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VUNCTEONS OF TWO COMPLEX
VARIABLES

Thesis for the Degree of M. A.
MICHIGAN STATE COLLEGE

Philip Lincoln Browne
i941

 

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\L' A

FUNCTIONS OF TWO COMPLEX VARIABLES

by
Philip Lincoln Browne

A THESIS

Submitted to the Graduate School of Michigan
State College of Agriculture and Applied
Science in partial fulfilment of the
requirements for the degree of

MASTER OF ARTS

Department of Mathematics

1941

MATH. ue.

 

be \I
"\3 (J,
“>0. \
\o

( , \ \2

 

‘ACKNOWLEDGEMENT

To Doctor James Ellis Powell
whose suggestions and help have

made this thesis possible.

CONTENTS

INTRODUCTION

CHAPTER I.
Functions.
Geometrical Representation.
Regions.

CHAPTER II.
Some Theorems on Continuity.

CHAPTER III.
Functions Regular in a Region.
ngood's Theorem.
Hartogs' Theorem.

CHAPTER IV. (
The Cauchy-Riemann Equations.
The Theorem of the Maximum.

CHAPTER V.
The Cauchy Theorem.
Cauchy's Integral Formula.
Taylor's Expansions.
Associated Radii of Convergence.

CHAPTER VI.
Singularities.

CHAPTER VII.
Zeros.
Non-essential Singular Points.
Mittag-Leffler's Theorem.
The Weierstrass Product Theorem.

BIBLIOGRAPHY

14

21

28

39

51

64

INTRODUCTION

The theory of functions of several complex vari-
ables has been the subject of study by mathematicians
for approximately the last fifty years. Publications
in the field began to make an appearance at about the
beginning of this century. However, most of the work
in functions of several complex variables has been done
since 1925, making it one of the newest fields in math-
ematics. Although this subject has never enjoyed the
tremendous popularity attained by some of the other
modern trends in mathematics, much has been accomplished
in the field, mainly by men in Germany and Italy. Among
those mathematicians who have made noteworthy contribu-
tions to the study of functions of several complex vari-
ables, mention might be made of the Germans, H. Behnke,
and his pupil, P. Thullen, Stefan Bergmann, F. Hartogs,
H. Kneser, and P.J.Myerberg, and the Italians, E. E. Levi,
and F. Severi, and a Frenchman, H. Cartan.

In general the investigations into the field of
functions of several complex variables have been made
along the same lines which were followed in developing
the theory for functions of one complex variable. That
is to say, in the former as in the lattaq studies have
been made of analyticity, continuity, the Cauchy integral,
power series expansions, singularities, zeros, and the

like. However, the results have been varied. In some

1

cases, theorems are transferable almost word for word
from one field to the other, while in other cases the
differences are so marked as to be rather astonishing.

The purpose of the following discussion is quite

shmple. We shall limit our investigations to functions
of two complex variables. This will simplify the state-
ment of theorems without too great a loss in generality,
since almost without exception,theorems which are proved
for functions of two complex variables can be generalized
to the case of‘n complex variables. Our purpose will be
to contrast or compare some of the differences and simi-
larities in theorems as they are stated for functions of
one complex variable with the corresponding theorems for
the case of two complex variables. In a few instances,
a proof will be given; more frequently examples will be
used. However, the main object of our discussions will
be to point out the differences or similarities between
corresponding theorems for the two cases.

In conclusion it might be stated that this work
uses few theorems that have not been found in published
form. An attempt will be made to give proper references.
Ilention.might also be made of the material to be found on
this subject. For a very complete bibliography of material
published prior to 1933, the reader is referred to Behnke, H.
and Thullen, P., Theorielg££_Funktionen,Mehrerer_§9mplexer

Veranderlichen, pp. 109-113. .A few more recent references

 

are listed at the end of this paper.

CHAPTER I
FUNCTIONS. GEOMETRICAL REPRESENTATION. REGIONS.

Functions. Fundamental to a discussion of functions
in some particular field is the definition of a function.
If for each pair of values (2.,2‘) of two complex vari-
ables, s.= xytiy., at: x.=iyz, where (3.,zz) is a point
of a region 3 of a 4-space of points (x,,x,,y,,y5), there
is determined a value or set of values for a third complex
variable w, then w is called a function of the two complex
variables 2. and I; [w = f(z.,z;g for the region 3. .As
an example, w“= 3z.+22IL is a function of the two complex
variables 2. and z, , since for any pair of finite values
given to a. and 2;, a value for w is determined. Here 8
could be considered as consisting of all finite values of
z. and 2;, respectively.

Geometrical Representation. Cur definition of
a function of two complex variables has introduced the
idea of a region. In order to clarify the concepts of
the regions we shall use, we must first see what type of
geometrical representation can be used for functions of
two complex variables.

