every arbitrarily small neighbourhood of (a,6),the para-meter X has values larger than a, the function 9(x) can be expanded for small positive values of x-a and if in every arbi-trarily small neighbourhood of (a,b) the parameter x has values smaller than a, the function cp(x) can be expanded for small positive values of a-x; also the case y = 6 is admitted;

ii: it can be represented by x = \\i{y) with \\i(b) = a; moreover, if in every arbitrarily small neighbourhood of (a,6) the variable y has values larger than 6, then the function \\i(y) can be expanded for small positive values of y-t and if in every

arbitrarily small neighbourhood of (a,6) the variable y has values smaller than fa, then the function ^(y) can be expanded for small positive values of 6-y; also the case x = a is ad-mitted.

It may be noticed that the expansions mentioned above are not

asymptotic expansions and that the functions c()(x) and \\i(y) are independent of w.

For instance the curve represented by the equation y = l-(l-x)^ is an expandable curve in the neighbourhood of the origin. With the aid of Theorem 18 of M R U one deduces the result that in every suitably small chosen fixed neighbourhood of the point (a,6) the expandable curve possesses a finite length.

A curve is called smooth if it can be represented by the equa-tions x = cp(t) and y = \\i(t), where t runs through a real inter-val a ^ t < b and where the real functions cp(t) and \y(t) are indefinitely differentlable such that {cp'(t)}2 + {v|j'(t)}2 > 0. So one has the curve R represented by

X = t y = e ^ ,

in a smooth curve. However, it is not an expandable curve In the neighbourhood of the origin, since

y «> e *

can not be expanded for small positive values of x.

At the other side the curve R expandable in the neighbourhood of (a, 6) can be represented by an equation of the form y = cp(x) or of the form x = t(y) and is therefore, in a suitably chosen neighbourhood of the point (a.fa) with exception perhaps of (a, 6) itself, a smooth curve; with the aid of lemma 2 of M R U , one shows that y = q>(t) or x = f(t) can be differentiated in-definitely for small positive values of t. so that

X = t y = cp(t) pr y = t x = »i/(t)

satisfy the conditions of a smooth curve; the condition l+(cp'(t))2 > 0 and l+(v|/'(t))2 > 0 is also fullfilled.

^ ^ W j ^ ^ i i j l i i i Pim j t ^ K

-UJ--C h a p t e r II

H O M O G E N E O U S F U N C T I O N S

Introduction

The purpose of this chapter is to examine the asymptotic behaviour of an integral I with the integrand g(üj, x,y)ei"^P(x,y)

extended over a certain region D; the function P(x,y) is a real homogeneous function of the form

I c, x"^ y'^-^ , (1.1)

h = 0

where the exponents a^^ > 0. .-ih > 0, but a^ + ^^ = '^ > 0 and where the coefficients Ch are real.

The calculus of asymptotic residues is based on the fact that under general conditions the asymptotic behaviour of a simple or multiple integral is uniquely defined by the behaviour of the integrand and the form of the integration region in the neighbourhood of certain points, curves and so on. These points, curves are called critical points, critical curves and so on. In the theory of double integrals there are critical points and critical curves. A critical point or a critical curve is char-acterized by a certain irregularity in its neighbourhood in the behaviour of the integrand or in the form of the integra-tion regions, for instance, a vertex of this region is always a critical point.

In this thesis three different kinds of critical points are considered; these points are caracterized by a singularity of the function P(x,y):

1. A point (x,y) is a critical point of the first kind with respect to P(x,y), if and only if at least one of the partial derivatives of P(x,y) with respect to x and y, does not exist at that point. This involves, that in the (u,v) plane the derivative — — d o e s not exist for a certain k.

2. A point (x,y) is a critical point of the second kind with respect to P(x,y), if and only if, at that point all partial derivatives of P(x,y) with respect to x and y exist and both derivatives of the first order are equal to zero.

3. A boundary point (x,y) of D is a critical point of the third kind with respect to P(x,y), if and only if, all partial derivatives of P(x,y) with respect to x and y exist at that point, the derivative of P(x,y) taken along the boundary curve of D at the point (x,y) exists and is equal to zero.

Some properties of critical lines 1. On the critical line of the second kind

Theorem 1. If a homogeneous function P(x,y) Is equal to zero

at a point not coinciding with the origin x = y = 0, then it is equal to zero at each point of the straight line passing through the origin and the point mentioned above.

