On the method of stationary phase for double integrals

On the method of stationary phase

for double integrals

1369

525

ON THE METHOD OF

STATIONARY PHASE

FOR DOUBLE INTEGRALS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP-PEN AAN DE TECHNISCHE HOGESCHOOL TE DELFT, OP GEZAG VAN DE RECTOR MAG-NIFICUS, IR. H. J. DE WIJS, HOOGLERAAR IN DE AFDELING DER MIJNBOUWKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP WOENSDAG 10 FEBRUARI

1965 DES NAMIDDAGS TE 2 UUR DOOR

PETRUS WILHELMUS MARIE BOIN

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR

PROF. DR. R . TIMMAN

C O N T E N T S

CHAPTER I

CHAPTER II

CHAPTER III

1 Introduction 9 2 Theorems for the one dimensional case 10

1 The contribution of a non-stationary interior point . . 18 2 T h e contribution of a stationary interior point . . . . 20

a. T h e elliptical case 21 b. T h e hyperbolic case 26 1 The contribution of an ordinary non-stationary

boundary point 33 2 T h e contribution of a critical boundary point 34

3 The contribution of a corner in the boundary . . . . 39 4 T h e contribution of a stationary boundary point . . . 43

5 Final remarks 46 References 47

C H A P T E R I

I.l Introduction

In several wave problems (waves in fluids, ligth and microwaves, see, for example, [1], [2] and [3]) one has to handle integrals of the form

/ jgi^^j) ^iri hfi^jj')^^4yj I jsi^jj') cos kf{x,y)dxdy ( L l - l ) 'D 'D

If A is the wave-length concerned, k stands for 27r/A, f{x,y) is a phase function and g{x,y) is an amplitude function. Under certain conditions for g{x,y) and f{x,y) and for small k the numerical evaluation of these integrals succeeded

reasonably well.

In 1949 VAN KAMPEN [4] applied the method of stationary phase, introduced by KELVIN and developed for single integrals by VAN DER CORPUT [5], [6], to the integrals mentioned above and obtained an asymptotic expansion for large k {k^> oo). However, VAN KAMPEN did not give a mathematical proof He based himself on the following heuristic principle:

The asymptotic series representation of (Ll.l) is determined by the points in which the function ƒ (A:, j ) is stationary. In other parts of the domain D of integration the integrand gives for large k, a fast oscillating sine wave with a relatively slow changing amplitude and so the contributions neutralize each other mutually.

A rigorous treatment of the asymptotic expansion of (1.1-1) put in the form

I = j je"^rMg[x,y)dxdy (A) 'D

was given by FOCKE [7] in 1954.

FocKE extended the use of a neutralizer as introduced by VAN DER CORPUT from single integrals to double integrals. The neutralizer is a function of .Ï and y which makes it possible to isolate various critical points so that one might determine the contribution of each domain with one critical point to an asymp-totic expansion of the double integral. Furthermore FOCKE assumed ƒ (jii:,^) and g{x,y) to be analytic in and on the boundary of the region D.

The method of VAN DER CORPUT was also used by BRAUN [8] who obtained explicit results for some types of interior critical points by assuming only that f{x,y) and g{x,y) have a finite number of continuous derivatives. He indicates how to employ his method for some boundary critical points. The assumptions BRAUN makes are weaker than those made by FOCKE but his results are, per-haps as a consequence, more complicated than the results of FOCKE.

and mostly connected with wave problems in physics, were obtained by several other authors.

Introducing the well-known Ó-function (see e.g. [9]) JONES and KLINE [10]

reduced the [problem to that of evaluating a single FOURIER integral by putting

M

^kf(.,y) _ le'''<S(^t-f)dt (1.1.2)

m

and applied a theorem, often called ERDÉLYI'S theorem [11], on asymptotic expansion of integrals of the form

b

l'e""h{t)dt (1.1.3) a

T h e method used by D. S. JONES and M. K L I N E seems to have several advan-tages above FOCKE'S.

" T h e calculation of the coefficients of the asymptotic series is simpler. Reduction of the problem of asymptotic expansion of higher dimensional integrals is immediately effected by that method. It is possible to predict the types of critical points from a knowledge of the rays in diffraction optics" [10].

In this thesis we will show that the contribution of several so-called critical points, in or on the boundary of the region D, to the asymptotic expansion of the integral / (A) can be found without making use of the notion of a neutralizer as introduced by VAN DER CORPUT or the (5-function as JONES and

K L I N E did.

We need therefore, among other things, three theorems. O n e of them is

ERDÉLYI'S theorem of which TIMMAN [1] gave a new proof by contour integra-tion. An other theorem, almost simultaneously proved by JONES and KLINE [10] and ERDÉLYI [12], concerns FOURIER integrals in which the integrand has a logarithmic singularity. For the proof of this theorem ERDÉLYI, JONES and K L I N E used a neutralizer.

In the next section we will give a different proof of that theorem by contour integration without making use of a neutralizer.

1.2 T h e o r e m s for t h e o n e d i m e n s i o n a l c a s e

To find an asymptotic series for the integral / (A) it is necessary to know the behaviour of the integrand in the neighbourhood of certain distinguished points, the critical points.

These points are (i) the points at wich the integrand or some derivative of the integrand is discontinuous, (ii) the points at which ƒ (A:,^) is stationary and (iii) the boundary points of the domain D.

Consider now in the one dimensional case the integrals Fi{k) = je'"g{t)dt a Fz{k) = fe'"{t-aY''{p-ty-'g{t)dt (1.2.1) (1.2.2) where A > 0 and fi > 0.

W e assume that there are no critical points between a and p.

If the number of critical points is finite the original integrals (1.2.1) and (1.2.2) are to be decomposed into a finite number of integrals each of which is free of critical points in the interior of the interval of integration.

Theorem 1

If |f(<) in (1.2.1) is N times continuously differentiable for — o o < a < < < ^ < o o , then Fv{k) = , S i v ( ^ ) - a i v ( - t ) + o f ^ j , k-^ oo with n = 0

Mk) = ^

For the proof of this theorem see e.g. [ H ] . Theorem 2

If^(<) in (1.2.2) is A'^ times continuously differentiable for — o o < a < < < / 9 < o o , then 1 Ft{k) =aN{k)+fiN{k) + 0

k^r

{{P-ty-^/m

{{t-ay-Yit)}

This theorem is ERDÉLYI'S theorem mentioned in the introduction. It can be proved by contour integration in the same way as TIMMAN [1] proved it, with only slight alterations in the assumptions.

Theorem 3

^if{t) for 0 < < < a can be written as follows f{t) = ao + aii+- • .+aA•^'^ + i^^'(0 and for RN{t) exist the first A'^+1 derivatives, while d^{RN{t)\ogt) then lim dt™- = 0, »2 = 0, 1, 2, . . . , N I = / ? ƒ (0 log t dt, {a > 0)

= Y,~^"HO)\y{m+l)-log k + ~jiy-^

Z

1 dr{z)

yj{z)

r{z) dz Proof

We write ao + ai< + . . .-\-aNt'^ = g{t) so that fit) =g{t) + R^{t) "('1) ;, 8(ir) ' i ^ Clr] 3 D f f H '. a) Fig. 1.

First we calculate

a

f«* g{t) logt dt (1.2.5)

Ö

by the aid of a contour in the complex i-plane (fig. 1) i.e. along the imaginary axis from A[iq) to B[ir) (/i), along a quarter of a circle with its centre in 0 from B to C{r) [h), along the real axis from C to D[a) [I3), next parallel to the imaginary axis from D to E{a-\-iq) [h) and then parallel to the real axis from E back to A [h).

