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CERTAIN EXPANSIDNS
IN A COMPLEX VARIABLE

m

THESIS FDR DEGREE OF M. A.

 

DONALD WARD WESTERN
1939

 

 

‘ uamv
Michigan Stat.
University

 

 

 

 

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CERTAIN EXPANSIONS IN A.COMPLEX VARIABLE

by

Donald'Ward'Wbotorn

Submitted in partial fulfilment of the requirements
for the degree of Master of Art: in the
Graduate School, Michigan State College,

Department of Ihthomatios

June, 1939

ACKNOWLEDGMENT

To Doctor James Ellis Powell
whose suggestions and help have

made this thesis possible.

121481

CONTENTS

Introduction 1
1. Major and.minor circles 2
2. The expansion for functions analytic in a regian 7
3. The expansion for functions with singularities 10

4. Discussion of the expansion 15
5. An application 20
6. Expansion about a.point not the origin. 29
7. Extension to functions of several variables 31

Bibliography 38

CERTAIN EXPANSIONS IN A COMPLEX.VARIABLE

INTRODUCTION

In a paper published in 1931 Flora Streetman and L. R. Ford
deve10ped a polynomial expansion in a complex variable.‘ The series
obtained by them.provides a formula for the analytic continuation of
a function, analytic at the origin, beyond the limits of the circle
of convergence of the Maclaurin's series. They introduced the con-
struction of major and minor circles for use in the definition of a
region of convergence.

In this paper we Obtain an expansion which.provides a method
for the analytic continuation of a function, having finite singular-
ities, beyond the limits of the ring of convergence of the Laurent's
series. we make use of the idea of major and minor circles in ob-
taining regions of convergence. Some new theorems are proved on the
interrelation of these circles.

The nature of the expansion and its region of convergence is
discussed and is illustrated for a particular function. Two general-
izations of the development are considered; the first, expansion about
any point in the plane, is quite obvious: the second is an extension

to functions of two or more complex variables.

‘

# Flora Streetman and L. R. Ford, A.Certain Pol omial ¥§Eu “s1
Anerican.lathematical Monthly, W<I9 3I;, pp.

1. 59.125 a_n_c_1_gi_i_r_i_o_r_ circles. For a given point z in the
complex plane the major circle is designated by Hz and is defined
as the circle with center at the point -hz and with radius
(l+h)IsI, where h is an arbitrary positive constant. Note that
the point z lies on the circumference of the circle and that the
origin is included within the circle for any choice of h .
and is defined

as the circle with center 77%;- i’ and radius -,/;,%-I%I J

The minor circle of z is designated by 1112

h being a positive constant. Again s lies on the minor circle,

and the origin is in its interior for any finite choice for h .
An important theorem connecting major and minor circles is

Theorem I. E z, is. outside, £11, 9!. inside the minor circle

 

 

3f :1 , then :1 is reapectively inside, an, 25 outside the major

 

circlegf- s, .
Let 2, = x+63fi , and 2,. = mil-[n .

We are given that

(1) z, - 4 2“ 2 Amt ,

We wish to prove that

‘21 fflé/I E 6+4.) Afll '

Using the values for s, and s1 , we can write (1) in the form

I: an}; —W§T(Wm)| 2 Lamina ,

fr: 4,

01‘ as

(2) [/ 7‘21/51" — Zrlm (MI/7) 74/5": ,4 my} ’- 2’!» 5‘24] + I? 2’

5' 5*“1/m’r‘h’] +4’KM’M’J

Dividing by (1+2h) and adding h‘ (:‘4- y‘) to both sides, we have
from (2)

(3) (”4/4/7377 I: (m+1zjz+[h+/Oyjz
Taking the square root of both sides of (3) , we obtain

“I la Hal (How,

as was to be proved.

The following two theorems are stronger than theorem I and are

useful in later work.

Theorem 11. _I_f_ s is outside the minor circle of sz _a_

 

distance )7 , then :1 is inside the major circle of. s, _s_.. dixtance

 

greater than 93; egal to )2
We are given that

 

(5) f "‘ 4 i" = I
I/ herd 2| 7:37 '2." +7 '

We wish to prove that
Ian Mal -‘-' (Mt/In: -7 .

From (5) we can write

 

 

- 4 Z .. 4 i = 5 V 3 z
\/( mm ”PC; H2}; ’7) ALI: MM +7 )

which, squared, becomes

(ii M): + {rat ”)1
-.J 25.19/72) + 17/773 W+ +237

net/'4

 

Using the same procedure as in theorem I we arrive at
{/M/ " k ‘77
-‘-' (Mi/27 z + [/7er z + 17/H-levr7 ZK/fizé/ .
This can be written in the form
(6) (Mi-l2“) 2' r [mp/z 5/21] ‘(rzr /-y (ff/M/Vr +7 +71
4/4;in W +27Z/7w/I/7f 7 (#247.

We now show that the quantity composed of the last three terms of (6)

is negative. That is

’77K/fl/W +17/fiflVZ—7ai/zflfi/

which amounts to showing that

IZ/I f PIA +7

This is evident since we know

4 .
|z,| — 5;: lrzl +7 *a—iz lei]

 

J

IzzI + 7

It then follows from (6) that

'2, HM]; ‘ :5 (mi/’12,] 2- 27/MJIz‘wl + 7")

which gives us upon taking the square root
If, 4-15,. 2"- [#015,] -7 I

as was to be proved.