Proof: Be (a,b) the point not coinciding with the origin such

that P(a,b) = 0 . A point lying on the line through (a,b) and the origin can be represented by x = Xa, y = Xb, where the para-meter X lies in the Interval 0 < X < oo,

From (1.1) it follows that

P(Xa,Xb) = xvp(a,b) = 0 and the theorem is proved.

Since the partial derivatives with respect to x and to y of a homogeneous function are again homogeneous, one has the result:

If both partial derivatives of the first order of a homogeneous function are equal to zero at a point not coinciding with the origin, then both these derivatives are equal to zero at each point of the straight line passing through the origin and the point mentioned.

From Euler's rule

vP(x,y) = X TT- P(x,y) + y-5-P(x,y)

dx ay

one finds t h a t a t each point of t h i s l i n e the homogeneous func-tion i t s e l f i s also equal to zero.

This conclusion can be formulated as follows:

I f a p o i n t (x.y) not coinciding with the o r i g i n i s a c r i t i c a l point of the second kind with respect to P ( x , y ) , then a l l p o i n t s belonging to the l i n e p a s s i n g through the o r i g i n and t h a t p o i n t (x,y) are c r i t i c a l p o i n t s of the second kind with r e s p e c t t o P ( x , y ) ; t h a t l i n e i s t h e r e f o r e a c r i t i c a l l i n e . A curve i s a c r i t i c a l curve with respect to P(x,y) if a l l points of t h i s curve are c r i t i c a l points with respect to P ( x , y ) .

2. On the critical line of the third kind

Iheoren 2. Let P(x,y) be the homogeneous real function defined

by (1.1). Be R a smooth curve starting from the origin and hav-ing the origin as a simple point. If at each point (9(t),vj/(t)) of R, not belonging to one of the coordinate axes, the relation

^P(q)(t),vi<(t)) = 0 (2.1) holds, then R lies on a halfline starting from the origin.

Proof: Consider first the case that each point of R lies on one

of the coordinate axes. The origin is the initial point of R at the same time a simple point of R, so that R lies on one and only one of the four horizontal or vertical hal flines starting from the origin. Therefore, it may be supposed that R contains at least one point which does not lie on one of the coordinate axes.

First step: One proves first that P(cp(t), i(/(t)) is

differentla-ble and that the derivative of P(cp(t), ^;(t)) is equal to zero at each point x where cp(x).\|/(T) = o.

As R contains at least one point that does not lie on one of the coordinate axes, then there exists an interval a < t < b, Where (2.1) holds.

Without loss of generality one may suppose fCc) = 0, so that «PC^) 4 0. If >f(t) /^ 0 at each point t j^ x in the neighbourhood of X, then (2.1) holds in the neighbourhood of T at both sides of 'T, the function P(9(t), y(t)) has, therefore, a constant value at both sides of T in the neighbourhood of -r, since P(cp(t),ij/(t)) is continuous at t = x, these two constant values are equal and (2.1) also holds for t = x.

It is therefore sufficient to consider the case that each neigh-bourhood of X cobtains infinitely many points t/x where \)/(t)=0. Then the function >|/(t) and all its derivatives are equal to zero at the point t = x.

In proving that P(9(t).y(t)) is differentlable at the point

t = X. one has to show that

P(y(t).,M'(t)) - P(<y('c).H'(^)) , t-x

_ P(y(t),v|/(t)) - P((p(^).Q) ^ P(<p(t).0) - P(cp(x).0) ^2.2) t-x t-x tends to a f i n i t e limit as t ;^ x tM)proaches x. Since i|/(t) and a l l i t s derivatives are equal to zero at t = t , one has for each positive

YH-Lin - ^ = 0; (2.3) t-x t - x

furthermore P(9(t),0) - P(cp(x),0) = 0 and from

P(cp(t).t(t)) - P(cp(t).0) _ " W t ^ ^ ^ h ^'^^^^^ "

it follows by (2.3) that the righthand side of (2.2) tends to zero as t approaches x. It is therefore sufficient to show that each Yh i^ positive.