Evidently

h+h+h+h+h = 0 (1.2.6) We substitute in /s t = u-\-iq and obtain

0

ƒ5 = e-^o j g{u + iq) \og{u + iq)e""'du

a

Now

\h\ < Ce~''<iq'''{log{a + q)+7il2}, (C independant of 9) so that

lim ƒ5 = 0 (1.2.7)

9 — ^ CO

In I2 we put t = re"'' which gives

0 0

h = rlogr /'«'•*"•'* girfyi'^dtp-r / ' / " " g{re)'''<pe''''d(p Ji/2 Ji/2 From Igftre"''! __ g-kr sin (p ^ i it follows that lim h = 0 (1.2.8) We have now 0 a + too I3 = I = — j ei'<:tg{t)logt dt - I ei''tg{t)logf dt (1.2.9) ICO a

In the first integral of ƒ3

0

A = — [ eiktg{t)\ogt dt (1.2.10)

ico

we put t = iu, and we get

CD

A = i I e-''^g{iu) i— + logu\du =

in v^ jm+i i' sr~\ z'^+i f = } ?(»») (0) / e-i'^u^du + ) g(™> (0) / «-*^«log u du

2 ^ m\ .' L-i ml J

0 0 "i = U 0

T h e first part of this equals

N .,

0

For the second part we can write, putting ku = z

Y CO v-1 t'»+i f z™ I 0 A' = — l o g A : ) - | —- ?(™) (0) / e-^z™ </z + /^\yi;/ ml 0 0 N + ) I - — |:<™> (0) / «-^2™log zdz = 0 ' Ó

= - l o g A 2 ^ y ^(™)(0) + 2 ^ y ^(™)(0)v(m+l) (1.2.12)

j5 = igika j e-''''g{a + iu)log{a + iu)du (1.2.14)

Ó We expand h{u) = g{a-\-iu)log{a-\-iu) N A<'»)(0) A(A'+1)(|) « = ()

In the interval (0.1) h^^+^>{u) is bounded. Furthermore

A^+l

j = 0

A(^+I)(M) = y (*'+')^W)(a + zM)log(^+W)(a + /a) =

; = ü

f/

So A(^+i>(a) is bounded on the interval (0, oo Thus we have " N f \ n 1 ld"^p(a + iu)log(a+iu)\ J A J m! \ < / M ™ /O i m=0

yt('^-siu)iogu\ / % , „ , , ^ ^ ^ ^

We now calculate from (1.2.3)

a

lR=\ei^^RN{t)logtdt (1.2.17)

0

By partial integration we find, using (c)

1 / ' 1 I } dRN(t)logt In = — Rif(u)\og u (/««*••« = — i?jv(a)e«Moga — — / ««*« (/< = 2A: ./ tk ik J dt 0 0 ^**" I D / M „ 1/</i?iv(01og<\ 1 jd^RN{t)logi gika I I /Q

= ^ p i v ( « ) i o g « - - ^ ^ ^^ / „ ' (zi)n ^^^

^ e ^ y l iVfd^R:,{t)logt\ /zWi / ^««^^v(01ogj

/ = 0 0

Here the last term is 0 1, ^ ^ oo.

If we take, for n = N, (1.2.18) together with (1.2.15), taking in account the sign of (1.2.18), we get

" (dr'^{f[t)lnt}\ ^ I I

"•i:p^r-i+«u *— («•-)

This combined with (1.2.11) and (1.2.12) gives (1.2.3). This completes the proof.

If a < o and

j . ^ dr-{Rr^{t)log[-t)] ^ ^

A dt"''

(JO

a similar consideration leads to

a ei''f{t)log{-t)dt = 0

= 2,/<™K0)|^'(^+i)-iog^--jy +

y/™(0)

W{m+l) -1°S'^ + -^|M

is usually called: the contribution of the point 0 to the asymptotic expansion of ƒ and mostly it is clear what is meant by that statement. As we will use this expression several times we will give an exact formulation of the meaning of it. 1) Let P{xo,yQ) be an interior point of the closed domain Di C D. If there occur in the asymptotic expansion of

/ i = l\ é''f^''''^g{x,y)dxdy

terms that are independent of the choice of the domain Di, but may depend on the values o[f{x,y) and its derivatives in the point {xo,yo), then we shall call these terms the contribution of the point P{xo,yo) to the asymptotic expan-sion of / i .

2) Let P{xo,yo) be a boundary point of the domain D. Let D2 c Z) be a closed domain of which the curve coincides with the boundary-curve of D over a finite length, such that P{xo,yo) is not an endpoint of the coinciding part of the boundary-curves of Ö2 and D (fig. 2).

P(«o.)'o)

That terms in the asymptotic expansion of h = \ \ é''f^''-'\g{x,y)dxdy

that are independent of the choice of the domain D2, but may depend on the values o(f(x,y) and its derivatives in the point {xo,yo), - provided/)2 satisfies the conditions mentioned above - will be called the contribution of the point P{xo,yo) to the asymptotic expansion of ƒ2.

C H A P T E R I I

II. 1 T h e c o n t r i b u t i o n of a non-stationary interior point

From now on we suppose that f{x,y) and g{x,y) are analytic functions of x andjc.i)

The term non-stationary point will be used for points {x,y) at which at least one of the partial derivatives/3;(;c,j) 2ind fy{x,y) is not zero.

Now let the point /"(xojjo) be an interior point of the closed domain D. Without loss of generality we can take for the point P(.ÏO,JO) the point (0,0) for a translation of the coordinates has no influence on the fact that P{xo,yo) is a stationary or a non-stationary point.

We choose our coordinates such that

/ . ( 0 , 0 ) > 0 ( I L l . l ) Since ƒ (A:,^) is assumed to be analytic in and on the boundary of Z) there

exists a closed subdomain E\ of Z) with the point (0,0) as an interior point such that for every point {x,y) of E\ holds

f.{x,y)^Q (II. 1.2) We now introduce the new variables u and v such that the equiphaseline f{x,y) = / ( 0 , 0 ) , which goes through the point (0,0), becomes a

coordinate-axis.

Therefore we put f{x,y)-f{Q,Q)=u

y = / . ( 0 , 0 ) t ; It follows from (II. 1.2) that

., , du dv dv du ^ ,-,^ , ,,

A{x,y) =~----^0 (IL1.4) ox ay ox oy

for a certain closed subdomain E2 of D. Consequently the determinant of

JACOBI of the transformation (II. 1.3) does not vanish for the points of £2-Let us denote the intersection of the closed subdomain Ei and E2 by EQ. The integral

/o = j 1' e"'^^''-'^g{x,y)dxdy (II. 1.5)

') I n most cases treated here it is sufficient whenf(x,y) a n d g(x,y) are A^ times differenti-able in the neighbourhood of the point in question. W e then obtain an expansion u p to the

I '

becomes

/o = e'*^(0'"' [Je-^-'G{u,v)dudv (II.1.6)

'F'O

The domaineFo in the w-n-plane is obtained by the transformation (II. 1.3) and corresponds with £'0. / ^

G{u,v) is the transform of ' obtained by said transformation (II. 1.3). A {x,y)

Since f{x,y)—f{0,0) - and so A{x,y) - and g{x,y) are analytic while A {x,y) j ^ 0 for points {x,y) of Eo we can write for G{u,v)

G{u,v) = go+giu+giu^^... (II.1.7) with gk = gk{v), k = 0,1,2,.. .

This expansion oï G{u,v) (II.1.7) shall hold for

0 < | M l < f l , 0 < | i ' | < 6 , a > 0 , b>0 (II.1.8) Furthermore we choose a and b such that the rectangle R in the a-y-plane, that

is determined by (II. 1.8), lies entirely inside the closed domain FQ.

T o determine the contribution of the point (0,0) to the asymptotic expansion of / (A) we now calculate

Ii, = ««/c.o' j j e''''G{u,v)dvdu (II.1.9)

R

a b

JkfiO.O) _{j e''"' [ G{u,v)dvdu}

= e''^^°-°^ [e''''L{u)du (ILl.lO)

— a

We can write for L{u)

L[u) ^ ao^aiu-\-.. .-\-aNU^-\-RN{u), | « | < a (II.1.11) taking in consideration that

aj = aj{b), J = 0 , 1 , 2 , . . .

Now we apply theorem 1 to (11.1.10) and we get the following asymptotic expansion for the integral IR

^ "-^- 1 IR = e'*^'"'"' ) {«««Z(«)(a)-e-««Z,(»)(-a)}e " " 2

11=0

+ 0\- ), A;^ 00 \A:«+2/'

boundary of the domain of integration, because IR only depends on a and ~ via a„ - on b.

So we see that an interior non-stationary point of Z) does not contribute to the asymptotic expansion if/ (A).

II.2 The contribution of a s t a t i o n a r y interior point

T o emphasize that we want especially the contribution of the point P{xo,yo) to the asymptotic expansion of I (A) we denote by Ip all integrals of the form (A), taken over a certain subdomain Do of Z), in which P{xo,yo) is the only stationary interior point. Also integrals taken over the image of such a domain as a result of certain transformations will be marked by Ip. Although we shall find in most cases handled here the contribution of the whole domain of integration to the asymptotic expansion of Ip we shall disregard that part of the expansion, which comes from the boundary of the domain, for the choice of the domain of integration depends more on the kind of the interior point in consideration than on the originally given domain D.