As in the theory for one complex variable, where
we conveniently adopted the idea of two complex planes,
one for the independent variable and one for the functional
value, so in the case of two complex variables we might use
three planes, a z.-p1ane, a z.-plane, and a w-plane. Now,

3

given any function of z, and 21, we have for each pair
of values (2.,z.) a corresponding value for the function.
Each such pair of values is called a point. For example,
the function w-*Sz¢2z;at the point (l+i,-i) has for its
w-value, w= 3+1. Hence, the corresponding values of 2.,
:2, and w can be mapped on their respective planes, as

shown in figure 1.1.

 

 

 

 

 

 

w-plane z.-plane zz-plane
‘1 7. Ya
efl:3+i .tal‘fii
’ 1%. 83
”‘g,‘ “A
Figure 1.1.

Regions. It is obvious that we might wish to con-
fine the discussion of a given function to sets of values
of z. and a, other then their whole complex planes. There
are several types of regions which have been defined in the
theory of functions of two complex variables, such as the
Reinhardt field, the Hartogs field, circular fields, and
a few others.‘ However, we shall use only three special
types of regions in our discussions.'“ These are the

generalized dicylinder, the dicylinder, and the hypersphere.
Generalized Dicylinder. The point (z. ,Zg) is said

 

to be contained in a generalized dicylinder I if its 2.-

coordinate belongs to a simply-connected region S. in the‘

 

" Behnke, H. and Thullen, P., Theorie der Funktionen

“ Mehrerer Komplexer Veranderlicfien, ppT'l-EU. {Hereafter
referred to as Behnke.)

"Bochner, 8., Functions of Several Complex Variables,

Part III, p. 161. (Hereafter referred to as Bochner.)

 

 

 

z.-plane and its zz-coordinate belongs to the simply-
connected region 3a in the 22-plane. Pictorially, S

s.-plane zz-plane

F‘\ ,. fix
4

\_._.J L

 

XL

 

 

Figure 1.2. .A Generalized Dicylinder.
might appear as shown in figure 1.2.
Diczlinder. A dicylinder about the point (a.,az)

consists of all points (z.,zz) such that
‘Z.‘al‘< A.) ‘zz'al‘<az.
In figure 1.3 we have illustrated such a region. The

s.-plane zz-plane
V. y&

/1 , /\

Figure 1.5. A Dicylinder.

 

 

  

dicylinder, we see, is a special case of the generalized
dicylinder where s. and 31 are now circles of radius d.,dz
about 2.: a. and 21: a;, respectively.

Hypersphere. A hypersphere about the point (a.,az)
consists of all points (z.,zz) such that

l2.—a.| + lav-Ad 4 A.
The value that z. may take to give a point in the hyper-
sphere depends on the value given to z,, or vice-verse.

Of these three regions defined, we shall use the first

two almost exclusively.
In the theory of functions of one complex variable

we often wish to consider the idea of a closed region,

that is, a region in which the boundary points are in-
cluded. In the case of a dicylinder or a generalized
dicylinder a boundary point will be a point (Z|,Zg) such
that,

a) 3. lies on the boundary of S. and z. is anywhere in 83,

b) 8; lies on the boundary of S; and z, is anywhere in 8.,

c) 3. lies on the boundary of 8,, z; on the boundary of 31.

We recall from the theory for one complex variable
that when we consider z=:x+iy, where x and y are real,
independent variables, we may then express
fix) 2- u(x,y)+ 1v(x.v)
where.u.and v are both real functions of x and y. Similarly,
in the theory of functions of two complex variables, if we
consider 3.: xrriy., and 2a? x¢+iyg, where x.,x,,y.,y¢,
are four independent, real variables, we may then express
f(l.,z;) = ulx.,xz,y.,y1)+-iv(x.,xz,y.,yg)

where u and v are real functions. From this we see that
a function of two complex variables is in reality a function
of four independent real variables. Therefore, in dis-
cussing the theory geometrically, we must think in terms
of a four-space. This, at times, causes some difficulty

in visualizing our procedures geometrically.

.Also, by considering z.= xrtiy. and zgnxi+iy3, as

mentioned in the proceeding paragraph, we may write
equivalent definitions for the dicylinder and the hyper-
sphere. Let a, =d.+ 15., and a..= d,+ 1.8.. Then the

dicylinder about (a.,az) is given by
‘L a. 1

(K. - “J‘ + (‘5:- Al)‘ < 6I ’ (XI. "' d.) + (at - £3)... < A; .
and the hypersphere about (a.,a,) is given by

(x.~q.)‘ + (g.~B.)‘ + (ma-cu)" + 03,- 130‘ < J‘.

CHAPTER II
SOME THEOREMS ON CONTINUITY.

In this chapter we shall take up some theorems on
the continuity of functions of two complex variables, first,
because they are of interest in themselves, and secondly,
because they will be of use in proving further theorems in
the following chapters.
To define a continuous function of two complex
variables we require, as for functions of one complex
variable, the idea of a limit. We say that a function
f(z. ,zg) has a _l_i_m_i_t_ P at the point (a.,a._) [kig‘.f(z.,z1)= P] ,

‘3'. i
if for every positive, real there exists a 3, such that

lflz.,z.) - 2| < e
for all z, and 22. such that
Iz.-a.|<$, lat-at|<5.
Further, we say that a function of two complex variables,
f(l.,z;), is continuous at the point (a.,a;) if

lim f(z.,z;) =- f(a.,a;) .
2,-ee.
{‘6 ‘g,

We have a theorem concerning limits in the theory
for functions of one complex variable,’
Theorem 2.1a.*‘
H1). 9.3.19.9. z=x+iy, air-«+16, P=A+ 1B,
and f(z)= u(x,y)+iv(x,y).