Theorem III. _I_1_:‘. s2 is inside the major circle of s, a

 

 

distance 5 , then s] is outside the minor circle 33 :2 a. distance

 

greater than _o_1_'_ egual to S .
/+ 21
We are given
(7) '2': +147] = [#4] [27,] - § -

lie wish to prove

I}, “—L 5'2! ->- .Lti 122! +_a?__ .

“‘14 “‘1’! H-zi

From equation (‘7) we can write

‘8’ (we/2 +(Hé’fl‘
=C/Wz/z’79 ~2f/M/W + 3“ .

Expanding (8), subtracting h‘ (1’4. y’) from each side, multiplying
through by (1-!- 2h), and adding to each side hz (m’+ n"), we obtain

(ow/z a) 2 {an}; 4,92 — 25m; flaw/7:;—
+fz//*7’OZ = (mar/79074492 .

1
Dividing by (l-I-Zh) and transposing, we have

(9) (z—T - 1772””) + (# xT—zfi ’1):

: 7%.) (M I”) + Elf-2013‘) ! 2", _(/r——;—4) '

Equation (9) can be written in the form

 

 

(lo) I, - 4 A 413 [#4 z-
\ mu; 2;:(7_/:1‘) I5I+WLI1I+Z7£W
+1s——— 3;“ In! -3_S_£_’_*i)_|a,_| .. raw/+4)

 

C/ *2 A); {/1524 Z]
We now show that the quantity composed of the last three terms of (10)

is positive. That is

1§é+1)\2,|_1£(z442121l _ 1§z(/+4). a: 0 )

 

(Ir-24) (/ 241‘ (Hzgj‘
which means
<11)(/+1/Ié,I + flaw] - IizI - 5‘ Z 0 .
Since by hypothesis

[(#4)|2'/A = I23 + if,‘ + f > libs-A 7" 3)

upon substituting in (ll) we have the desired result

Mafia + filial jzl -X *- 0~

It is now evident from equation (10) that

 

 

 

2,—42 e. H4 in + 5
A #24 1 #24 A ‘| 7:27: 1

as was to be proved.
One more theorem which will be useful is

Theorem IV. I3 ls,| > Is,| , then le contains m2, completely.

 

 

We designate by t,~ the points on mg; .

It then follows that

 

 

- Ii... - .—
If. #M 2’ ' 5:2}; I?" 2' lf‘AIZ‘én—‘I‘h
which allows us to write
(12) [4| *3 7—2: IEzI fir?) '24 2 ”2| '

Designate by t, the points on 11,, .
From the definition of Hz! we have

I2? *ffil 1' (”‘0 IE/I 5- IZ‘II 7‘ fllz'll J
which gives
(13) \nl Z owns/I - [It‘d =- I24 .
Therefore from (12) and (13! it follows that

If“ '5 lizI 4 If/I f IT/I J
and

1er 4 It/I

as was to be proved.

2. 2.139. expansion £0315 function analytic in. a region. We
shall now consider any function, f(z), which is single-valued and
analytic in and on the contour of a region 8 . By analytic is meant
that the function is defined and has derivatives at each point. The
region S is such that it is bounded by two regular curves about
the origin, AC, and 04 , where every point on c, is less in absolute
value than any point on C .

Construct the major circles for all points t1 on 0‘,z and
the minor circles for all points t, on C, . Since lt,‘ 4 \t2Ifor

all t, and t1 , then by theorem IV all the minor circles, mt, ,

are completely inside all the major circles, Mt Therefore

2 0
there exists a region, R, bounded on the outside by the major circles,
Mt: , and on the inside by the minor circles, mt, , such that R
encircles the origin. Also R is contained in S .

For any s in R we have by Cauchy's integral formula

(14) Pa) = :7" c (7% air _j___/ of?) alt

”I
-1; xiv oar //‘__ m as
”4 6’20”“ 2-) 1”" (z- -z‘)

For the first integral of (14) we use the expansion

 

(15)_._/_ = / J
z"? (Ht/2‘ - (24-417
2 Z
: __4 [/ + sir—42‘ +(mr + ..---+ ms +—---
z‘rm,‘ 2‘1‘42‘ r+4r 1+4:

Since s is in R it is inside the major circle of every t
on 01 . The distance of z from the center of the major circle of
any t on 0,, is at least some value, 3 , less than the radius of
the major circle. We have, then

Mil—+179] f éf-A')I7I " g )

or

(16) FL” :5 1 —-.___§__ = r', < 1 .
rue/it— (/¢Aj|fl

Relation (16) holds for all values of t on 02. .
It follows that each term in expansion (15) is less than or

equal to the corresponding term in the convergent constant term

_._.l___. -
series .23; (I+4/\z"’\ )
where lt/l is the maximum value of It) on 01 . By the Weierstrass

 

t E. J. Townsend, Functions of a C lex Variable, p. 75.
(Hereafter referred to as Townseng;

test expansion (15) is uniformly convergent in t on Cg. for a
chosen s in R .* Since f(t) is bounded for t on c , the
uniform convergence is not affected if the series is multiplied by
f(t). We can then integrate term by term and obtain

(17) I'év air _ f (“40“ My 42‘
41(2'?) ’ ..s 0, (teens "*1

For the second integral of (14) we use the expansion

(18) / a, /
(3-1.) {/ffijfi -Ct-*‘z) ’

/ + 2+ ---- +--- .
2+ Ila-f (anew-flit 9+4:- (ft—+4: if

since 2 is in B it is outside the minor circle of every t on

 

c, by at least some distance 7 . Then by theorem II any t on C,

is inside the major circle of s by at least I] . This means
‘
l/rggl +7 .. [is-stfle-I )

01'

(19) £15.