Now let the interior point P{xo,yo) of the domain D, where we want to find the asymptotic expansion of/ (A), be a stationary point for the function ƒ (Ar,^), this means

f.'{xo,yo) =fy'{xo,yo) = 0 (11.2.1) Furthermore we assume that there exists a closed subdomain A) of D such

that P{xo,yo) is the only stationary interior point of Do.

We introduce a transformation, such that the equiphaselines ƒ (A;,J))) = con-stant become the lines u = concon-stant, viz.

f(x,y) —f{xo,yo) = u ]

(II.2.2) v{x,y) = V J

The function v{x,y) will be more precisely indicated later on. By this trans-formation (II.2.2) we get for Ip

Ip = «*(''•••'•«' / / e"" - ^ ^ - dudv (II.2.3) G{u,v) is the transform oig{x,y) and

du dv dv du

A= (II.2.4) dx dy dx dy

We assume the coordinate-axes already to be rotated such that we can write for u

T h a t this expansion is possible follows from the analyticity oïf[x,y). We shall now investigate two cases, namely:

«20"02 > 0 and «20^02 < 0 (II.2.6) In the case that «20^02 = 0, which occurs for example when the equiphaseline

through the point P{xo,yo) has in (xojjvo) a doublepoint, it sometimes is possible to use the same method as is used hereafter. Mostly this method breaks down while the Jacobian for the transformation (II.2.4) is equal to zero.

II.2a The elliptical case

1.

U20U02 > 0

U20 > 0 and «02 > 0 (II.2.7) To determine the function v{x,y) we shall require that the curves v{x,y) =

con-stant are the orthogonal trajectories to the curves u{x,y) = concon-stant. Therefore we put dv du dv du and so du dv du dv dx dx dy dy T h e Jacobian is du dv dv du A = = dx dy dx dy

i/duY (duV

A = 4p{uhox^+uh2y^) + Z{x,y)

Z[x,y) is a function in which terms of a degree lower then three (in x and j» together) do not occur.

We have now for (II.2.4)

dv dv

{-2uo2y-Yix,y)}- + {2u2ox + X{x,y)}~ = 4^p{uhox^ + u^o2y^) + Z{x,y) {U.2AI) ox oy

X[x,y) and Y{x,y) are functions without terms of a degree lower than two. As we are interested in the behaviour oï v[x,y) in the neigbourhood of the stationary point, here the point (0,0), we first omit the functions X, Y and Z from (II.2.11) and we shall try to find a solution y'(A;,j))) ofthe partial diflFerential equation,

dv dv

-2uo2y-- + 2u2ox — = ^piu^2ox^+uh2y^) (II.2.12) ax oy

When v'{x,y) is known we have an indication for the solution ofthe equation (II.2.11). If _ {U20U02.y'' 2{uhox^ + u\2y^) we have a solution

v'{x,y) =avctgi — \'-^ (II.2.13)

\U2o' X

With this v'{x,y) is

xX+yY A = («20^02) "'M 2 + du U20X^ + U02y^. «20 — dy. y .

If we put in v'{x,y) in stead of- we obtain a solution of (II.2.11) which du X

«02 —

ox

in first approximation coincides with (II.2.13). And so

du

. = a r c t g f e f f (II.2.14)

\ « 0 2 ' du

dx

To this V corresponds a function/)(x,j>') which also in first approximation coin-cides with (II.2.13). p{x,y) is different from zero in a certain neighbourhood of the origin.

From this we derive

dv {u2oUo2y'' I d^u du d^u du dx (du\^ I duY \dxdy dx dx^ dy

"""idxl + Hdy)

and

dv {U20U02)''' (d'^u du d^u du dy IduY (duY \dy^ dx dxdy dy

HVxl-^"'Xdy)

We then find for (II.2.4)

(M20"02)''" id^ufduV d^u du du d^u/duX^ (duY (duY [dy^\dx/ dxdy dx dy dx^Xdy,

This gives with (11.2.5)

, ,^, 8{u2ox^+uo2y^) + T{x,y) / T T O I T N

A = M20«02 '" J ^ 2 1 2 I AT I I . 2 . 1 5 4 («20X2 + «02^2) + N{x,y)

In T{x,y) and iV(x,j) terms of a degree lower than three do not occur. We now introduce the transformation

r

X = — r r cos w U20 ''

y = ^sin(p (II.2.16)

«02 ' '

This means that v in first approximation has been replaced by 99, as follows from (II.2.13).

We can write now for (II.2.15)

1 ^ ' l + ^ i ^ + . 2 r 2 + . . . A 2iu2oUo2)^'l+bir+b2r^ + ...

In this formula a* and bk [k = 1,2,...) are expressed in the coefficient «;,,„ of

u and in cos 99 and sin 99 in such a way that there only are terms of the form fi,m cos'95 sin™99 with l-^m = A + 2.

And so 1 1 :r = ïï7 ^ (l+cir+C2r2 + . . . ) A 2{U20U02)'' Ck = Ck{cos'(psm'>'(p), l+m = k + 2 (II.2.18) While U = «20X2 + « 0 2 ^ 2 + . _ _ it follows

« = r2(l+a'ir+«/2r2 + . . . ) , 4 = 4 ( c o s V sin"»?;), l+m = k + 2 (II.2.19)

whence

u'' = ^ , / ( r ) = ( l + 4 r + é r 2 + . . . ) - ' ' '

ƒ ( ' • )

If we now apply LAGRANGE inversion formula [13] it follows that

r = u'''+e2{u'')^ + e3{u'--)^ + . .. (II.2.20) Where the coefficients Ck are given by

We shall have to examine the coefficients Ck more closely. If we put 1

-—- = ao + air+a2r2 + . . ., « 0 = 1

then is

{l+dir+d2r^-\-. . .)''' = ao + air+a2r2 + . . .

From

{l+gir))''' = l + Ci')gir) + a')g'ir)+. . .,

with

g{r) = dir+d2r^ + ...,

one finds that a* is a sum of terms, of which each consists of a product of the coefficients dp(p=l,2,...,k) such that the sum o f t h e indices o f t h e factors dp, appearing in each term, is equal to k.

Therefore

=y%.'

«fc = } «t „ cos'99 sin™99

with l-\-m is even when k is even and l-^-m is odd when k is odd. If now

f{r) = Po + pir+P2r^ + . . . = — j , ^ . fio = I ao + air+a2'''^ + . . .

then it follows from

y « ; | 3 , _ y = 0, k=l,2,... ; = 0

that we can write for /?*

Pk = / i?*'" cos'93 sin™99

where k and l-^-m are both even or both odd. In

if{r)Y = {Po+Pir+p2r^ + . . .)* we can evaluate the coefficient of r*^i.

There comes

= y t - l fifc = y r X ' / ^ ; . . . with /1Ó1+/2Ó2+.

V

Finally we have then k-2

ek= y e^„ cosV sm^(p, k> 2 (II.2.22) r = 0

/ + m = A: + 2 i ' + I

We now get for (11.2.18) 1 1

A 2(«20«02)'''

with

(t-i

i^ {po+piu''+p2tt+p3u''+....), po= I (II.2.23)

pk = y A.vCosV sin'»^), l+m = k + 2 + 2v (II.2.24) Substituting (II.2.20) in g{x,y) gives

G{U,V) = 9o + ^i«''' + ^2« + . . with

k-?* = y k-?*,.cos'99 sin™9?, Z+wz = ^ + 2^ (II.2.25) r = 0

T h e above mentioned expansion in series are valid in a neighbourhood Di of Pi - the image of Z* in the «-ü-plane, viz. (0,0) - on behalf o f / ( x , j ) and ^(x,^) being analytic so that for the determination of the contribution of the station-ary point to the asymptotic expansion of/ (A) we can evaluate

a 2n

Iv = e"^"""' ffe''" ^ . ^ . , {Po+Piu'' + . . .) • {qo + qiu'' + . . .)dudcp =

././ 2(«20«02) '" OO a ikf(x„,f„) I plku

y ^„(«•/')»^«,(«>0) (IL2.26)

T h e circle in the «-99-plane corresponds with a certain domain in the x ^ - p l a n e . For that domain holds (II.2.26). Sn consists of integrals ofthe form

2JI

/ 5„cosVsin'»99(/9), l,m = 0,1,2,... (II.2.27)

0 '

-From (II.2.24) and (II.2.25) it follows that l+m is odd when n is odd and l+m is even when n is even.