 

’ Townsend, E. J., Functions of a Complex Variable, p. 27.
(Hereafter referred to as Townsend.)

” We shall use this form for the statement of our theorems
in order to bring out the differences or similarities
between the theory for one variable and that for two
variables. 8

01). The necessary and sufficient condition that
lim f(z) 9 P
2+6
_i_s_ that

19:31‘11(I,y)= A and lilaflxq) = B.
3+8 g-v-AS

We shall prove the corresponding theorem for
functions of two complex variables.
Theorem 2.1b.
H1). _(';_i_v_gx_1_ 1,: x.+iy. , 1,,- xz-riy‘, a.=°\.+iB., a3: ate-1,6”
p a 14-13, an} f(z. ,z‘) - u(x. ,x,_,y, ,y,)+iv(x. ,xvy, ,y.).
01). The necessary and sufficient condition that
as”. N = P

3"")3;
is that
£149¢P( 1' ’xhy' 'yav) . A _____and 1;}E“y(xl 912.7. eyg) ‘ I e
x .“‘
9f+80 :11}:-
sa-na, “rm“

We prove the necessity of the condition first. Ie
have given then that

}.1.£“..f"""’ a 1"
23-53:,

By the definition of a limit this means that given $370,
there exists a 8g (8 dependent on e) such that for
|z.-a.|< S and [2,-a.I<8,
it follows that
|f(s.,z,_) - P‘< 1%: .
This in turn means that
(1) lu+iv - A - iB‘<i§-§
for
(xv-“0" + (y. -8.)z'< S1 and (xg-Ohf‘? (ya-8.)}? 5:

10
By a theorem for functions of one complex variable‘

(2), |“*1V‘A'A§| g lu-h'i-iv-BI .

 

. Vi
Substituting this in (1) we get
(a) 'u-Al+'v-B|<€
for

(x.-o(.)'+ (y.-o.)‘< 8‘ and (x1-«.)‘+ (yr/3,31 S" .
Now, for the same conditions on x, ,xby, ,yz,
(4) lu-Al<€ and Iv-Bl<€.
From the definition of a limit, (4) states that

lim u= A and limv= l.

Krfid. X05“.

K‘9q, “*3

“0‘55! $.Q’3-

51:50; atfifit

For the sufficiency part of the proof, we have given

_that

1imu=A and limv=3.

X.-)d. Ke"‘l

xg-pa; M‘O‘h

9e+£| ‘61-’50

urea. ns-vfl‘
By the definition of a limit these mean that given gz->O,
there exist 3.(u) , SJu) , S.(v) , 84v) , such that
f In - AI < S;-

or a g 7. a z "
(Iv-W) I (Yovan)<[$.(‘% and (123%.) "' (7s'33)<[sz(“‘e
and
Iv - Bl<§j

for

1. 2.
(1.4. )‘+ (y. -o.)‘<[gefl and (1,- «J4 (y,-o,f<[s,(v)].
Adding together |u - A| and Iv - BI, we obtain
(6) A ‘u-Al+'v-B|<E

for a, a.
‘ (1|‘qlf + (y. ‘8.) < 8; and. (1... «Ja + (y‘-£‘)3< S

Where 5 is the minimum of 5.(u), 5.,(u), 84v), SJv).

 

"‘ Townsend, p.10 .

11
We know that
|f(z.,z,,) - I":'(u+iv)-(A+iB)|=‘(u—A)+i(v-B)'< lu-Al+|v-B| .
Substituting in (5) we get

lfh. .11) - P|< e
for

(x,-a.f+ (y.-s.)l< 8'I and (x,-o(,)"+ (y,-s,)"< S‘,
This is nothing but the definition of the limit
gig. f(s.,z,_) = P.
2r».

Our main purpose in proving the above theorem was
to make possible a theorem on the continuity of a function
at a point. The theorem for functions of a single complex
variable is ’

Theorem 2.23. _
H1). 31133 z=x+iy _a_r_1_d_ f(z)- u(x,y)+iv(x,y).
01). 31.9. necessary and sufficient condition M fl!)

be continuous 33 3333 £321}. s= a 33M u(x,y)

and v(x,y) a M continuous 93?. a.

The corresponding theorem for functions of two
complex variables is almost identically stated.

Theorem 2.2b.
1H1). ELIE}! l,= x.+iy. , I‘nXgI’iy‘, fl
flz, ,1.) = u(x. ,x,,,y. ,yz) + iv(x. ,xuy, ,y‘).
Cl). 1113 29.229.53.11 £n_d_ sufficient: condition £1133 fh. ,3...)