Assume for a moment that at least one of the exponents Yh is equal to zero. Then the sum representing the function P(x,y) has one term of the form cx^ with c ?^ 0, so that

P(y(t).0) - P(cp(x),0) . c[(cp(t))v - (cp(x))^]

t - x t - x This expression has the limit cv(cp(x)) ^"^cp'(x) as t ?^ x tends to X. Since »|/(x) = ^''(x) = 0 it follows that cp(x) and cp'(x) and therefore also the obtained derivative is different from zero. In this way it has been proved in this case

(I) that it would be possible to find in the interval a < t < b at least one point x with \j/(x) = 0,

(II) that at each point x with a < x < b and v(T) = 0 the deriva-tive o f P(cp(t), ij/(t)) exists and is different from zero. Consider an arbitrary p o i n t x' in the interval a < x' < b with v)/(x') = 0. A c c o r d i n g to t h e r e s t r i c t i o n s . I n t r o d u c e d at the beginning of the first step, this interval contains a point t „

with f(to) / 0. The interval contains, therefore, a point x with ^/(x) = 0, lying at the same side of to as x' such that its distance to to is the smallest possible. In the interval X < t < t^ or t„ < t < X one has \)/(t) j^ 0, so that (2.1) holds

and that P(cp(t). v|/(t)) has a constant value in x < t < to or in

to < t < X.

Prom the continuity of P(cp(t), v|;(t)) one deduces again that P(9(t),v|/(t)) has a constant value in x < t < to or in to < t < x; therefore the derivative of P(cp(t), >f(t)) at t = x is equal to zero contrary to the result found above that derivative is unequal to zero. So the proof of the first step is complete.

Second step: According to step 1 the relation (2-1) holds in

the whole interval a < t < b, so that P(cp(t),((/(t)) is constant in that interval. If t tends to a, then cp(t) and ^/(t) tends to zero so that also P(cp(t),f(t)) tends to zero. Therefore at each point of R the equation P(x,y) = 0 is satisfied.

start-ing from the origin. Consequently each point (x.y) of R belongs to one of these hal flines. From the fact that the origin is the initial point and at the same time a simple point of R, it fol-lows by the same arguments as above that R lies on one and only one of these hal flines.

The critical point (homogeneous) 3. A theorem

Theorem 3. Be P ( x , y ) a r e a l homogenous f u n c t i o n of p o s i t i v e degree v of the form

" 3 Y P(x.y) = 2 ChX^V \

h = 0

where the coefficients Ch are fixed real numbers ^ 0 , H is a non negative integer and the exponents satisfy €he relations Ph ^ 0, Yh ^ 0 and ph + Yh = V > 0.

Let R and R' be two curves expandable in the neighbourhood of the origin; it is assumed that on these curves the variable x has only non negative real values.

Be S(et,X) the set bounded by R and R', whereupon x > 0 and |P(x.y) I < Qu?--^,

with X < 1 and 9 suitably small chosen fixed positive numbers. On E(9, X) and Its boundary, only the origin is a critical point

with respect to orf'(x,y) and only at the origin the value of

P(x,y) is equal to zero.

Let a and b be two real numbers larger than -1.

Assertion: The asymptotic residue I o

ƒ / N(oüi->-P(x.y)) X" y" ei"^(''y) dx dy E(9,X)

can be expanded asymptotically for large positive values of to.

Proof: Without loss of generality one can suppose P(x,y)

posi-tive everywhere on E(9, X), the origin excepted, for the case P(x,y) negative on E(9, X) can be dealt with in the same way.

First step: Introduction of new integration variables.

It is possible to choose the value of oo so l-arge that each point (x,y) of E(9, X) has at most a distance r to the origin and that the functions representing the boundary curves can be expanded on the interval 0 < x < r resp. 0 < y < r. Furthermore it is possible to maie r so small that the boundary of E(9, X)

not coinciding with the o r i g i n and having a distance to the origin smaller than r, e i t h e r has no point in common with the y-axis or the boundary of E(9, X) l i e s completely on the y-axis. As new integrationvariables are introduced

U = ooi-'^P(x.y), (3.1) P = y . x - ^ (3.1) Be P the image of E(e,X) in the (U.p) pLane.

Since P(x,y) is a homogeneous function of positive degree v one can put

P(x.y) = x^Q(p), where Q(p) denotes a function of p only. Also

U = wi->^x^Q(p).