T h e integrals ofthe type (II.2.27) are equal to zero when / and m are not both even, so

Consequently we can write for Ip according to (II.2.26) a Jkf{,x„y,) We put ^^ = 7T7 ^ I ^"" y «»""^« (II.2.29) 2(«20«02) 'V ^ 0 " = 0

F{u) = y a««»

= y am«» + /?„

Therefore we have for « = 0

dRn d«Rn ^ Kn = -T— = . . . = — = 0

du du^

According to the theorem TIMMAN proved in [1] we find

_ ê'*^*'»'^»' Ï y ami'"+'^r{m+l)

" "^ 2(«20«02)''' 2 J k^l +

m = {)

glka y/i\ldlF{a) I 1

The contribution of the stationary point ^(xojjVo) to the asymptotic expansion o f / ( A ) is

- " " " , y - - - " " ^ ^ - + ^ 1 . ^ 0 0 (II.2.31)

2(«2oao2)^'^ ^™+i

2.

«20 < 0 and «02 < 0 (II.2.32) T h e contribution of the stationary point P{xo,yo) to the asymptotic expansion

of Z (A) is the same as (II.2.31) except that the sign of it is reversed.

In the case just treated «20'' and «02''' are replaced by |«2o|''' and |«o2p' while

putting in (II.2.15).

—« = U20X^ + U02y^ + . . .

Taking in (II.2.29) —« = w one arrives at (II.2.31) multiplied by —1. 2b The hyperbolic case

Let the interior point P{xo,yo) ofthe domain Z) be a saddlepoint of/(x,j)') such that, next to,

dx dy

holds

T ^ ? - ( ^ T < 0 for x = xo, 7 = 7 0 (II.2.34)

ox oy \dxdyl

For sake of simplicity we take in (II.2.5) —«02 instead of «02 and we put ƒ (x,7) —ƒ (xo,7o) = « = «20x2 —«0272 + . . . , «20 > 0 and «02 > 0 . (II.2.35) There is no need to distinguish the cases «20 $ 0 and «02 $ 0, as we did in 2a, for this amounts to interchanging of the x-axis and the 7-axis which is of no influence on the result.

We introduce a coordinate transformation I = x + a2o^^ + 2aiir»' + ao272 + a30.ï^ + . • •

r] =y + b2ox^ + 2bnxy + bo2y^ + b3ox^ + . •. (II.2.36) such that «(x,7) becomes

F{i,r,) = U2oi^~uo2V^ (II.2.37) For the transformation (II.2.36) we require that the curves f = constant are

the orthogonal trajectories of the curves »/ = constant, therefore

Now di dt] di dri dx dx dy dy du di = 2 « 2 0 f -dx -dx du di = 2 « 2 o l -oy -oy = 0 - 2uo2rj — dx 5'? - 2«02»? — dy )

(II.2.38)

(II.2.39)

T o determine l(x,7) we eliminate ri, — and — in (II.2.38) and (II.2.39) and dx dy

so we obtain the partial differential equation for i

l/ai\2 |s^\2^ du di du di _

\\dx/ \dyl J dx dx dy dy

di di Likewise we eliminate from (II.2.38) and (II.2.39) I, — and — and we get

dx dy

{

We have

i (0,0) = 0 rj (0,0) = 0

^^(0,0) = 1 ry^(0,0) = 0 ^,(0,0) = 0 r/,(0,0) = 1

Substitution of (II.2.36) in (II.2.40) and (II.2.41) gives the relations to determine the coefficients a « and bik.

We give — 3«21 4«20 + 2«02 — 3«12 «30 «20 = „ 2«20 3«21 4«20 + 2«02 3«12 2«20 + 4«02 Furthermore we have ^20 = èll = ^02 = 2«20 + 4«02 — «30 2«20 zl = | ^ ^ - ^ ^ = \+c,ox+coiy+c2ox^ + ... (II.2.42) ax oy ox ay

T h e flrst coefficients here, cio and CQI, are

«30«20 + 2«30«02 — 3«12«20 CW Coi = «20(«20 + 2«02) 2«30«20 + «30«02 — 3«21«20 «20(2«20 + «02)

If we invert the equations (II.2.36) and we substitute the result in A (II.2.42) there comes

A = l+cioi + covj+c'2oi^ + - • •

(II.2.43)

G{i,rj)didrj, p> 0, q> 0 (II.2.44) ~p -1

We observe that the terms gj.iihj' {J,l = 0,1,2,...) of G{i,tj) = goo+gioi+ +goir] + ... only than contribute to the integral (II.2.44) when j and / are both even. And so 1 A we / . = 1-have = / ' -cioi- -coirj + .. . for our integral /(ï.,.V.)

p 1

So we can write for Ip

Iv = 4«-' {goo+gwi^+go2V^+gioi* + . . .)didrj (II.2.45) T h e domain of integration for Ip, which is a rectangle, is replaced by the do-main enclosed by the |^-axis, the rj-a.xis, the line i

bola U2oi'^ — uo2rf = —a (Fig. 3). «20,

and the

hyper-• " . 0

Fig. 3.

W e put

« = «20^2 —a(,2»;2

V = u'\oi

T h e Jacobian of this transformation is

du dv du dv ,, , ,., , ,,, A = —- V^TT = 2«02« ''20^/ = 2(«20M02) ''{v^~u) '' oi orj arj ai T h e integral Ip (11.2.45) becomes <)Jkf{x,,y,) Ip = («20«02,

'•'

ff

1

T h e domain of integration D' is enclosed by the «-axis, the parabola « = ^2 and the lines u = —a and v = a''% a > 0 (Fig. 4).

C-(-J.O)

By dividing the domain D' in two parts we can write for the integral Ip (II.2.47) 0 a Ii a al-i 2e' ikf{x,„ya) («20«02) ' ' ," H(u,v^) "«*« ' —^ Y^dvdu (v^-u)" -a 0

Evaluation of integrals of the form

^J J (v2^uyi' dvdu

H{u,v^) /" V V ^^' -TT dv ^ } ) hj^i lU^^^ rr dv v^-u)'' ./ Z_j Z_j ^ • {i;2-« '/' leads to the evaluation of

; = 0 ; = ( ) [v^ — u)

Ii

vil

[V^~U)

-T^dv, / = 0,1,2,. We use the following reduction formula

„21-1

n =

21-1

^ ( ^ ' - « ) ' ' + - ^ ^h-i, I > I, which can be obtained by partial integration.

We have now ' Z;2-h.i^ 0 {v'^ — u) T, dv .. 1 ^ 2 / - 1 . . . 2 / - 2 A + 1 21 ^ ' 2 / A J ( 2 / - 2 ) . . . ( 2 / - 2 1 ) ( 2 / - l ) . . . l a'' + {a-uy'^ -\ a' loff Ti , / > 1 ^ 2 / ( 2 / - 2 ) . . . 2 ^ ( - « ) ' ' xi-x-^U (II.2.48) « a a~u] ' 1 , 0 log a'^+ia — u)''' T ( V A V , , ( 2 / - l ) ( 2 Z - 3 ) . . . l , , « • H ( « - « ) ' ^ ' , ^ , i i , = (a—«1 " > a, i« H «' loa; n , / > 1

1,' I ^ ^ M -r 21(21-2)...2 ^ - « '' In the same way we obtain

a'l' V^' ki = v^ — u Yrdv ,'/a / - I

(a-«)'''y a;^;,«;.

and

/o o = log

d''+[a—uy''

We join, in (II.2.48), the integral from —a tot O and the integral from O to a. This gives («20«02) / - I

v; / ^"" y y ^-'-'^^

In the case that / -'>•! must be omitted and the coefficient of «', in the last

A=0

term between brackets must be taken equal to one. We put

2 J "•i+p,p = ^j.i' ^j-i,i Pj.i = yj-i,i

and

p = 0

;=o

( 2 / - l ) . . . l 2/(2/-2)...2 and so we can write for Ip (II.2.49)

a

h=TT—vv, / «**"(<^-«)''' y y r;-/,/«'"''^«

+

_ / g««ln(a''' +(a—«)''") > djU^u —

-a i=0 gikfix^y,) r ~ -yr / «»•*« > Ó^«^ logluldu («20«02) ' V 4 - J — a ; = U (II.2.50) Now the point (0,0) is not a stationary point for the first two integrals of (II.2.50) so that these two terms do not contribute to the asymptotic expansion of/p as for as P is concerned according to section 1 of this chapter.