_b_e_ continuous it. the point (a. J.) _i_s_ that u(x.,xg,y, ,yz)

 

and v(x.,x,,y, ,y,) no both continuous 91 (oi.,°k,6.,3.).
The proof follows immediately from theorem 2.1b.

 

’ Townsend, p. 35.

12
The next theorem on the continuity of functions
of two complex variables has no counterpart in the theory
for one variable. Instead it corresponds to a theorem
of two real variables.

Theorem 2.5.
By). _I_n_ 3 given region I, fix. ,3.) _i_s_ continuous _i_n z.
uniforme with respect _t_o_ 2;.

H2). flz. ,z:) _i_s_ continuous 3:11 2,.

Cl). f(z. ,z‘) _i_s_ continuous _i_n both variables together.

The final theorem of this chapter concerns sequences
of continuous functions. For functions of one complex
variable we have *

Theorem 2.“.
H1). fll),f,(z),f[z), . . . _i_s_g1 sequence 3f functipns
converging uniformg _t_9_ g limiting function flu).
H2). M rim _ig continuous _i_n 2 region S.

 

01). fl!) is. continuous _i_g S.

For functions of two complex variables" we can prove
a similar theorem.
Theorem 2.4b.
H1). #3,,2‘), f!’.,",, fgzuzg), . . . 3.33 seguence
23 functions converging uniformly fl respect .t_o_
_b_o_t_h s. .923. z‘ 193 limitig function fill. .13.).
[2). M f3”, ,zd _i_s_ continuous _i_n 23th variables 3.};

3 region 8e

01). fls. ,st) _i_s_ continuous Ln. both variables _i_n S.

 

 

 

 

" Copson, E. T., Theory 93 Functions 9_i_’ _a_ Complex Variable_ ,

n- 9‘. (flares er referral in as annann -

13
To prove the theorem, suppose we are given an 6 .
Then consider
‘f(l.,zz) - f(z,+p,zz+q)‘ .
This can be written
(8) If”. .33) - flz.+p,z,,+q)|
= |f(z.,z;) - f.(z.,z;) + fh(z.,z;)

- fh(z,+p,z‘+q) + £h(zl+Pesz+Q) " f‘8.+P.zlfi’Q)I .

(8‘) <|£(‘.,Zz) - fn('.,st)l+|fh(‘. ’2‘)
- £.(-.+p.s.+q))+If.(l.+p.s.+q) - fll.+p.s..+q)| 0
Consider the terms of (8a) separately. By H1 of the theorem

there exists an H such that

‘9) ‘1‘”: .21) "’ fu‘“: 932)) < E;
and

E
(10) )1.(2.+p.8..+q) - fiz.+p.2.+q)) < ‘3‘

for n) I. Also, since each f.,(s,,s‘) is continuous in 8,
then by the definition of continuity, given ‘3- there exists
a 3 (dependent on § and also on both 2. and 2,.) such that
(11) If..(s,,z,,) - f..(z,+p,s,,+q)l < '65

for

‘pl<3 and |q|<5.
Hence for each 2. and za in 8 and for any positive 6 there
exists a 3 which is dependent on E and also on s. and 3;,
such that by substituting (9), (10), and (11) in (8a) we have

|f(z.,zz) - f(z,+p,z,.+q)| < 5‘5- + g; + §- 7- e

for

'p|(5 and lq)<$.
This is merely the definition of the continuity of f(z,,zg)

in both variables, and hence our theorem is proved.

CHAPTER III

FUNCTIONS REGULAR IN A REGION.
OSGOOD'S THEOREM. HARTOGS' THEOREM.

Functions regular E _a_ region. For functions of
two complex variables, we define a function to be regular
_i_._n_ _a. region 8 if the derivatives of all orders, iterated
and mixed, exist and are continuous and bounded in every
region interior to s.‘ A function which has a region in
which it is regular is often referred to as an anagtic
function.

It should be noted that when we refer to a function
of two complex variables as being regular or analytic, we
mean that the function is analytic in both variables to-
gether. Osgood's theorem and Hartogs' theorem express
conditions on a function for such regularity.

Oggood's theorem. In order to prove Osgood's theo-
rem we require several preliminary lemmas.

Lena 1.”
H1). fh) 3.; regular in. 2 region 3.
H3). 1(8) _i_s_ bounded _i_n 8 mgmflgfil.
H3). A _i_s_ 3 region 11:33 closure 33 interior £9 3.
H4). at 331133 distance 3393133193 boundary 2; s. .0.

_i_s_ the length _o_f the boundary 23?. 3.
Cl). For I and uh interior to A _i_t_ follows that:

m_

 

‘ BOOhnOr, Po 164s
" Bochner, p. 1.62.

14

 

1). |f(l+h) - 1(3), é u'lhl'l

 

 

ZTl'ot"
1&1
2 I 1' “_-
> I MK NW.
3). lf(s+h) -f(z) _ , ‘ s-|h|-£ ,
h I") s ZTrot?’