3441!-

21-7 :rvzzz,
W

Relation (19) holds for all values of t on c, .

 

 

Again by the Weierstrass test, series (18) is unifomly con-
vergent for t on c, and any chosen s in R . wiltiplying (18)
by f(t) which is bounded on c, and integrating the series term

by term, we have a ‘1
(20) /*_1 I6" 12‘- : (2‘44”?) 43/ afi”

 

! Townsend, p. 220.

10

Making use of series (17) and (20), we can now write equation

(14) in the form

(21) P52) = 1:2 770/ (ii-Mr)” [596%
30 d

 

, (2‘ +42“ ’”’

7‘ Z / Q‘rle-X flé‘jw/z“,

45” {fit a, <E+flahfl

 

Since f(s) is analytic throughout the region S, the contours of
integration, 0, snd 0,. , may be deformed into any regular curve, C,
lying completely in S and encircling the origin. For any s in

R we can write

(22) /[’[U = Z {IV—(’- C! ffjjfjau [067%

+2.: / aria)" Wane“ .

hzo ‘5”! a (2+!Ejh4pl

3. _T_hg_ expansion £25 mnctions it}; singularities. We now
oonsidsr any function, F(s), having a denumerable number of finite
singularities. It is desired to apply eXpansion (22) to such a
function in a properly defined region.

Select any one of the singular points which is not a limit point
of singularities of greater absolute value. Let us call this point 0’- .
For all points, -- - 'JY' P) ‘H which are less than or equal to A in
magnitude, construct the minor circles, - - - -) MUM. , mg . For

the remaining finite singular points, a, b, c, - - - - , construct the

11

major circles, Ha, my, Mo, a- - - - , Since we have established
that — - - 'JIVI) “Us” < |a|,|bl,|c|, - - - - , then by theorem IV
.. - - -J my) Mp) Md are all completely inside Ma, Mb, Mo: - - — -.
Therefore, there exists a domain, D, bounded by the minor circles on
the inside and by the major circles on the outside in such a way that
D encircles the origin. Figure l, pagelha, illustrates such a region.
By the nature of the definition of the region, D , we see that
F(s) is analytic in D but not on the boundary. If expansion (22)
is to hold for any 2 in D there must exist a region, J', contain-
ing s and encircling the origin such that F(z) is analytic in and
on the boundary of‘ 0’ and such that s is outside the minor circles
of all points of the inner boundary of 0" and inside the major circles
of all points of the outer boundary of 0/ . Figure 4 , page A3 ,
displays the constructions used in the definition of this region 0’ .
Pick any point z in the domain D . Construct the major circle,
ll” of s . Since 2 is outside the minor circles, - - - -) my,mp,ma)
by at least some distance '2 , then.by theorem II the points, - - -
”Y, Q, d. , are all inside M, by at least ‘1 . However, Hz
may cut some or all of the circles, - - - -) MY , m9, ma. 0
There exist circles, - - - -’fifim-fi1 , concentric with
- - - ’ymr ,mG ,WHL respectively with slightly larger radii which
do not include the point z . Circles with centers

- - — - 4 Y’ 4 5 A.
J H‘ZZ ) H‘s/I @ use)?

and with radii

..__...._. /+4H-|+1)/+/z #4 1+1.
) ”-14 1 ALI/t m‘,‘ 2 ”’12' | 2

J

 

 

)

12

 

 

 

:1 .
Figure 1. The domain, D 3 singular points, e ,cL , a, b, c;

13

 

 

 

 

Figure 2. Definition of the region, 0/ .

14

respectively have this prOperty. Likewise there is a circle, N; ,
concentric with IIIz but of slightly smaller radius which still
includes - - - -) V, (a, 0k in its interior. A circle with its
center at - hs and with radius (11-h)lzI- % possesses this proper-
ty.

The arcs of the circumferences of the circles, - - - -)7YI-;,W)
We; , exterior to each other plus the part of It; intersecting
these circles form a regular curve, 0,, which is inside 11, . There
are no singular points on 0,. It follows that F(z) is analytic
on 0,, and by theorem III, that s is outside the minor circle of

every point of C, by at least 20:22 A) .

 

Next construct the minor circle, mz, of s . Since 2 is in
R , it is inside the major circles, Ma, Mb, no, - - - - , by at
least some distance 3 . Then by theorem III the singular points,

a, b, c, - - - - , are outside mz by at least __i____ .

(/ + 24) i
There exists a circle, Til"z , concentric with mz but of slightly larger
radius which does not include a, b, c, - - - - . A circle with

center ‘4 £- and radius ___/_"_’{.|z| ,l— g
/s‘2fl /+zfi 2(14-1A)

 

 

possesses this prOperty.

Denote by C: the curve formed by the arc of if, exterior to
0, plus the part of O, exterior to Ti" . Thai 02 is completely
outside the minor circle of s and has no singular points on it.
Therefore, F(z) is analytic along 01 , and s is inside the major

circle of every point of 01 by at least g
g(/ + 1,1)

 

15

The region, 0’ bounded by C, and CL and including the
common part of G, and Oz encircling the origin, satisfies the
condition fulfilled by the region, S , used in developing equation (22).
The point z fulfils the conditions for region R used previously
since it is outside the minor circles of C, and inside the major
circles of C, . However, since 2 was chosen as any point in the
domain, D , the conditions necessary for the use of expansion (22)
are satisfied for any point in D .