The third term of Ip (II.2.50) can be written as follows

0 a

,'kf{x„y„)

( M 2 0 « 0 2 ) ' ' '

ei'c^D{u)\og{ — u)du + / e^''«D{u)logudu

0

If we now apply the theorem 3 proved in Chapter 1 paragraph 2 we find: _g'kf{xa,y„) r r—, g'limin-i} "2o«o2)''' [ ^ A;»+i { - V 2 m - l o g y t + ^ ( « + l ) } . Z ) ( « ) ( 0 ) + n = 0

+ Ln^i^^^'''''' - l°g^ + y>{n+l)}-Din)[0) + 0[^,k^ oo

C H A P T E R I I I

m . l T h e contribution of an ordinary n o n - s t a t i o n a r y b o u n d a r y point

Let us label as ordinary boundary point the points of the boundary of D at which at least one oifx a n d / j / is not zero. The boundary curve is assumed to be analytic, and ƒ (x,7) = const, is neither tangent to nor coincident with the boundary.

All other boundary points will be called critical boundary points.

Let the boundary curve of the domain D be given by the analytic function r(x,7) = r(xo,7o) (III. 1.1) T h e point P(xo,7o) shall be a non-stationary point neither for the boundary curve r(x,7) = r(xo,7o) nor for the equiphase line/(x,7) = / ( x o , 7 o ) .

Furthermore we suppose df dr df dr

F F - F r ^ 0 '^ (^'-^) = (^°'>) ("^-1 -2) ox ay oy ax

This means that the equiphase line f{x,y) =f{xo,yo) through (xo,7o) is not tangent to the boundary curve r{x,y) = r(xo,7o) (fig. 5).

r(x,rt = r{x„,y„)

Fig. 5.

We introduce the transformation fix,y) -f[xo,yo) = u

r{x,y)—r{xo,yo) = V (III.1.3) and choose a closed subdomain containing P(xo,7o), where

du dv dv du

A { x , y ) = - - - ~ ~ ^ 0 ( i n . 1 . 4 ) ox oy ax oy

If the domain Do is the part of Di that belongs to D then we have to consider, according to ( I I I . 1.3)

Ip = I j e''^^"''"^e'''"G{u,v)dudv (III. 1.5) with

——- = G{u,v) A{x,y)

The notation Ip is used analogous with the use we made of it in Chapter I I (page 20).

We take for the domain of integration the rectangle in the «-n-plane deter-mined by

0 < « < a, 0 < |z;| < é, a>0 and b>0 (IIL1.6) such that G[u,v) is analytic in and on the boundary of this rectangle.

This gives

a b

Ip = e*^^^"'^'"' ie"" \{go+giu + . . .)dudv (III.1.7)

Ö - b

with

gk = gk{v), ^ = 0,1,2,... and so

a

Ip = //('«•>«' fe"''-{ao + aiu + .. .)du (III.1.8) &

with

ai = aj{b), j = 0,1,2,... ( i n . 1 . 9 ) If we put /ƒ(«) = ao + aiU + ... we can apply theorem 1. on (III.1.8) and so

we find

z , - . - / < - ' { y . „ r ( « + i ) ( ^ ) " " + . - y e--^^'^.^"^) ^^}j (iii.i.io)

n=0 n=0 "

It is clear from ( H I . 1.9) that there are no terms independent o f t h e choice of the domain of integration.

So an ordinary non-stationary boundary point does not contribute to the asymptotic expansion of I.

We further can derive from (III.I.IO) that the first series is related to the contributions ofthe points A and B (Fig. 5). This is caused by a corner in the boundary of Do, a case that we shall investigate in section 3 of this chapter. The second series involves the value of a.

i n . 2 T h e contribution of a critical b o u n d a r y point

We consider a non-stationary boundary point P(xo,7o) such that the boundary is analytic at P(xo,7o) and such that the equiphase line f[x,y) =f{xo,yo),

through P(xo,7o), is tangent to the boundary curve r(x,7) = 0 in the point -f(^o,7o).

We write and

r(x,7) = rio(x —xo)+?-oi(7—7o)+r2o(x—xo)2 + . (111.2.1) ƒ (x,7) = ƒ(xo,7o) + / i o ( ^ - ^ o ) + / o i ( 7 - J o ) +f2o{x-xo)^ + . .. (III.2.2) Here we suppose ƒ10 ^ 0.

T h e point Z'(xo,7o) is neither for ƒ (x,7) nor for ?'(x,7) a stationary point. S i n c e / ( x , 7 ) =/(xo,7o) and r{x,y) = 0 are tangent to each other, it follows that

rwfoi-roifio = 0 (III.2.3) rio and roi are not both equal to zero, so we assume

no ^ 0 (III.2.4) T h e equation ofthe tangent to the boundary curve in the point P{xo,yo) is

t: rio{x-Xü)+roi{y-yo) =0 (III.2.5) and the normal ofthe boundary curve has for equation

n: roi(x —Xo)—rio(7—70) = 0 (III.2.6) We now introduce the transformation

rio(x—xo)+roi(7—70) = a«, a^Q (III.2.7) roi(x—Xo)—rio(7—70) = f'^> b ^0

T h e point Z'(xo,7o) becomes the point (0,0) in the «-ö-plane. The joint tangent (111.2.5) of both curves becomes

« = 0

T h e normal (III.2.6) becomes V = 0

T h e choice of a is such that the positive «-axis, points into the domain of inte-gration (Fig. 6).

e derive X V from —Xo = —ro = (II1.2.7) no r2io + r2oi roi aü aü

+

?'2io + r2oi r2io + r2oi

Then, iff, f and g denote the transforms off, r and g,

f[a,v) - ƒ (xo,7o) = fioü+f2oü'- + 2fnüv+fo2V^ + . . • (III.2.9) f{Ü,v) = noM + f20«2 + 2fii«i5 + fo2Ö2 + . . .

g(ü,v) = |00+|l0«+.?0lÖ+|20«^ + . . .

The terms of the first degree in v do not occur for foi = 0 and it follows from (III.2.3) t h a t / o i = 0.

Now the boundary curve will become the f = 0 axis by

f = noM-+r2o«2 + . . . (III.2.10)

Ij = V

'— ^ 0 for a surroundings of (0,0) because no =^ 0. d {ü,v)

We now invert (III.2.10) and substitute the result in (III.2.9). Then we have

f{i,rj) = aioi + a2oi'^ + 2aniii + ao2ri^ + . . . 1 l(f>'/) =gOO+hoi+g01>l+g2f>i'^ + .-. j

w i t h (210 7^ 0.

We further assume that the curvature of the equiphase line through (0,0) does not coincide with the curvature o f t h e boundary at the point (0,0). Then

«02 7^0 (III.2.12) There are now two possibilities, viz.

ao2 > 0 and ao2 < 0 We take first:

2a.

«02 > 0

We introduce the transformation

awi = « + <2«2 + <3«3 + . . . (III.2.13) aioao2ti = —a\\u + a\_oao2''v + Ö2ou^ + 2èiiuv + . . .

with

di drj di drj

doo + diou + doiV + . 1

„'/.

du dv dv du aioao2' The coefficients tk and dik in (III.2.13) are determined such that the transform

of f {i,rj) becomes

f*{u,v) = u + v^ (III.2.14) For the integral Ip we take as domain of integration a subdomain of D which

has as boundary a part of r(x,7) = 0 on which P(xo,7o) is situated (Fig. 7a and 7b). ('o-Xo) Kx.y) = 0 /•("•rt = fKïa) Rx.y) = f(X||.y,) r(x,y) = 0

Fig. 7a. Fig. 7b.

For a neighbourhood of P(xo,7o) it shall depend on the sign of «lo whether the curveƒ(x,7) = ƒ(xo,7o) lies inside the domain D or not

We consider first: aio > 0 This means that the curve ƒ(x,7) =f{xo,yo) lies outside the domain D, and likewise the curve ƒ*(«,&) = 0. Now we have to deal with

" \ik(u + v')

Ir, gik/(x.,yo) j I g.kiu + .V T{u,v)dudv _(III.2.15)

A

where T[u,v) is the transform of ^(x,7) multiplied by the Jacobian ofthe trans-formation.