These are readily proved by the use of Cauchy's formula.
Lemma 2 deals with functions of two complex variables.
Lama 2.
£1). mun) _i_s_ defined _i_n__a dicylinder P(Gg,r); £32.!
_i_s_, _t_h_e_dicylinderz |z.-a.|<r, l.1"‘z‘<re
H2). |f(s.,s‘)l$l £03; (2.,11) _i_n P(a;,r).
Hz). Emm (8'. ,I1) _i_g P(a;,r), f(s. ,s;) _i_g
analytic in s. ,_i_'9_r_ (z, .33) _i_n P(a;,r) 251.91. f(s.' ,s..)
is analytic” in s‘ for (a: ,z‘) i__n P(ag,r).

01). I_r_1_ a_ny dicylinder P(a,,r,), where r,< r, 6f(s.z , z.)

and 3f”. 133:...) exi____s__t _a_n_d are analytic in each
321
variable separately.

It will be sufficient to carry through the proof for 3f .

of- 3"
To prove that 3;- is analytic in z| , we recall that
I

by the definition of a partial derivative at a point, we

need consider f(s.,z,) as a function of 3., only. Since

by hypothesis we know that f(z.,zz) is analytic in s. for

(8.,1‘) in P(a;,r), then by the theorem on derivatives of
69

functions of one complex variable, 3-;- exists and is analytic

in s. for the same dicylinder P(a;,,r)."

6F
T"
the following device. Let h, be a sequence-of complex

To prove that— is analytic in a; we make use of

 

" Townsend, p. 77.

1‘6":
quantities approaching zero as n+oo . Then consider

the sequence of functions

 

I (8.,zt) :: W) " f(z.,s,_)

0‘

Considering f(z. ,z;) as a function of 2, alone, from (3)

of lemma 1 we get
g|< n lh.| 9.
’a - \ a Tr“
in a dicylinder a,P(a-,,r. ), where r.<r. This means that

r.(:. ,1.) converges uniformly in both variables to g;— ,
since In»)! is independent of the values given to

Tl'°(
s, and :1. Now, considering f(s. ,s,) as a function of

III alone, we know by hypothesis that f(s, ,z‘) is analytic
in 3;. Therefore, P.,(s, ,zg), considered now as a function
of s,’ alone, is also analytic in 3... By the Weierstrass
theorem for functions of a single complex variable, since,
in the dicylinder P(a; ,r| ), 'F,‘(z, ,zz) converges uniformly
to -g-£:- in st and each 1",, (s. ,s;) is analytic in 3‘, then
the limiting function -%—- is analytic in 1,, for (3. ,st)
in r(ei.r. )."‘
We may now proceed with the statement and proof of
Osgood's theorem.”
Theorem 3.1.
H1). 1(3|9.g, _i_g defined _i_n _t_1_1_e_ dicylinder P(a; ,r).
H3). If“. ,zfll S 1! £95 (3|,Ig)_12 P(a;,r).
2,). materials}. (-3.4) _i_g than. run-’2) is.
analytic _i_n. is. £95 (z. ,z;) _i_n_ P(a;,r) 33g f(s'.,s.,)
_i_s_ analytic in 3,, $33; (3.35,) _i_g P(a;,r).'

 

"' Cepson, p. 95.
”Bochner, p. 163.

17
01). Then for rd<r the derivatives 9.3 all orders,

iterated and mixed, exist and are continuous

 

and bounded _i_n the dicylinder P(a;,rd).[_I_z_1_
other words, f(s,,z;) is analytic in both
variables together.]

First we shall show that the iterated derivatives
of all orders exist in P(a;,r‘). Considering £5
for example, by the definition of a partial derivative
at a point, we need consider f(z.,sg) as a function of
the single variable 2., only. Applying the theorem on
derivatives of functions of one complex variable, we know
that all the derivatives with respect to s. exist at any
point in P(a;,r), and hence at any point in the interior
dicylinder P(a;,rd).‘ Corresponding reasoning would prove
the existence of the iterated derivatives with respect to 8‘.
To show that the iterated derivatives are bounded,

consider f(s.,s§ as a function of 3. alone. From (2) of

lemma 1 we have then

 

 

‘ OF 11.1
Oh \ 2Tl"c(‘~

for (s.,za) in P(a;,r.). This bound is independent of the

value of s. and also of the fixed value assigned to 3..

Hence ~9—- is uniformly bounded for (z.,z;) in P(e;,r.).

or.
In a dicylinder P(a;,r,) we know now by lemma 2 and from

what we have just proved that €%ET satisfies the hypotheses

 

of both lemmas. Considering-géf- as our function, now, we
can apply (2) of lemma 1 to show that ::i_ is uniformly

 

’ Townsend, p. 77.