The Jacobian (^ P) has the absolute value vtt)'"'^x^"2Q(p) and is, therefore, on E(9, X) with the exception of the y-axis, finite and positive, since

x^-»Q(p) = x-2p(x,y) > 0. Now the following conditions are satisfied:

i: the boundary of E is rectlfiable;

ii: from (3.1) one deduces

X = to(>--l)-^"'ü^''Q-^'\p) > 0, (3-2X y = u,(X-l).v-ipüV-iQ-v-i(p)^ (3.2) so that one has an (l.l)-correspondance between the points of

E(e,X) and P;

Hi: on the interior of P the derivatives of x and y with res-pect to U and p exist and they are continuous;

iv: the Jacobian is continually positive of the interior of P. Following theorem 17 of [2] wplied with r = 1 and z coinciding with the origin the integral

^.ljX-l)2v-i ƒƒ N(ü)x«y''Q-»^'\p)U-i+»^''ei'^" dU dp (3.3) F

e x i s t s and i s equal to l o .

Second step: In t h i s l a s t integral ü runs through the interval 0 | U g 9 . If U i s given, p runs through an interval ^(u)a)^Ti(u),

where t h e extrerae v a l u e s 5(u) and T)(U) depend on u = to^'^U. In t h i s p a r t of the proof ^(u) and TI(U) are determined as func-t i o n s of u and i func-t i s proved func-t h a func-t func-these funcfunc-tions can be expand-ed asymptotically for small p o s i t i v e values of u.

If the curve R represented by y = <p(x) belongs to the boundary of E, then one considers the following l i n e s :

Since R i s expandable in the neighbourhood of the o r i g i n , the function cp(x) can be expanded for small p o s i t i v e values of x. The case t h a t the curve R i s r e p r e s e n t e d by x = \|/(y) can be d e a l t with in the same way.

The function P(x,y) i s a sum of a f i n i t e number of terras of the form ChxP^yYh, wherein the exponents Ph = 0 ^"d Yh = 0 - so t h a t P(x.cp(x)) can be expanded too for small p o s i t i v e v a l u e s of X. So P ( x , 9 ( x ) ) can be expanded a s y m p t o t i c a l l y for small p o s i t i v e values of x, a l l exponents occurring in t h a t expansion are p o s i t i v e and there are c o e f f i c i e n t s unequal to zero. Using lemma 2 of [2] the d e r i v a t i v e — P(x,cp(x)) a l s o e x i s t s , where

dx

a l l o c c u r r i n g exponents are more than - 1 ; so t h e r e e x i s t s a number p such t h a t

{^ P(x,cp(x))}-* = O(x-P)

dx

for small positive values of x.

One applies now theorem 9 of [2] with f(x) = P(x.cp(x)) that satisfies also the condition f(x) - 0 if x - 0.

Therefore, the equation

f(x) = u

possesses one and only one continuous s o l u t i o n Xj(u), tending to zero as u approaches zero; t h i s s o l u t i o n can be expanded a s y m p t o t i c a l l y for small p o s i t i v e v a l u e s of u. The function

^(u) = 9 ( x , ( u ) ) . x [ i ( u ) (3.4)

can also be expanded asymptotically for small positive values of u.

In the curve R' represented by y = H'<x) belongs to the boundary of E(9,X), one has on an analogous manner:

The equation P(x, (j/(x)) = u possesses one and only one conti-nuous solution X2(u) tending to zero as u - 0; this solution can be expanded asymptotically for small positive values of n; the function TI(U) defined by

can be expanded asymptotically for .small p o s i t i v e values of u. Without l o s s of g e n e r a l i t y one may suppose ^(u) 5 i\(u) for suf-f i c i e n t l y small p o s i t i v e values osuf-f u.

If the p o s i t i v e y - a x i s belongs to E(0. x) one puts r|(u) = oo and i f the negative y - a x i s belongs to E(0, X) one puts r,(u) = -m. Third step: End of the proof.

Putting

X = v"^ ( 2 + a + b ) ,

that is a positive number since a and b are more than -1. one has

lo -^' v-1 ^0.-i)x j ' ^ N(U)ei"'''" u-i+x j(U)du, (3.5)

0

Where J(U) = j p'' Q'^ (P)dp. (3.5) According to theorem 3 of [2] the integral J(u) can be expanded

for small positive values of u = UcoX-i, since p''Q-'^(p), ^(u) and Ti(u) satisfy the conditions of that theorem. So one can expand J(u) asymptotically for large positive values of M and, therefore, also the asymptotic residue l^; one has

asymptoti-cally P_^ /'N(U)e-''-"U-i-^ M ^ m ^

0 to '^

where c denotes a fixed real number > 0, following theorem D of Chapter I.