We choose for the domain of integration Do the domain enclosed by the r)-axis [t = 0) and the parabola t+v'^ = a, a > 0, (Fig. 8a).

By the transformation

d(t,v)

t+v^= w ~ ^ = 1 (III.2 16) d{w,v)

V = V,

the domain Do in the ^-y-plane is transformed in a domain in the w-v-plane with boundary: the line w = a and the parabola ^2 = w (Fig. 8b).

So

a w'l'

Ip = «'*^(''»-'«' /• j'e''"W{w,v)dv dw (III.2.17)

«1 = 0 —w '^

The odd powers of y in W{w,v) do not contribute to the integral. There comes a Ip = ^'V(--^») / g''"" J f i dw (III.2.18) J w'--0 with L{w) = k,w+l2w'^+kws + . .. . (III.2.19) therefore Z(0) = 0 (III.2.20) If we now apply theorem 2 of Chapter I on (III.2.18) we find the contribution

ofthe point Z'(xo,7o) to the asymptotic expansion of Z (A)

' 7^(« + 3/2_) ,„,,,J_ y J [n-\ Z J t(n--2) w H»/, Q T H + I L(w) (/w»+i

+ Ö 1 J T I , M ^ ^

(I"-2-2n

Now we consider aio < 0. Consequently the curve/(x,7) = ƒ (xo,7o) hes inside the domain D for a neighbourhood of Z*(xo,7o).

In (III.2.13) we replace a by —a, then we have to deal with

Ip = e'^f^-""^'-^ Ije-"-'" •"'' T{u,v)dudv (III.2.22)

'D„'

For the domain of integration we choose now the domain enclosed by the ö-axis (« = 0), the lines v = a and z; = — a and by the parabola —u + v'^ = —a, a > 0, (Fig. 9a).

We choose new coordinates w, v by — u + v^= w,

V = V (III.2.23) The domain of integration becomes the domain enclosed by the lines w = —a,

y = a and y = —a and the parabola v^ ^ w (Fig. 9b). The integral Ip can be split up in two parts

o* a (^ I f ' ' 2

lp = g*/(-«-'.) I ƒ• je""''W{w,v)dvdw - f le""°W{w,v)dvdw\

-'a -a o' - » ' / ü

For the contribution of P(xo,7o) to the asymptotic expansion of lp we can neglect the first double integral, while the second double integral is treated before in the case aio > 0. Hence the final result is the same as (III.2.21)

2

(O.a) ^ ^

r\

l (a'.O)

(0,-a) ' ^

Fig. 9a. Fig. 9b.

2 b .

fl02 < 0

Instead of «02 in (III.2.13) we put now |(Zo2| and we proceed in same way as we did in 2a.

We distinguish between a\o > 0 and «lo < 0. The result is that the contri-bution o f t h e point P(xo,7o) to the asymptotic expansion of/ (A) is (III.2.21) apart from «•"<'"+'/«"2 being replaced by £^('"+"/='/l

n i . 3 T h e c o n t r i b u t i o n o f a corner in the b o u n d a r y

Let Z'(xo,7o) be a point ofthe boundarycurve ?'(x,7) = 0 at which the direction ofthe tangents to the boundarycurve changes discontinuously but such that the direction of the tangent to the equiphaseline ƒ (x,7) =f{xo,yo) through the point Z'(xo,7o) does not coincide with either the right or left hand directions of the tangents to D at (xo,7o).

We shall determine a transformation such that P(xo,7o) becomes the origin (0,0), and such that the boundary of D coincides with the positive x-axis and the positive 7-axis for a neighbourhood of Z'(xo,7o).

Let the boundary arcs on either side of Z'(xo,7o) be given respectively by (p{x,y) = 0 and %p{x,y) = 0 (Fig. 10).

Then we have as a result of the discontinuity of the tangent to D.

Since the tangent to the equiphaseline ƒ (x,7) =/(xo,7o) does not coincide with either of the above tangents, we have also

(px{xo,yo)fy{xo,yo)—<Py{xo,yo)fx{xo,yo) ^ 0 and (III.3.2) Wx{xo,yo)fy{xo,yo)—ify{xo,yo)fy{xo,yo) ^ 0 !';(''./) = 0 n^.y) = fK.Co) Fig. 10.

Consider the linear transformation

X—Xo = ay)y{xQ,yQ)x — b(py{xo,yo)y (III.3.3) 7 ^ 7 0 = —afx{xo,yo)x+b(px{xo,yo)y

and choose a and b such that the positive x-axis and 7-axis coincides with the tangents to r(x,7) = 0 in (0,0).

The expansions ofthe transforms of 99(x,7) and y^{x,y) are now

^{x,y) = (^iox + <^2ox2 + ^iiX7 + . . . . (III.3.4)

^(X,7) = V01'^ + V20-^^ + V l l ' ^ J + . • •

a n d / ( x , 7 ) becomes

f{xj) =f{xo,yo) +fiox+foiy+f2ox^ + .... (III.3.5) Because of (IIL3.2), ƒ10 and ƒ01 are both ^ 0.

By putting

^lof = ^iox + ^2o.^^ + . • . (III.3.6) ipoifj = V'oi7 + V'2ox2 + . . .

both parts of the boundary curve of D shall coincide with the |-axis and the »j-axis.

We invert the equations (III.3.6) and substitute in (III.3.5). T h e n

F*{i,rj) =foo+F*ioi+F*oVi + . . ., F*ro ^ 0, Z-^i ^ 0 (IIL3.7) a. We suppose first Z'*io and Z'*ÜI are both positive. We transform the family of curves F* {i,i]) = constant into a family of straight lines by

F*ioi = «(l+/)io«+j&oiz'+/'2o«2 + . . .) (III.3.8)

For F* {i,rj) in (III.3.7) we have now

Fiu,v) =f{xo,yo)+u+v (IIL3.9) We now choose the domain of integration enclosed by the «-axis, the y-axis

and the line u + v = a, a > 0, {fig. 11a).

Fig. 11a. Fig. l i b .

If we put

u + v = t v — w

(III.3.10)

the domain of integration in the t-w-Tp\a.nc is enclosed by the straight lines

t = vu, t = 0 and t = a (Fig. l i b ) .

The integral Ip becomes

Ip = ^'*/<-«.>«) \\e''"T{t,w)dwdt (III.3.11)

0 0

e*^<'"-"">(/Ti(0^^

with Ti[t) = tit + t2t^ + . .. and so r i ( 0 ) = 0.

According to theorem 1 we find the contribution of a corner in the boundary of D, to be

e ikf{x„,y,) ^ 1

d-+^n{t)\ ^ ^ / 1

/-j{ik)n+^\ dtn+^ 1,^,

k^+^r

b. Are Z"*io and Z'*oi both negative, the contribution of a corner of the boundary of D to the asymptotic expansion of I (A) shall be the same as

(III.3.12) for we have only in (III.3.11) a to replace by —a.

c. Now let us suppose that Z'*io > 0 and Z'*oi < 0 in (III.3.7). If we now take the transformation

F*ioi = u{l+piou+poiv+p2ou^+.. •) Z'*oi»? = —v{l+qiou + qoiv + q2oU^ + ...)

we get for F* (i,rj) in (III.3.7)

Z'(«,z;) =f{xo,yo)+u^v (III.3.14) Here we choose the domain of integration enclosed by the «-axis, the w-axis

and the lines u = a and u — v = a, a > 0, (fig. 12a).

w A t+w B = 0 \ W = 3 0 t = : a

Fig. 12a. Fig. 12b.

Analogous with (III.3.10) we put u—v = t

V = w (III.3.15) T h e n the domain of integration in the t-w-p\a.ne is enclosed by the lines ix' = 0,

t = a, w = a and t+w = 0 (Fig. 12b).

To evaluate Ip we add to the domain of integration Do the domain D\ in the figure the domain ABO. If a is suffieciently small the function f{x,y) is analytic in and on the boundary of Z)i. If we now subtract from the integral taken over Do + Di the integral taken over Di we shall obtain the correct form of Ip.

So

a a 0 - (

Ip = g»/Uo'.)j ƒ• lg''='T{t,w)dwdt - j le''"T{t,w)dwdt^^ (III.3.16) T h e first term of this difference is not wanted to find the contribution of the

point P(xü,7o).