18
bounded in a dicylinder P(a;,r,.) , whore r,_<r, . Using

this and also applying lemma 2 on the function égé- ,
‘L |
we see that -%h§: now satisfies the hypotheses for both

lemmas. This enables us to carry on the same discussion
a

for gig-5 . Hence, by continuing in a similar manner and
I

using a sequence of dicylinders, each one of which is
contained in the one proceeding, we can show that any
a“F
or."
dicylinder P(a;,rh). Moreover, we can show that any

iterated derivative

 

is uniformly bounded in a

iterated derivative satisfies the hypotheses of both
lemmas in P(a;,rn), a fact which we shall use in further
parts of this proof. We have confined our proof to the
iterated derivatives with respect to 2, since the proof
for the iterated derivatives with respect to s‘ follows
through in exactly the same way.

To conclude our discussion of the iterated der-

ivatives, we must show that they are continuous in both
61?
of lemma 1 in P(a;,r“) as shown in the proceeding para-
3‘9
dew
and applying (1) of lemma 1, we have

6"F(s.+%,-'e.) _ 6" He... 1;) < _

 

variables together.’ Since satisfies the hypotheses

graph, then considering as a function of I. alone

 

 

 

where.K is a constant depending on the sequence of dicylinders
a“?

 

used. From (4) we have that 6%., is continuous in a. uni-
I

formly with respect to a;, since the right hand side of the

inequality is independent of the fixed value assigned to 8,.

Next, since we can show that any iterated derivative is

 

’ The continuity in both variables could also be shown
by using theorem 2.4b.

l9
c)“F
analytic in 3,, by the use of lemma 2, then ———,; is analytic

 

 

as.
h
in z... The continuity of 3:“ in z,, follows from its
.‘ I
analyticity. 3:“ now satisfies the hypotheses of theorem
I

2.3 and therefore is continuous in both variables in a di-

cylinder P(a;,rgfl). A.similar discussion would verify the

continuity of the iterated derivatives with respect to z;.
We shall next discuss the mixed derivatives. As

22L

bit”

satisfies the hypotheses of both lemmas in a dicylinder

we have already shown, any iterated derivative

 

a
P(a;,r,,). Applying lemma 2 to the function if“ , we
0

 

 

 

have the existence of 3:31,. and moreover, its analyticity

in each variable. Also for the same function, from (2) of
lemma 1, we hairs that 32:22, is uniformly bounded in P(a;,r.,,,).
We now have 6:. ghostisfying the hypotheses of both lemmas

in P(a;,r,,,,). Consequently, we can apply these lemmas to

a”... F ah‘l'l. F
considered as the function and show that ——————
at" 31: at.” a 1':

exists, is analytic in each variable, and is uniformly

 

bounded in a dicylinder P(a;,r.,,,_). By repeated application
am
[f both lemmas we can show that any mixed derivative 3:3;—
." ti"
is bounded in P(a;,,r,,,n). Furthermore, we _can show that
0”“):
at.“ eta.“

 

satisfies the hypotheses of both lemmas.

 

 

an“?
Finally, we must show the continuity of -—;.-—-—-_.
“h as. at.
As mentioned in the last paragraph :2”) ... can be shown
| i;
to satisfy the hypotheses of both lemmas in P(a,;,r.,,,,,).
he
Hence, considering 3 hp has a function of 2. alone,
as.“ as,

from (1) of lemma 1, we 562....
(5) 3"" Kuwait) _ a “2"“)

3'1.“ 655;“ Wait” 3*...» g X... h.
where K, is a constant depending on the sequence of di-

20

mm.
cylinders used. Hence, from (5) we have that 56—13%-
z.‘ {"
is continuous in z, uniformly with respect to 5.. ‘Also
ah-OIMF
since we can show -—————- to be ana tie in z , it is
at.“ 6-1.1“ 1y ‘
necessarily continuous in 2;. Using these two facts and
he“;
applying theorem 2.3 we have the continuity of 5L————
dares?

in both variables. in a dicylinder P!a;,rw~“J.
Now we see that we have shown that any iterated

or mixed derivative, all of which can be represented by
a“:

6176*?
in both variables in a dicylinder P(a;,,r,,,,_,,). The se-

, exists and is uniformly bounded and continuous

quence of dicylinders used was such that each one was
necessarily contained in the preceeding one. We can there-
fore select any dicylinder P(a;,r¢) interior to P(a;,r)

and show that Osgood's theorem.holds in P(a;,r‘) by making
(r- r!)

awe-nun °

the decrease in radius from one dicylinder to the next
Hartogs' theorem is less restrictive than Osgood's
theorem in that it does not require the property of
boundedness. We hereby state Hartogs' theoremQ‘
Theoren.3.2.
El). fix. ,5,) _i_s_ defined _i_n_a_ region 8.
Hz). £93m M (21,151) of s, fh. ,s’z) _i_s_. analytic

for (s. .31) _in_ 8 and fix: ,2.) _i_s_ analyticfg (I: ,3.) in S.

 

01). Ln 3, fh, ,s‘) _i_s_ analytic in both variables.

The proof of this theorem is too long and involved to be
presented here. We shall use Hartogs' Theorem frequently

to determine the analyticity of various functions.

 

‘ Bochner, pp. 164-172.