Remark: i: If the two boundary curves R and R ' are hal flines

through the origin, one has

Tdx ±

-^r-lo - v-'tAj-'^J r(x) e ^ . (3.6)

where J = ƒ'' p'' Q-'^(p)dp. (3.6) 5 and n being fixed real numbers and the plus or minus sign is

chosen according to the fact P(x,y) g 0 on E(ö.X) or P(x,y) i 0 on E(9, X ) . This answer is independent of X as could be expect-ed: it is namely possible to choose X arbitrarily small, while the answer has to rest the same.

ii: This theorem also holds if E(0,X) contains points with x < 0 and P(x,y) is real on E(e, X); one can give a similar proof for the part of E(e, X), where x < 0, separately.

4. Problem 1

Let D be the region of the (x.y) plane given by 0 < x2 + y2 < 2

We want to determine the behaviour of the asymptotic residue II N(ooi-\x2 + y2)2)(i+<,,x2y'«)-ie'"J(''^+y^)^ dx dy.

D

S o l u t i o n : The f u n c t i o n s

(oP(x.y) = to(x2 + y2)2

g ( w , x . y ) = ( 1 + tüx2y^)-i

have a high degree of symmetry since x can be replaced by -x and y by -y without any change in the values of the functions. The required asymptotic residue is also equal to four times the asymptotic residue of (1 + a)x2y*) "ie'"^(x2 + y2^2 ^t the origin on the quadrant x ^ 0. y ^ 0 .

Using the development

GO

g('",X,y) = 2 (-l)''u)*'x2hy*»>

h=0

and using theorem 3 of this chapter, we have

lo "^ 2 Jhr{(3h+l)/2}e-(''-')^'/'»oü-(h+i)/2,

h = 0

where Jh represents the integral

Jh = r p**'(l+p2)-3h-i dp. 0

By means of the substitution p2 = v one obtains

Jh = 14 ƒ " v2h-«(i+v)-3*>-i dv = iir(2h+^)r(h+^){r(3h+l)}-^ 0

and so

I„~2-i7f« 2 r(h+i4)r(2h+i4){r((3h+2)/2)}-'e-<''-i)"'/*2-3hu)-(h+i)/:

h=0

The conditions of theorem 3 of Chapter III are satisfied as the reader will check easily.

5. Problem 2

ƒƒ (l+oaV)-'ei"»«J'<«+') dx dy,

D

Where the region D is contained between the two arcs of the

circles (x-l)2+y2 = i and x2+(y-i)2 = i that begin at the origin and finish at the point (1.1).

Solution: The equation of a circle passing through the points

(0,0) and (1,1) is given by

ux2 - 2x + uy2 + 2(l-u)y = 0, where u is an arbitrarily parameter.

First let us consider only Dj, the part of D contained between the circle (x-l)2 + y2 = i and the line y = x; the treatment for the other part of D is sigiilar. The parameter u has values in the interval 0 $ u ^ 1.

We choose the new variables

Ü = OL>xy(x+y); the f i r s t one being already used above. The Jacobian i s equal to

2oo{(x+y)2+2xy}. (x-y> "^ being equal to i n f i n i t y for x = y .

One c a l c u l a t e s

dx u ' dU to(3x2u2-l2ux+4)

the roots of the polynomial 3x2u-12ux+4 being > 1, so t h a t dif-f e r e n t i a t i n g with r e s p e c t t o U g i v e s a dif-f a c t o r aj^i. The same holds for y .

Let be Uj = uj^'^xy(x+y), neighbourhood of the o r i g i n -Uj = 3 ( o ' " \ l - x + l - y ) , -neighbourhood of x=y=l-X being a p o s i t i v e s u i t a b l y small chosen number < 1.

The c o n d i t i o n s of theorem A of Chapter I a r e s a t i s f i e d by the i n t e g r a l

-LII {i-N(uo}{i-N(U2)} , 'y-']' Z'"o .^

2ü) „- (l+<jox^y*){(x+y)2+2xy)} °i

where D ( is the region o < U < 2od, o < u < 1, being the image of D in the (U,u) plane.