For the second term we can write

0

j'ei''tTi{t)dt (III.3.17) Again we have Z"i(0) = 0.

Hence the contribution of the point Z'(xo,7o) is here of the same form as (III.3.12).

d. Finally we take (III.3.7) Z'*io < 0 and Z'*oi > 0. Instead of (III.3.13) we use the transformation

F*iol = -u[l+p,ou + ...) ( n L 3 . 1 8 ) Z"*oi'/ = v{\+qiou + . . .)

Likewise as in c. this case yields the result in the form (III.3.12).

III.4 The contribution of a s t a t i o n a r y b o u n d a r y point

Let Z'(xo,7o) be a point ofthe boundary curve, here given by

/(x,7) = 0 (III.4.1) The boundary is supposed to be analytic in a surroundings of P(xo,7o).

a l .

We first deal with the case that for (x,7) = (xo,7o) holds

/ . ( x , 7 ) = 0, /,(x,7) = 0 (III.4.2)

fxxfyy-fxy^ > 0, fxx > 0 and fyy > 0

SO/(x,7) has in the point P(xo,7o) a minimum. Just as in Chapter II.2a we put

f[x,y)-f[xo,yo) =u{x,y) (III.4.3) We rotate the coordinate-axes such thatyij,(xo,7o) = 0 and moreover such that

the positive x-axis points into the domain D. For «(x,7) we write

«(X,7) = «20X2 +«0272+ «20^^ + . . . (III.4.4) and for t{x,y),

t{x,y) = twx+toiy+t2ox2 + .. . (III.4.5) If the coefficient of7 in /(x,7) viz. ^01 is not equal to zero then we make a further

transformation (see [7]),

{thoU02 + t^Olll2o)x = tioUo2Ï—toifi (III.4.6) (^^10"02 + <^01«20)7 = ^l«2ol+^10^

where t'^ioUo2 + t^oiU2o 7^ 0 as follows from (III.4.2).

The result of this transformation is that the coefficient toi of | m /(|,^) is equal to zero and that ü{ï,fj) begins with two squares as follows

ü{i,fi) = Ü20i^ + Ü02fj^ + a30Ï^ + . . . (III.4.7) and

t{i,fj) = iioï+hoï^+tiiïfj+... «20 = U20U02, «02 = 1 and ho = I

Fig. 13.

We now make the boundary curve i{ï,ri) = 0, (Fig. 13), coordinate-axis by the transformation

X = i{i,v), y = V and so

f = aioX + a2ox^ + anxy+ao2y^ + . .. fj=y

By substituting this in ü{ï,rj) (III.4.7) we find W{x,y) = W20X^ + W02y^ + W30X^ + . . .

We arrive now likewise as in Chapter II.2a with the aid of r « 2 0 ' r cos <p (III.4.8) (III.4.9) (III.4.10) (III.4.11) y «02 rj-jm(p

at the following integral .ikf{x„,y„)

a nil

2{W20WQ2) ' \

Tj-^ I j e"«'(^o+/'iw''' + . . .) {qo + qiw'^' + . . .)dwdcp,

0 - i i / 2

{a> 0) (III.4.12) Here the integration to 99 is not taken from 0 to 2n so now we have not, as we had in Chapter I I (2.28), iS'2j,+i = 0. In the integral

Jkf{x„,y„) Ip =

2(W20U)02) '\. / e^''"'F{w'')dw (III.4.13) we split

F[w'') =fo+fiw''+f2W + ... in two parts

F{W'') =fo+f2W+fiW^ + . . .+fi_w''+f3wl'+fw'' + . . . = F2(w)

= Filw)+^^ (III.4.14) w''

We now apply theorem 2. and we get as contribution of the point P(xo,7o) to the asymptotic expansion of/ (A), observing thatZ'2(0) = 0,

gtkf{xa,y„) r / j \ n+1

Z"i(»)(0) +

+

y ^ ^ ^ ^ .'-•/.W.Z'2(«.(0) - 1 , 1 + o ( l ) , k~> c^ (III.4.15)

LJ nl ' ' ' /t"+'/'

1

a2.

Obviously this is a expansion in powers of JTT , beginning with y .

If in (III.4.2) fxx < 0 and fyy < 0, we get the contribution of the point ^(•^0,70) to the asymptotic expansion of / (A) analogous with Chapter II.2a, in the form (III.4.15) with a reversed sign.

b . Now take that for (x,7) = (xo,7o) holds

fx = 0,fy = 0 and fxxfyy-fxy^ < 0 (III.4.16) We introduce the same sequence of transformations used above to obtain

w{x,y) = W20X^ — W02y^ + W3oX^ + ... (IIL4.17) In the same way as in Chapter II.2b we come to

p 1

Ip = /«-••^.' [ le''^"»''-""''^G{i,ri)did7j, V2Ü > 0 and z;o2 > 0 (III.4.18)

f = 0 'ri=-'l

We see that we have only to consider in G{i,v) = ^00+^10^+^01»? + . . .

the terms gj.ii^rj'{j,l = 0,1,2,...) with / even for other terms do not contribute to the integral

Introducing the transformation

u = V2oi^-Vo2rj^ (III.4.19) V = V2o''i

we get to explore the following integral

2g'kf{xo,yo) rr 1

Ip = -^, rr e''"H{u,v) -— -rr du dv (III.4.20) U20"U02"JJ (V^ — U)

We observe the difference with (2.47) in Chapter II where we had H{u,v'^) instead of H{u,v).

Proceeding as we did in Chapter I I we arrive at integrals ofthe type (2.49) but also at integrals involving an algebraic singularity coming from the terms with &', / odd. In the same way we find for the contribution of a saddle-point on the boundary of D to the asymptotic expansion of Z (A) the formula (III.4.15) multiplied by Iji.

III.5 Final r e m a r k s

It is clear that the problem of finding an asymptotic expansion of the integral Z (A) for the whole domain D is not completely exposed here.

Other points, that should be marked as critical points, are left out of con-sideration. Of special interest in optics is the case that an equiphase-line has a cusp.

It is also possible that there are points on the boundary of the domain of integration e.g. double points that deserve further exploration.

In the hyperbolic case. Chapter I I , the domain of integration was chosen in such a way that an equiphaseline partially coincided with the boundary curve. In that event the length of the coinciding part is involved.

It may occur that the domain D has a whole line of critical points. O n this subject some investigations were done by BERGHUIS [14].

The results achieved in this thesis can be resumed as follows.

A non-stationary interior point of the domain D does not contribute to the asymptotic expansion of Z.

T h e same holds for a non-stationary boundary point.

An interior point which is an elliptic or hyperbolic point for ƒ (x,7) gives an expansion in power series in k~'^ beginning with k^^.

If an equiphase-line touches the boundary curve the contribution of the point of contact is an expansion in power series in k~^ beginning with A^i and multiplied by A^'''.

A corner in the boundary gives a expansion in power series in k~^ beginning with A:~2.

For an elliptic or hyperbolic point situated on the boundary we find an expansion in powerseries in k^'^' beginning with k~'^.

REFERENCES

[1] TIMMAN, R . , T h e wave pattern of a moving ship; Simon Stevin, 35 jrg., I - I I , (1961). [2] W O L F , E . , T h e diffraction theory of a b e r r a t i o n ; Reports on Progress in Physics, vol.

X I V , (1951), T h e Physical Society, L o n d e n .

[3] BREMMER, H . , Diffraction problems of microwave optics; I . R . E . T r a n s , on A n t e n n a s a n d Propagation, Vol. A P - 3 , (1955).

[4] V A N K A M P E N , N . G . , An asymptotic t r e a t m e n t of diffraction problems I, I I ; Physica, X I V no. 9, (1949) a n d X V I , no. 11-12, (1950).

[5] V A N DER C O R P U T , J . G., Z u r M e t h o d e der stationare Phase I, I I . Compositio M a t h . 1, (1934)-3, (1936).

[6] V A N DER C O R P U T , J . G., O n the methods of critical points I. Ned. Acad. Wetensch., Proc. 51 (1948).

[7] FOCKE, J . , Asymptotische Entwicklungen mittels der M e t h o d e der stationare Phase; Berichte ü b e r die V e r h a n d l u n g e n der sachsiscHen Akademie der Wissenschaften zu Leipzig, Bd. 101, Heft 3 (1954).