We can now write for any 2 in D and any regular curve C

in D around the origin

<23) mm = 2". Referrer Home

 

 

has 0 (T-l’fltj‘fl”
7‘ Z 27/7/ (”Mn Fey as- .
Its-o a @+K&)“+/ -

4. Discussion _o_1; the expansion. We denote the two series in

 

the expansion of F(s) by

(24 __ "’ / ‘1
) fl ‘ A42; 1,," £34K?) FY29 0%)
a (er-41y "W

 

 

and

(25) _ °° __/_ Quiz)” Fr 05(-
5 — 42;; {We'd (2- *4“?th (J J

scthat

FC2)=/4+<9

16

In the general term of series A we make the substitution

fléj : (rs-+417" Fé-‘J .

We have then for the general term

(26) ’ .I_ We M

{/74}? ”H 2774‘ a TM"

Since from Cauchy's integral formula

aw 2170:...”

 

O

we can write (26) in the form

/ @WCO) = / {3: (2+4f2nFé‘J .

”/(nghfl /7/(/+Aj ”‘H t'=0

 

 

Thus it is evident that each term of series A can be written as a
polynomial in s g
In considering series B , we write the general term as

/ f @442) F(r) 412-,

(2.4%,? n+1 1TH

 

Examination of the integrand shows that s occurs only with integral
powers from the binomial, (t+hz)n, the highest power being 2” .

Results of integration by residues or otherwise will then give a

/

polynomial in s which, combined with the coefficient (3 4"?) ”H

gives for the general term a rational fractional function of s .
It is of particular interest to note the effect upon series
(23) and upon the region of convergence of the choice of different

values for h .

17

First we consider the case for h=0 . Substitution of this

value in equation (2 3) gives us

do ‘1 .
F62) = ”Z“ TTfit/c‘ 22:44 1‘7“} dz?
1‘- 0‘ / r’l
£2,771 ani—l Fé/W )
9° ‘1
Z if ._/__ Fé‘Jafi‘
e

 

 

so 2" ”*7

-00 2h
1;, 1172/_/_+-_— 24-" Fé‘jfl)

__ *0” n
:Z:~(1m// tn“ Féjazn "- Z: C(hZ

h move

3

 

1.

 

The final form of this two-way series is identified as the regular
Laurent's expansion of F(s) about the origin.‘ Notice also that
the region of convergence is now defined as the ring of convergence
of the Laurent's series. It is bounded on the inside by the circle
with center at the origin and radius (acl . It is bounded on the
outside by the circle with center at the origin and radius the mini-

mum of laI,lbI,|cl, - - - - . This is evident since the minor circle

4 d /___/'___4 ‘x
of (l with center at m 0C an radius 7—“ A

becomes, for h = O , the circle with center at the origin, radius (LII .

Likewise the major circle of a , having center - ha and radius
(1 +h)IaI , becomes the circle with center at the origin and radius

IaI .

 

e Townsend, p. 278.

18

Next we observe the effect of assigning large values to h .
The region of convergence, D , is increased in size with larger h .
This is true since the minor circles, - - - -Jm")m9)md' become
smaller and the major circles, Ma, lib, Mc' - - - -, become larger.

Consider, for example, the minor circle, m“. . Since

lieu—L4 =44.

4"- H-afi. 9- J
and
1+4. =
a... 7177‘“ 26"”)

then m‘ approaches as a limiting position for increasing h the
circle with center % GI. and radius é-IUU . Likewise for the major

circle, Ma, of a , we have

%(—Aa) == oo 1

and V
hC/+x)lal :. 00 )

This shows that the major circles increase in size without limit

for increasing h . Thus, in its limiting form, the domain, D ,

has an inner boundary composed of the circles through the origin

and the singular points, - - - - , Y) Q’OI, . Its outer boundary

is composed of straight lines through the singular points, a, b, c, -
— - - , perpendicular to the radius vector of that point. Figure 3 ,
page If , illustrates this limiting position for D in comparison
with the Laurent's ring.

To study the effect of different choices for h on the ex-

pansion itself, we notice that in series (14) and (18) on which (23)

depends, the ratios determining the convergence depend directly on h .

19

 

 

 

Figure 3. Limiting position of D , and the Laurent's ring.

20

In (14) we note that

._ .. f
V’ " " (#4)qu )

and in (18) that

_ __ ’2
V1 - 1 {/MJIEI

It is evident that the two series converge most rapidly for h = O
and less rapidly as h is increased in value. Here we see that the
convergence of the Laurent's series, where h = 0 , is more rapid
than the convergence of (23) for any other choice for h . The ad-
vantage of using (23) lies in the fact that the region of convergence
is enlarged for larger h , and we have a means of analytic contin-

uation of the function into a region beyond the limits of the Laurent's

ring.

5. An. application. Let us consider the function,

/
We) .-: (z- -/j(-?"2) A

having singularities at s : 1 and s = 2 . We shall write the ex-

 

pansion for the function for any s in the region bounded by the
minor circle of z z 1 and the major circle of s 2 2 . Figure If ,
page 1/ , displays such a region for h = i- .

From equation (23) the eatpansion in general form is

°° n M
‘2'” ”a = Z 2-76; (3:52 -
a (hf-fl z""" (5'4) (1‘ 2)

 

'50

Z / ’7 néé
* -a 1”". @4473]

”’ g (2+4?) (I‘d/(9‘2)

21

 

 

 

 

 

Figure 4. Region of convergence for FZII:

 

, , / . h _._
(we-2)

and the Laurent's ring.