So one only needs to determine the asymptotic residues at the origin and at the point (1,1).

i: the origin: One puts

u = xy(x+y) = x'p(p+l), y = px,

1 .± .i

80 X = u^p 3(l+p) 3. The equation of the first boundary curve is

y = 1 - (l-x2)» = ii x2 + ± x* + J^ x« ... .

The function Xj(u) i s determined as the continuous s o l u t i o n of the equation

(J4 x3 + 1 x 5 + jL x7 . . . ) ( x + 14 x2 + I x * + i x « . . . ) = u.

that tends to zero with u.

Prom -1 X* + 1- x' + 1 x* + 1 x^ ... = u

* 4 o 8

one obtains x .11 J. . Ü 1

Xi(u) = (2u)* (I-2" •• u* - 2' ^ u' ...)

1 1 1 9

~ "J 7 "?"

and y,(u) = 1 - (l - x2(u)) = 2 u - 2 * 2'^ u ... .

y (u) ' ' - — ^ T

Further e(u) =-i^-L = 2 * u^ - 2"^ u' + 31.2 * \i* ...

Xi(u)

Xi(u) 1 .1 -2i. 1

and Ti(u) = = 2* u *\ 2'^ - 29.2 ^ u* ... .

yi(u)

The following expansion holds:

(l+tüX^y*)-' = 2 (-l)''üD»'x3hy*h_

h=0

Be I„ the asymptotic residue of the function (1 + toxV)-*e*<^'''(''+''> • on D taken at the origin. Then

lö = II N(a)»-M(l + wxV)-^ei'^«'(«+y) dx dy.

D

X is a positive number < 1. We'use the transformation

u = xy(x+y), p = y.x-i;

following the method used in theorem 3 of this Chapter, we ob-tain using theorem 17 of [2] the relation

lo ~ 1 2 (-O))*- feiW-u'^^S" /n(") p(Sh.2)/3(i^p).(7h + 2)/3dp du. ' h=0 0 ^(u)

For the calculation of the first two terms one needs

i 1 1 1 = {r(l/3)}2{r(2/3)}-» - i^^"^ p'^(l+P)"^dp - r p ' ^ ( l + p ) ' ^ dp 0 T l ( u ) = { r ( i / 3 ) } ' { r ( 2 / 3 ) } ' / ^ ^ " ^ p ' ^ { i 2 p , | p 2 . . . } dp -0 3 9 - f p-3{l - I p - » + | p - 2 . . . } dp T ] ( u ) 1 1 1 J_ = { r ( l / 3 ) } ^ { r ( 2 / 3 ) } ' ' - 3 . 2 ' ^ u"^ . . . - 3.2'* u*2 . . . = {r(l/3)}*{r(2/3)}"* - 3.2* u*^ + O(u^) . , 2 3. . 3 So lo - 3 - ^ ^ 1 / 3 ) } e^^/3^^'3-2*r(3/4)e3^^/''a) * + O(oo-i).

ii: the vertex (1,1): First one puts X = 1-t, y = 1-v. SO that the integrand becomes

{1 + to(i-t)M-v)n-^i'^(l-*>(l-^><2-t-v). The two boundary curves of D are given hy

t2+(i.v)2 = 1 and (l-t)2 + v2 = 1. using further u = 3(t+v) = 3v(l+p),

t = pv,

V = 3'^ u(l+p)"'.

So t h e f i r s t term of t h e asymptotic r e s i d u e I j of the function (l+oox3y4)-iei'^^J^(^+y) on D a t t h e v e r t e x ( 1 , 1 ) i s equal t o

ei"^-i(l+oo)-» f*e-'^" u/"^^"^ (1+P)-^ dp du.

where 5(u) ~ ^ u • ...

T)(U) ~ 6.U** + ... .

So the first term of Ii is equal to

e*"»-'^*3'^00-2 (1+00)-»,

being of much smaller order than the asymptotic residue !„. The reader remarks that the procedure followed up till now, is always the same:

first: one tries to find a subregion of the region of integra-tion D, whereupon the integrand g(oD,x,y)eif("^.x,y) multiplied eventually by terms of the form {l-N(u)} is asymptotically ana-lytic and

second: the treatment of the asymptotic residues;

In the following pages we will not mention any longer the first step of this procedure. In these special problems the critical points are mostly given as the points:

1°. where the derivatives of fase-function f(a),x,y) with res-pect to X and to y are equal to zero;

2°. where the derivative of f(w,x,y) along the boundary of D is equal to zero;

3°. where one of the derivatives of g(oo,x.y) to the order b

with respect to x or to y does not exist;

4°. the vertices of the region D. 6. The critical line

Problem: For reasons of comparison the asymptotic behaviour of

the integral //(l+ü)X^y*)-*eiwxy(x+y dx dy is determined, where

D

the region D is contained between the following curves: the x axis from the origin to (1,0); the vertical from (1,0) to (1,1). and the arc of the circle (x-1)' + y2 = l from (0,0) to (1,1).