CHAPTER IV

THE GAUCHY-RIEMANN EQUATIONS.
THE THEOREM 0? THE MAXIMUM.
‘ggg’cauchy-Riemann Equations. The extension of
the Cauchy-Riemann equations to the case of two complex
variables leads to several interesting results.
The theorem dealing with the Cauchy-Riemann e-
quations for functions of a single complex variable is:'

@901?” 4e1.e

H1). _I__n_ 2 gm finite region 3, u(x,y) 511g v(x,y)
353 :19. gal, single-valued functions _o_£ the
3333... variables, x and y.

01). The necessary and sufficient condition mm
complex function w=u(x,y)+ iv(x,y) _b_e_ regular

in s g that the partial derivatives 2.2L, 3“

 

 

 

 

 

bx by ’
.%%— , 953’ , exist and are continuous in 8 and
3 fl “ — .— #
satisfy the Cauchy-Riemann equations,
3% _ 3V . an = __ 9.3..
bx 33 03 3x

Immediately, using Hartogs' theorem, we can state
a.corresponding theorem for functions of two complex vari-

0.19183.
Theorem 4.lb.
El" .12. a £9.22 £239.: ..._____.r°61°n 3. “(1. any. ad 229.

v(x.,x,,y.,y,) are two real, single-valued functions

‘2; the real variables x,,x,,y.,yi.

01). The necessary and sufficient condition that the

 

‘
owns and , e 83 e
T p 21

22
complex function we- uix. ,xuy, ,y,)+ iv(x, ,xuy, ,yz)
33 regular in both variables _i_n 8 _i_s_ that the

 

artial derivatives 32.. is. , £24.. , .291. (i=1,2
L“"' 5x; ' by; ex; 33;, ’ )
exist and are continuous in S and satisfy the

partial differential equations,

 

ea _ av 25 =__§_!. (1:1,:2).
bx; - 5‘5‘ ’ 35; bxi

The proof of this theorem is made quite obvious
when we recall that in Hartogs' theorem the analyticity
of a function in each complex variable separately is a
necessary and sufficient condition that the function be
analytic in both variables together.

We have a further interesting theorem dealing with
the application of the Cauchy-Riemann equations to functions
of two complex variables.‘

Theorem 4.2.
31). ‘gizgg_a real, single—valued function v(x,,x1,y,,y;)
M _i_s_ continuous _i_n _a. region 8.
Hg). 21. 3, flag second-m 2295.1. derivatives of v 3321,

are continuous, and satisfy the relations,

 

m 6‘: ,. 2111. = o (i= 1,2)
bx; 3‘51

(b) .21!— = _QLL.
Etnagt 3X; 5‘5.

OK. at“ + 68‘ 6‘51.-

02’ ° $9.29. 19.25:: 2.5.3.939. 2 £211. “110““ “(1: Jun J.)

 

such that the complex function w = unriv 12,

regular 33:1. Se

 

 

* Bochner, pp. 176-177.

23
To prove this theorem, suppose we have w: u+iv,
a function which is regular in a region S. Then applying

theorem i.lb we see that necessarily,

 

 

 

 

bu - 3v
(1) OX. ' 6‘5.
bu z __ 6V
(2’ be. ex.
bu __ 6V
is) an - New
(4) ’3‘; = —' “Tr: .

Therefore, if we are given a real function,
v(x,,x,,y,,yt), in order for some other real function,

u(x, ,xuy. .yz.) . to form with v an analytic function 11+ iv,

u must satisfy the four partial differential equations,

(1), (2), (3), (4). The necessary and sufficient con-

dition that these possess a solution for u is that the

second-order mixed derivatives are independent of their

order of differentiation. The second-order mixed der-

ivatives, independent of their order of differentiation, are,
a‘u a‘u b‘u b‘u 6"u 3%,

Our procedure will be to take each one of these
mixed derivatives, obtain it in as many ways as possible
by differentiating (l), (2), (3), (4) above, equate the
proper second-order mixed derivatives of v, and see what
conditions we get on the function v.

Differentiating equations (1) and (3) with respect

to x, and x,, respectively, we get

a‘u _ a‘v
(1‘) ex, ex. - ex. 65.
I. atv
3‘ LL = ————-' .
( ) but. 3X1. 3" 33‘

Bquating the right hand sides of (la) and (3a), we obtain

3'"! = O‘v
all. 3‘5. bx. 6‘51. .
Differentiating (2) and (4) with respect to y, and

 

(5)

y,, respectively, we get through a similar procedure,

.211... - .92....
a3;ax. - aguaxl

which is merely a restatement of (5).
Differentiating (l)with respect to y; and (4) with

respect to x,, we get.

 

 

a‘v ___ o‘v
or 3%..5‘5: anal;
a‘v o‘v -
(6) "— + """' '°

6:. 6x, 63. 65.. '
Differentiating (3) with respect to y, and (2) with

respect to x,, we get,
a‘v _ a‘v

 

 

or 63.63.. ‘ exeax.
a‘v + c‘v g
bx. 6x; 3‘516‘3‘

which is a restatement of (6).
Differentiating (l) with respect to y. and (2) with

respect to x,, we obtain,

 

 

_ga‘; : e— a‘v ’
or 63} 6 Ex?