22

Consider the general term of the first summation which is

(28) / (aw—r)” ad?!-
1772 £ (/+fij”*/ 2211+! (5" '// (12-2]

 

To evaluate this integral we make use of the theory of residues.‘
The integrand function of (28) has a pole of order one at t = l ,
and a pole of order (n+1) at t :0 . The value of the term
will equal the sum of the residues at these two points.

The residue at t = 1 can be obtained as the coefficient of
the term involving 21—57- in the Laurent's expansion about the

point, t = 1 , of the integrand function,
(EU Kt] /’
[/Hy ”H 7”“ (f-UKr-Z)

Write the function in the form

 

I

 

 

/ /
pfljh+/ X 2“”) X RC?) 1
where ( )‘I
_ zit/ff
[f(t) . fhft Cf-Zj

Mending R(t) in powers of (t - l) in a Taylor's series, we find
the constant term is
”(4) 7" -- (3+1) [1 I

Thus, the coefficient of erL/j— for the integrand function and,

therefore, the residue at t = 1 becomes

(29) - (24-4)}? ,
(he I] ”"' /

 

 

1' Townsend, pp. 284 - 289.

23

The residue at t ='O can be Obtained as the coefficient of

the term involving -7%— in the Laurent's expansion of the integrand

function about the origin. Write the function in the form

/ .X .__JL__ Ix C?K?) J

{/M)h+l raw

where

0&9 : 'fiwfl-fjh

1-31- 42"
we need the coefficient of the temm tn in the expansion of Q(t).

To determine this coefficient we write

éHf/h = (2-3r+z'7(z4. 15/4,?“ + —~—-+ A}, z"’+ ---).
Equating coefficients of like powers of t , we obtain
Zfla :- 2” )
M, - 3/4,, = (7) 2""fi ,
Mi 44, +40 = (2)2“4‘ )
(:50) 2’4! __ 3/4: + ,4] = (:1 24-: fly)
I I 5 I
I I I I
I I I I j

jflh'l ’7/411-2 *flaq : 6:7) 3 girl J

24;, -(?/4}1—/ +flH-2 = {h

£?/4;r/ "I7 /?,1 ‘* /Zfl-/
/ I

I I

I I I I
/ I ‘ I

J

II
o
\.

We wish to find An , the coefficient of tn . From (30) we find

 

 

__ h
#0 "' f )
:. (f1 "" 3 h)
fl’ if; 4 * ‘2' ”5 J

{42 g)i:‘zh-zfil 4' 1% fl/ “—1144

= (1’) 2“"47' +- 1 [7) 2""4 + _Z a"
z ‘7’ r J
3

A»; -'-’ 1’22"“? +2242 -—2L4/ J

.—.. (flaky: 4 1. ($7)2""41 4-?2 flak/4 +1.5 2“ ,
z 5’ 4

Assume that the coefficient A]: , k 1‘ n , is given by

h = (Ivy-w + 1;. ,z, W
1.

+ ,é (flJah'KH 4"” + —-——

+ 2"-/ (DEW/i + 2""’—/ z" .
2K 2K‘f'l

We know from (so) the relationship

(31) 67/“, = K7)2”-k-/4K+l +1”; __ _L [4,“! '
2 Z 7‘

25

By substitution in (31) for Ak and Ak + 1 , we obtain

«4/ ”(#73 “KHAN, + 16:)? K A"
2

7’3;qu I)?» mug/«4+ #(anz h-K-zmz‘ + -_-_

+_-22.x2__:_ “/(7)2"/4 +31“ g 1—! 2"

1K4!

2,411,) WK — gm) We-

 

Collecting terms, we have

(32) n . -
flA’f-l = (MN 3;“ ‘4/(*/ + :7: (I? 2"-ng

~ I . - -
+£6rlz’ 9'“ “*l'ifll +-/é£[k?z alk+24/(2+—___

+ +[3(1"-/) _. WW4} (,7) a“
2K

zK-o-l

522k+z zk+l

11+!) 2K_// 2'“ .

26

However, in the last two terms of (32) we see that

 

 

 

 

 

 

 

367/34) _ 2"" -/
2K‘/ 2K
2 K k
_ _C’x’ -Q@ ‘4) " 2 +2
— 2K1‘I
: 1(1x¢/____[1/r+//)
{1‘44
vk-H
_ 2. -/
- 2K+I )
and that
[ 2K1“! 1‘44 2mm.

Then we can write

”my :- 3) 2:444KH * 73—6?) 2""‘4“ +

+ 7 ___:_/ WNW + 2“'-/ 2"
_;———-k+/ 2K+z )

which in summation notation becomes

1"] —/§:: 2"..‘1H-j (1’1) Zh-c'flc.

15'“ch

Since this derived value for Ak+1 agrees with the assumed value

for Ak , ksn , the proof by mathematical induction is complete

and we have

A)”: 2:: 111- (f/_/ (fl) 2'" 17-444 ‘

:20- (4V

 

27

This gives us the value of the residue at t :: 0 which can be

written as

n .
Z 1”"*/—/ ’7 ’7 ‘ ‘
(33) (”J/m7 [:0 21”” (g) 2 i

 

The sum of the residues at t :0 and t.=l is, then,

from (29) and (33)

"‘_.z_;__L"’—- L—Ll—(a) 2,. .__ (2%)”

/
/?“j *I [:0 251-1. 4'] C/f-A/ "4'!