Solution: The x axis and the point (1,1) are critical.

1°. Contribution of the x axis: Putting u = xy(x+y),

one has - p = x(x+2y) and

y = (2x)-» {(X* + 4 u x ) " - x*}. Furthermore, there holds again

OS

( l + o o x V ) * ' = 2 (-tt))''x^»'/''.

Let Ig be the asymptotic residue of the function (l+ua3y*)-*ei'^^(^*J^) on D along the x axis:

IR = II N(toi->^u)(l+<oxV)-*ei'^*J'(^+J^) dx dy. D

X being a positive number < l.

We use the method of the example of Chapter I.

the asymptotic residue Ig of the function (l-Kox^y*)-'e^"^^(*+J^) on D along the x axis is equal to

» 0+ I I R - 2 (-00/16)»' ƒ e'*^" ƒ x7''-2{(l+4ux-3)''-l}*h(l+4ux-3)-«dxdu h=0 0 ^(u) . 1 1 .S. I where ^(u) = 2 ^ u ' - 2 * u* + 2'* ii ... . P u t t i n g h = 0 one o b t a i n s , . i . 1 ƒ x-2(i+4ux-3)-« dx ~ - 1 + 2** u-»* + 2 * u * + 0 ( 1 ) . TTi _S 37ri _3_

So IR ~ 2'*r(14) e * to-» + 2 ' * r ( 3 / 4 ) e ^ oo''* + O(oo-i).

2°. Contribution of the p o i n t ( 1 , 1 ) :

In a s i m i l a r way as before one can prove t h a t t h e asymptotic r e s i d u e I j of the function (l+0Dx3y*)-iei(öxy(x+y) a t t h e p o i n t (1,1) i s of the same order as ix>'^.

Comparising of t h e problem 2 of s e c t i o n 5 and t h i s problem gives the r e s u l t t h a t the c r i t i c a l l i n e R c o n t r i b u t e s of l a r g e r order than the c r i t i c a l p o i n t 0 .

7. A problem

Wanted is the term of highest order of the asymptotic residue of gi(ü(A x«yP + B xYyS) ^ j ^ ^ ^ ^^^ critical line y = 0, on the region D, given by 0 < x < 5, 0 < y < px; p is a finite number > 0, p > 0 and 5 > 0. while a and Y *re non negative real num-bers, such that a + p = Y + 5 = 'v and 5 > p .

Solution: Let I be the wanted asymptotic residue, so that

I = II N(uoo»->-)e*<^ dx dy D

with u = Ax°^P + BxYyS

Pollowibg theorem 17 of [2] one can apply again the transform U = ool->^ (Ax»yP + BxYyS),

p = y.x'^ .

The answer, however, has to be independent of X and therefore, we choose X = 1. Then we have

I ~ v ' V e»w««u-i + 2V* /P (ApP)-2V*(i+BA-ipS-P)-»v"* dp du 0 P(u)

where p(u) denotes the smallest continuous positive root of the equation

A p P + B p S = u.^'^, that tends to zero as u -• 0.

According to theorem 8 of [2]

p(u) = uP''^-vp-^A-P"' § Ch(P.6)ii.»'

h=0

With n = B.A-8r*^-v(5-p).r' uv(6-p).r'

and Ch(P,6) suitable coefficients dependent of h, p and 5. One can distinguish the following three cases:

1°. p > a or 2p > v:

In this case - 2 p v * < -l, so that the term of largest order of the expansion of I is determined by the behaviour of the inte-grand

(A pP)"2^'' a * BA-ip5-P)-2v-'

at the lower limit p(u). After a few computations one finds the first term equal to

(v-2p)-r(p-)e^^(^P-^^P" A-^"\"^" .