[8] BRAUN, G . , Z u r M e t h o d e der stationare P h a s e ; Acta Physica Austriaca, Vol. 10 (1956). [9] V A N DER P O L , B . a n d H . BREMMER, O p e r a t i o n a l calculus; C a m b r i g d e U n i v . Press,

C a m b r i d g e (1950).

[10] JONES, D . S . a n d M O R R I S K L I N E , Asymptotic expansion of multiple integrals a n d the method of stationary phase. J o u r n . of M a t h , a n d Physics Vol. X X X V I I (1958).

[11] ERDÉLYI, A., Asymptotic expansions; Technical Reports 3, O . N . R . (1955). Dover Publications.

[12] ERDÉLYI, A., Asymptotic expansions of Fourier integrals involving logarithmic sing-ularities, J . Soc. Industr. Appl. M a t h . Vol. 4, no. 1, (1956).

[13] W H I T T A K E R , E . T . and G. N . WATSON, M o d e r n analysis, 4th ed., C a m b r i d g e 1946, § 7.32.

[14] BERGHUIS, J . , T h e method of critical regions for twodimensional integrals and its applica-tion to a problem of a n t e n n a theory. Proefschrift T e c h n . Hogeschool, Delft. Excelsior, 's-Gravenhage.

S A M E N V A T T I N G

Dit proefschrift bevat een onderzoek naar de asymptotische ontwikkeling naar k{k-^ oo) van de integraal

Z = jl^'^^'''^g{x,y)dxdy

waarbij ƒ (x,7) en^(;i;,7) analytisch verondersteld zijn.

Het gedrag van de integrand in een omgeving van bepaalde punten, de zgn. kritieke punten, speelt hierbij een belangrijke rol.

Kritieke punten zijn: (i) punten waar de integrand van Z of een of meer van zijn afgeleiden discontinu is, (ii) de punten wa.a.rf(x,y) stationnair is en (iii) de randpunten van het integratiegebied D.

Bovengenoemde ontwikkeling kan gevonden worden zonder gebruik te maken van het begrip neutralisator zoals dit door KELVIN werd ingevoerd en door V A N DER CORPUT en FOCKE werd ontwikkeld.

I n hoofdstuk I worden enige stellingen geformuleerd die betrekking hebben op d e enkelvoudige integraal

P

/ i = l'ei>'tg{f)dt a

Speciaal wordt de asymptotische ontwikkeling afgeleid voor het geval dat de integrand een logarithmische singulariteit bevat.

Daar ^(^,7) analytisch verondersteld is zal de asymptotische ontwikkeling van / voor inwendige punten van het integratiegebied hoofdzakelijk afhangen

v a n het gedrag van ƒ (A;,7) in die p u n t e n .

Uit hoofdstuk I I blijkt dat een inwendig punt, zo het voor ƒ (.Ï,7) een niet-stationnair punt is, geen bijdrage tot de asymptotische ontwikkeling van / levert.

H e t is mogelijk een zodanige transformatie, waarbij f{x,y) = u gesteld wordt, uit te voeren dat het bepalen van de asymptotische ontwikkeling van / neerkomt op de asymptotische ontwikkeling van een enkelvoudige integraal v a n het type / i . Door geschikte keuze van het integratiegebied komt men dan tot het volgende.

Is een inwendig punt, van het integratiegebied een elliptisch of hyperbolisch p u n t v o o r / ( A ; , 7 ) d a n is de bijdrage van d a t p u n t tot de asymptotische

ont-wikkeling van / een ontont-wikkeling naar gehele machten van k^^, beginnende met k~^.

In hoofdstuk I I I worden enkele gevallen onderzocht die betrekking hebben op de randpunten van het integratiegebied D.

Geen bijdrage tot de asymptotische ontwikkeling van / blijken gewone niet-stationnaire randpunten te geven.

Die punten waar de kromme ƒ (^,7) = c raakt aan de randkromme - deze laatste wordt analytisch verondersteld - geven als bijdrage een ontwikkeling die begint met k~'' en met gehele machten van k~^ op klimt.

Zo de raaklijn aan de randkromme in een bepaald punt discontinu ver-andert, een hoek van het gebied D, bestaat de bijdrage van dat punt tot de asymptotische ontwikkeling van Z in een ontwikkeling naar opklimmende machten van k~'^ beginnend met k^"^.

Tenslotte is in hoofdstuk I I I nog bepaald de bijdrage tot de asymptotische ontwikkeling van Z van een randpunt, dat voor de randkromme een gewoon punt maar voor ƒ (^,7) een stationnair punt is.

Deze bijdrage is een ontwikkeling naar opklimmende machten van k~^' be-ginnend met A:~i.

S T E L L I N G E N I

De in dit proefschrift gebruikte methode ter bepaling van de asymptotische ont-wikkeling van Z (A) kan toegepast worden in geval

f{x,y) =f{xo,yo) +a2o{x—xo)^+a3o{x—xo)^+a2i{x—xo)^{y—yo) + +ai2{x—xo){y—yo)^ + ao3iy—yo)^ + ..., «207^0, a o s ^ O , dus als het punt (^0,70) een keerpunt voor ƒ (^,7) =fixo,yo) is.

De bijdrage van het punt {xo,yo) tot de asymptotische ontwikkeling van Z (A) heeft dan de gedaante

I.ank-''"'^''' + 0{k-^), k-^ 00

n = 0

I I

Bij de dubbelintegraal (II.2.44) van dit proefschrift kan, evenals bij enkele andere in dit proefschrift voorkomende integralen, direct worden overgegaan op de herhaalde integraal. De in hoofdstuk I I gevolgde weg verdient echter de voorkeur.

I' /w-'^fe)''"''^'

I I I

A en B reëel, O < p < l, O < q < l,

dan is het functiesysteem \f{z)\, a.rgf{z), voor z = {\+ci)t, t > O, c-gelijk verdeeld mod. 1.

I V

Zij ƒ(/) c-gelijk verdeeld mod 1 en laat ƒ(<) alle waarden tussen O en 00 aan-nemen.

Geldt voor alle q en t {q > O, t > 0)

f{q)-f{t)=Af{qt),f{t)^0, A>0,

dan isf{t) c-gelijk verdeeld mod p, (p > 0). V

Bestaat er een natuurlijk getal no zo dat voor alle n> HQ geldt Pk{n) = nLi{n)

Un+l

k k

Il — 2 2 Z,(«) > 5 > l, k geheel, positief, (5 constant) dan is de reeks S a„ convergent.

VI

Bestaat er een natuurlijk getal no zodat voor alle n > no geldt:

R^in) = ^ ( f A - l ) - y ^ > ^ > 1' ^geheel, positief, Lkin) \ ' \u„\ I j-J Lk{n)

{S constant) dan is de reeks S Un convergent.

" = ' co

Is voor alle n > no Rk{n) <, 1, dan is S u„ divergent.

Deze en voorgaande stelling zijn aanvullingen op de convergentiestellingen van CAUCHY en D'ALEMBERT en laten zich, evenals deze, in limietvorm for-muleren.

V I I 2, 3 , . . . volgt

V I I I

ledere vorm van studiebegeleiding dient er allereerst op gericht te zijn de voor de studie nadelige invloeden van bepaalde facetten van de structuur van de huidige maatschappij zo veel mogelijk te elimineren.

I X

Voor het volledig benutten van de mogelijkheden die de centrale bibliotheek van de Technische Hogeschool te Delft biedt dient deze bibliotheek in de T.H.-wijk gevestigd te zijn.

X

De reorganisatie van het Hoger Onderwijs zoals deze door Prof. Dr. A. D. DE

G R O O T wordt gedacht, kan voor vele studenten nadelig zijn.

A. D . DE G R O O T , Propaedeuse nieuwe stijl. Universiteit en hogeschool, juli 1964, nr. 6, j a a r g a n g 10.

X I

Het is waarschijnlijk noodzakelijk maar zeker gewenst de regeling tot het ver-krijgen van vrijstellingen voor de wiskundevakken bij het eindexamen H.B.S.-B ook voor het eindexamen gymnasium-/? te laten gelden.

X I I

Het in boek of film voorkomende sadisme is een sterk motief tot wettelijk ver-Uit <„ = S tktn-k, tl = l, n =

k=i

tn 2 ( 2 r e - 3 ) ! {n-2)lnl