X .z“"*’—- / (1'7) 5""? ‘. -- f: (29 {if}

zit-[fl

f. (*7) ark-‘2‘ @‘“""’ " 2"'"*’\

:CH‘J/‘Hb/ i=0 2 h-4‘4-I /

._._ I " £6963”?

2040"” 1'7- 0

__ €- +fi)h _—__—_ __ z+zfl)h

{(#4)}:4‘1 <1+1£jh+l

1

_ /
_ [HA] *

 

 

 

 

From this it follows for the first summation of (27) that

:271’*”1 ”5“ :_:E£47);T.
“CM/1“)", r“*’(r-)(t—z) "" (24-14

For the second summation in (27), consider the general integral

 

term,

(ff/é)” ah"
EFL; (2+léj"*’<t")(z“ '2)

27

This gives us the value of the residue at t : O which can be

written as

’1 17-4'+/
(33) / Z 1 -/ (’7) 2'7‘4" ,-

fMj ”’7 Km 2 lr-z H

The sum of the residues at t = 0 and t :1 is, then,

from (29) and (33)

, :L—L 'm— _Z____LW ‘I ;.. (2%)”

 

 

W a. a ,m», (who
/ -u-/ 17 n"; h“?
:(/fl/* £11h-‘srl/ ()2 —‘2::()2 It]

 

 

f; (’1) Zn-lflz I“! /_ 2h-l'*/\

:64!)/H+/ ":0 2‘1- ‘+' /

'1 . .
,:1 "-4 z
2540“” 1'1; we???) 4

_ (2‘- +49)” _-_-.. .... (2%)"

1{/+K)h+l <1+tljh+l

 

 

From this it follows for the first summation of (27) that

(in-1ft) A41." _____ __ Z (£- +2é)h ,
Z—[CI/(‘fi‘Jn‘fil tn+I<t'O({’2) 4:0 (24.45

For the second summation in (27), consider the general integral

 

term,

(fol/é)" ah”
2775‘ (2+‘e‘jh+/(g—-1)(z~-z)

27

This gives us the value of the residue at t : 0 which can be

writtenas n .
(33) Z 1""+/—/ (’7) 2”".‘2‘ .,_

(H), ”'7 i=0 2"“ r/

 

 

The sum of the residues at t :0 and t :11 is, then,

from (29) and (33)

H
/ z:_,;__L H“... W“)? .__, (2144)"
E/v‘fijhfl 4'20 251-; 4-! C/f-fijn'H

2: .2 lr-u-/_ 6.7)?" “4‘ ”[4,: (fl akigt]

:(/"A/ :11- url

/ I: (tn) and?" é:-4°rl_/ _ 2h-[4/\

:<fi‘)n+i [=0 2h-‘.+’ /_

-_-_ I Z (5%;- ”"2"

2040*” t =0

—_(-z"i *5)” Caz/z)”

— {[H“ n+1 <1+11jh+l

 

 

 

H

 

From this it follows for the first summation of (27) that

00

Z 27/7 {“4“}: fl : -— Z 1%. o
m C’K/HJMI r"*’(t-)(t—z) "” (24-14

For the second summation in (27), consider the general integral

 

term,

(fafiéjh flflf’
I??? <2+lajh+I<t-/)(z--z)

28

The integrand function has a pole of order one at t :1 . To find

the residue. which is the coefficient of the term (7'17)- in the

Laurent's expansion, we write

(wife/’7 _ /

 

 

 

(2+4t)”*/(z‘—//Ct-2/ " €+4aj"+/ st—LI " P61 J
where
Pf] :: £ie+/)+(z‘—Q7n
-/ +(z‘ -/j

'1
The constant term of the expansion of P(t) by division is __(flerl)

which gives us for the residue at t = l ,

(34) __. yaw/)4 ,
(?+ 42/7”

It follows from (34) that we can express the second summation

 

 

 

 

as w
i _L, Cer/” 4"“ __ __ (LE-H)" ‘ ,
Its: 1774 c (i#l?)”*/<l"/)Cf ‘2} - ”=0 Lin‘fié ”Fr
The final form for the expansion of the function
F6?) :2 /
(Fm/<2 -2)
becomes a h
0‘
(35)F(2j .=_- -2: £23.”? __. Ziesya .
1"" <1+24 n20 (114"?)‘14’,
This can be written in expanded form in a two-way series,
F69 :______ __ zufl ” _____ __ /
(2%“!th (1+:de- (2,574)

_. / _, (Zn/Z _ (fat/217 __-..- .
(2+5?) (2’49); ---— (21“?)115‘1

29

We note here that for h : O the expansion resolves into

 

‘1
F6?) =- --..__ __. 2’79”] ----— f?- — é-
“ ELL— #- ----—'-,%--"- )

which is the Laurent's expansion in the ring about the origin for
any s such that l <Iz|< 2 .1' Observe also that by the discussion
on page I8, for a sufficiently large choice of h the orpansion (35)

will hold for any finite z where z = x + iy, such that

|;_7-—7/Z\47

and

6. Egansion about a point not the origin. The foregoing
work has been concerned with an earpansion which holds in a defined

region that encircles the origin. The extension to an expansion hold-
ing in a defined region about any other point, so, in the complex
plane is immediate.