The term of largest order of the wanted asymptotic expansion is found to be equal to TTi a

Av-1 p-»[;f { I M f i . } ]

C, = A-2V-' /P {p-i(i+BA->pR-P)->v-* _ p-I} dp. o

3°. p < a or 2P < v:

The term of largest order of the asymptotic expansion for I is now determined by the integral over p. Be Cj the constant value

A-2V-' /p-2^v-\i,BA-'p5-,^)-2v-« dp, 0

then the term of largest order is equal to V-» C, r(2v-') e^^''

to-2^*'-8. Two critical limes

Problem. A simple example of a problem with two critical lines

cutting each other at the origin of the real (x.y) plane is the following:

Be D the twodimensional real region 0 < x < ^. 0 < y < ii. where % and •t\ are fixed positive numbers.

Wanted are a few terms of the asymptotic sum that is asymptoti-cally equal to the asymptotic residue I R of the function

gXy(l+X) "* giodx2y2

on the region D along the two critical lines x = 0 and y = 0.

Solution: For reasons of simplicity we shall take the number

X occurring In the proofs and solutions above equal to one; if one likes, the proof {an be made exact by introducing again the number X.

(tee has

IR = II N(u)'->^x2y2)eXJ^(l+x)-»elu)x2y2 ^^ ^j^^ D

For small values of |y| and also for small values of |x| there holds:

exy(l+x)-» = 2 - ^ x " y''(l+x)-'' . k=o k!

Substituting u = x2y2 into this expansion we find: 0+ k

l^^y, 2 - 1 / e i - u 2 d u / ; . , - ^ k=o k' Ó »\-^ xd+x)"

Suppose one has to compute the first three terms of the asymp-totic expansion for IR:

+

0 +

^ ƒ e'"»" u-« {log (^Ti) - Vi log u} du =

o ^ . Tri 1

= 2-».7:2-\e"^.log(5Ti).to-«-4-i[-f{^^^^i^^}],=« .

ds u)'

Por k = 1 one finds for the first two terms:

0+

Ü ƒ e^'^ {log[^Ti(l+^)->] - i/i log u + u« Ti-i - . . . } du =

0 TTi» ini

- tó logftT,(l+5)-i}e-^.co-' - i [ l { Ï X i ) ± J - } ] . J ^

4TIOJ2

and for k = 2

0+

1 / e*'"" u»{logfen(l+5)"*] - '/4 log u - 5(1+5)-^ . . . } du =

.-_ . _ TT i s

= f{log[(^T,)(l+^)-'] - ^(1+^)-!} e * u . - 2 . 1 [ A { I ï s i ! _ } ] ^ ^ ^ , . . The sum of these terms gives the first three terms of the asymp-totic residue I R . The asympasymp-totic sum is of the same order as oo"* and each following term has a factor w"* more. The co-efficients of these expansion appear to be rather complicated.

9. An example of an improper critical line

Wanted is the asymptotic residue I R of the improper critical boundary line x = 0 on the region D of the function e^^^^y; D is defined by

-1 < y < 1; 0 < X < a.

Solution: The line x = 0 is a critical one. We should like to introduce xy as new variable but this variable is equal to zero for y = 0. Therefore the line y = 0 is called an improper cri-tical line. In order to overcome this difficulty we choose a fixed positive number H « a.

The asymptotic residue IR is according to its definition IR = II N(üoi-^x) e''""'' dx dy

P

with X a fixed positive number < 1.

Since N(u)i-Xx) = o for values of uji-Xx > l+e, where e denotes a positive suitably small chosen number, there exists near the y axis a small subregion E having non vanishing values of the integrand. This subregion is now enlarged along the x-axis: The

boundaries of the enlarged region E are given by the y-axis, y " 1. y = -1 the curves

jji-X xy = e and w*'^ xy = -9

and X = H; 9 denotes a suitably small chosen fixed positive number.

Now one computes the asymptotic residues of the two parts of E, where y > 0 and y < 0. The sum of these residues is equal to the residue IR, that has to be independent of H.

So it follows that

IR = I°*I^ , e ' ^ y » dy du + J ° V ' , e"''^ y » dy du.

0 u.H'* 0 uH*'

The calculation gives:

•nit . I ^ _ [•d{r(s)e"^^] ^ Jog H.e"^ o.'

-" as ..,» -_. d s ,„- , = 1

[^{iXËifll}] + logH.e'^to-» =

u) » = 1