It is first necessary to set up definitions of the major and
minor circles with respect to the point so . For any point z the
major circle will be defined with center at (lffi) i1, '- 4 2'
and with radius Q;4)| z -—2,l . The minor circle of 2 will

be taken with center at /7‘ 3 >556 ..i— i- ’
”Iii ”'24
and radius /+£ ‘2 “Z ‘ ,
”2% '

 

*- Townsend, p. 283.

30

Major and minor circles defined in this manner have the same relation
to the point so as the original major and.minor circles had with
the origin. Also theorems I, II, III, and IV carry over directly.

’ The procedure is parallel to that used in deriving expansion
(22). we consider a function, f(z), single valued and analytic in

and on the contour of a region 8' about the point z This region

0 .
is bounded by two curves, C} and C; , such tha t every point of C}
is nearer to so than any point of G; . Construct the major circles
for all points of c; and the minor circles for all points of C}
The region, R' , thus defined lies in S' and encircles so .

For any 2 in R' we have again by Cauchy's formula that

:: (27 .Afif -‘£~' a? .
#6:) 2177/54— wm “245%

For the first integral we use the series

..JL. :’ I I
24.! [/1“/(t‘&'1} — 5'90 4" (rue‘j ,

 

: /
(”I/(749} (#4) #4.} LAM} 0—24,]

This series is uniformly convergent in t on G; for any z in R'.

For the second integral use the series

, _ /
2—2" — [/*4)(a-s,)-[r-z,+4(a-zj )

 

: / / 7‘ (fat) +4(z-a.) ,1. - -- + (2'-2.)+I{a-z,
<¢£€Afl?-£§Q Zhaéjcz-gz) <z*€)(zosaa

 

 

'

H
/7" £72,)7‘ 5/2"") + ~“+Ei'7l)*‘(r'a+ --- p

Q.

31

which is also uniformly convergent in t on G; for z in R’.

It follows then that the expansion for f(z) can be written
in the fomm

flea = 5: _/_. Zé-w ”(r-J” 46a?”

. 2771
his C év‘ijfl'l [2._£_0)h+/

+ gfif— [Zr-as) #fls- 23;? /‘[/a2‘

p—ijjhf/ée- -fyj‘fi‘f'l

The integration is taken along any regular curve C which lies in

S' and encircles the point so .

The extension of this expansion to hold in a region defined
by the major and minor circles of the singular point of a function

follows exactly the same steps used in section 3 to justify equation

(23) .

7. Extension to functions of several variables. Consider a

 

function, F(z,,z1,- - - ,2") = F(z) , of n independent complex

variables, such that F(z) is analytic in and on the boundary of a

2n dimensional domain, B . By analytic is meant that the function

is defined for every point of B and that the partial derivatives,

mixed and iterated, exist. This domain, B , is a generalized poly—

cylinder, which is a region of points, (z,,z,,— - - ,zn) , defined by
a) a) a)

s,inA,zzinA,---,zninh(", where A ,j=l,2,---,n,

are regions on the respective coordinate planes bounded by regular

32

éfl
curves, C .* Then by Cauchy‘s extended integral formula, for any

point, z , in B we can write

(36) F6?) :: {—97% ”#1 " "' " Féfl fl”
.1777 am am 5911,? (I? ‘ 54)

 

where t7,t2,- - - ,t” are the parameters of integration taken in a

1')
positive sense along C(. *i For a simpler notation we write (36) as

Q?) =(74_)/ (IL/3W W s

i)
we now define B by choosing A( as regions on the respective

 

coordinate planes such that each A?) is bounded by two regular curves,
C?) wd CQU , about the origin, where the absolute value of any point

on C?) is less than the absolute value of every point on the corre-

sponding C?) . We next define regions, Raj, in the respective

coordinate planes such that each ét’is bounded on the inside by the

minor circles of C?) and on the outside by the major circles of C?” .
These regions, each one being completely within the corresponding AU),

define a generalized polycylinder, E , which is contained in B .
For any point, z , contained in E , we have by Cauchy's extended
integral formula

(37) For)- _CTZ)/(r :jgw flfl/(ziém

For the first of the integrals in (37) we can write
/

 

= T— : .4 ..— .- — o
r '9 :7 aa- 1;) ct, 1.) " (PI-L5: 2 " " (61- e.
t S. Bochner, Functions of SeveralC mlex Variables, p. 161.

" Goursat, Mathematical—m m1 sis, o . , mp. 225.

33

By means of expansion (15), each fraction can be eXpanded in the form
I = / /+ E'M’z‘ r————+—<=’u—4sz‘o "+—-)
(fl. -2“./ E/f'Z‘JrZ( C/S‘ I‘M/Z“ /+l¢/Z“)

If we form a generalized Cauchy product of these 11 series, we have

 

 

DO
__ /
z—Z-r-u — {News Z— J°« ,

where K A’s-1 a, 4. rd.
-. 2 2 “ - " 2 *fl 1" 2: f(at‘ .. -
FK - 4..., :0 434:6 (I :0 (ff-[‘1 ti 4*48) 2";

(1'
Since a is in E and t,‘ is on C J for each j: l,2,- - -,n ,
then we have by (16)

zahfl'fll) 5- r; < z , (a) .
(/+41j1f,°l

Choosing Q equal to the maximum r; then, since there are

(I) fK-IL/ terms in pk with the sum of the exponents being k ,
(M)! K!

we can write

(PK 5 in +/<-/)/ (3"

(Iv-w K/

Thus, series (38) is dominated by the series,
”

/ Z (fifK-IL/ 6K .

K
. . h-/ K 4/
‘2?- [H‘lc/Z‘t K30 ( j!
Using the ratio test for convergence, we obtain

may (mu/(«~01 e" =
K=w (h-Q/ K" (n+x-zj/ QM"

 

 

$3201. (JLfKK-l) (3 2. Q 4 1 .

Therefore, series (39) is convergent and it follows that series (38)
'J
is uniformly convergent for t,‘ on Ca and s in E e
We have considered (38) as a series in which the terms are

. .I
from a Cauchy product. In fact the 1:“ term consists of 53%

terms of the form

/21 f/(Zi)@'(2’4 ff; f sz __. .. ..A/Zn *flhf/‘Aeh o
)(h‘lrj 2: {/*‘:j t“ \(l’bl’jtfl 7

(40)

Since (39) is a positive term series, we can express the series by
summing each Q“ ) W times.* The series in this form
with terms, Q“ , is still convergent and still dominates the series
obtained from (38) by writing it as a sum of terms of the type in

(40). This amounts to removing the parentheses from the Cauchy pro-
duct terms and then rearranging terms as desired. Multiplying this
series from (38) by F(s) and integrating n times, we obtain for

the first integral of (37)

(41) n a
/ as a __.. f: / .77— was”; 54/45:,
a; c, “’ ‘

 

(If—E) €«“‘?n ‘0 ' (2"; *Aitc)

 

1' Pierpont, Functions £3 leex VariableI pp. 66 - 67.

35

In a like manner for the second integral of (37) we can write

__.L = ’ = / / __ /
(z 47 Affirm-j (3,15) X(z,-:,) " ' "‘(z,, 4:7) ‘

 

Using the expansion of (18) for each of these fractions and forming

a general Cauchy product, we have again

“21—— 2f)
/ ..
_.7T(/:'14)24

(2 -z‘) K =0

 

where x “M

E E K- da-
./ -—-- 25%?! __f't‘, Ha x---x.___a. (5,444”? I
5/ ( -13.,20 (II-1'0 x134 é’lJé’ (”‘43)31 ((/*4n)£‘1

(1'
Since t,‘ is on C,J and z is in E , we have from (19) that

Iii. fiVZZJJ f r11. < 1 (J) I
6* 1UP?" X
This again is sufficient to show that (42) is dominated by a conver-
gent positive term series and, therefore, is uniformly convergent.

(.
Then for t,- on C,” and s in E , we obtain
00

 

(43) '
Fir; d? z E 77(1); af‘ )fléjfl‘
[’1 <3 '2‘) €1“'én1'0£77’<2fl‘4s 2) €‘ 1r/
If each Cf” and C({j is deformed into a regular curve, Cw ,

U

which lies in A and encircles the origin on that respective coor-

dinate plane, then from (41) and (43) we can substitute in (37) and

36

obtain for any s in E

 

(44) E‘
/ Z- __.." -——/-d ’7 7'7 12: #4: f) :/\

+6: (T'#z)/77 (25 //,g-‘)@‘ \szcff.

ql"'gh"0 (3‘ fig ECJe‘J-j

In considering the extension of (44) to functions of several
variables possessing a denumerable number of singularities, it will
be necessary to restrict the study to those functions whose singular-
ities are due only to the occurrence of the variables separately and
not to the occurrence of two or mere variables together. This re-
striction allows us to confine our attention to the coordinate planes
in defining regions in tenms of the singular points. These regions
in turn define a generalized polycylinder for the domain of convergence.

We take a function, F(z) , possessing the singular points,
alpoaaa"sama"3311.311.9"s'3-zns"! asosaus"
e , a”,, - - 3 - - - - 3 an,, an,, - - , ann, - - e The first sub-
script denotes the variable from which that set of singularities
arises. The method of defining and proving a region of convergence
on each coordinate plane follows that of section 5 . Such a region,

If”, in the s,‘ plane has an inner boundary composed of the minor

37

circles of a“ and all singular points ajL' such that laji) :3 Iaij .
Its outer boundary is formed by the major circles of all singular
points such that lafil 2. Iain]. The only restriction on the choice
of aJ‘x is that it can not be the limit point of a sequence of sing-
ular points which are of greater absolute value than it is. These
DU)

regions, , j = 1,2,- - -,n , define a generalized polycylinder,

D , such that for any 2 in D , expansion (44) will hold.

BIBLIOGRAPHY

Behnke, H. and Thullen, P., Theorie der mnctionen Mehrerer
Komplexer Veranderlichen, Berlin, VerTag von Uulius
springer, 1934.

 

 

Bochner, 8., Functions of Several C lex variables, Notes by
H. W. IIexander,1. '1'. Mtin, W. . made, 8. Kylie,
Princeton, 1936.

Goursat, E., translated by Hedrick and Dunkel, mathematical Anal sis,
Vol. II, Part I, Functions 23 §_Cgmplex variable, New' or ,
Ginn and Company, 1513.

Pierpont, J., Functions 22mg Complex Variable, New York, Ginn and

Company, I513.
Streetman, Flora and Ford, L. R., A.Certain Pol omial Egpansion,
American.Mathematical Monthly, Vol. 3 .

Titchmarsh, E. 0., The Theory 2; Functions, Oxford, Clarendon Press,
1932.

Townsend, E. J., Functions of §_Complex Variable, New Yerk, Henry
Holt and c—‘—"Iompany, 9T5. ................. ,

38

 

M'CITI'I‘WEITHMII[lifijtijfjijil'l’ljflfliljlauflfllflfl