.—————- = o
(7) bx:- + O \3?’ o

Differentiating (3) with respect to y, and (4) with

respect to x,, we get,

 

 

a‘v _ __ a‘v
or 0‘5: but
atv OEV
-—-——- ——-—- 2 o
(a) an; '* sq; '

Thus we have the conditions (5). (6). (7), and (8)
for v in order that a function u exist to make w=u+ iv
regular in 8. These conditions are stated in the theorem

as (a), (b), and (c).

25

Let us apply this theorem to several examples.
Example 1. Consider

v = 2x, y, + 2x,y,+ x,y,+ x,_y, .
This function v is real, single-valued, and continuous for
all finite values of z, and 2... Hence the first hypothesis
of theorem 4.2 is satisfied. Computing our mixed derivatives,
we find that

a a ‘- a ‘ a I.
fl=°’&=°.il_=o’_é_xrgo'_a_x_= ’31.. ’ggo’fl—go.
bx‘, big} a 3‘; a 3,, snag, 68.65. and!“ 63.6%,

These derivatives exist and are continuous for all values
of s, and 2,. Substituting these values in the equations
(a), (b), (c), we see that H2 is satisfied. Hence there
must exist a function u, such that u+ iv is regular for
all finite z. and 2... ‘Such a u is
ti: 1} - y.‘+1.x. ‘ y.y;+x§+yt 0
coming from the function
we 37+ z.z.+ 2‘;

= (x.+ iy. )‘+ (x.+ iy. )(x..+ iyl) + (x..+ iyi)‘.

=(x,‘ ~y.‘+ x. x,-y. y,+ xt+y,_‘)+i (2x.y,+2x,y,+x.y,_+x1y. )

=u+ iv .
This function, w= 23+ z. 2.4- at, is regular for all finite

values, as predicted by our application of the theorem.

Example 2. Consider
V = In I; " lax: .

x: + y:
Immediately we see that v is not defined for xv O, y,,= O,
that is, for the point z._= 0. Hence, any region 3, which

we consider must exclude hue . Our second-order mixed

26

der ivat ives are

 

 

 

 

 

a“! = o 6“ - 2 :I.-6 . :ng's gx. x:+2 .1:
bx.‘ ' but ' I: + Y: ’
a‘v =‘ O ’ a‘v z ~21.x§+6y,x.x;v6y.y:x,-2zix.
by? 63‘; (x‘; + y:_)3 ’

 

 

a“! “Ia i " In. 21: a. “axe t.
snag. gm'a dude. 3.1%” emu, =GT§£LFJ 53,65, :T-‘ng-F
Substituting the proper derivatives in the equations (a),

.(b), (e) of the theorem, we find that all are satisfied.
Thus, by the conclusion of the theorem there must exist a
function u.such.that Unhiv will be regular for all finite

values of z. and 3, except for 3,: 0. Such a u is

11 __ XiX;+ I. 2a
(11* it)

coming from the function,

' I. x.+ ix. x. x91, y. 12. 1.3231.
If? ya. +

5: I:* Y: 31;" Y;
As predicted, the function w = ~34:- is regular in any
finite region which does not include points with their
1,-coordinate equal to 0.

It might be interesting to investigate an example
which does not satisfy the hypotheses of the theorem. A1-
most am v chosen at random would fail to satisfy the
conditions of the theorem. For example, v = xf+ x,y.,

would give 2 ='- 0 for equation (a) of the theorem.

The Theorem 9; the Maximum. For functions of one
complex variable the theorem of the maximum is "
Theorem 4.3a.

El). f(z) 3.3 regular _i_n _a_ region 8 and continuous

 

. Oopson, 1). 162a

27
an. the boundary 2; 8.

01). There exists _n_o_ point, s-a, interior to 8 such

that

 

(1(2)) 4 law)

for all 3 within 3, unless f(z) E. f(a).

The corresponding theorem for functions of two complex
variables follows almost word for word.‘
Theorem 4.3b. ,
3}). fix. ,s..) _i_g regular _i_n _a_ region 8 and continuous
93333 boundary _o_f_ 3.
01). £11933 exists 22.29.12}. (a.,a,), interior 3.3 8,
such that
)f(s.,z,)‘ é |f(a..ai)l

_i_‘_o_1_'_ 21.1.. (s. ,s;) _i_n 8 unless f(z. ,z,_) E f(a.,a,_).

 

Expressed more simply, both these theorems state that a
function which is regular in an open region and continuous
on the boundary of that region attains no maximum value
in that open region unless the function is a constant.

The extreme similarity between the theorems for the two

. cases is due to the fact that the theorem for functions
of two complex variables is proved by considering each
variable separately and applying the theorem for functions

of one complex variable.

 

‘ Bcchner, p.175.

CHAPTER'V

THE CAUOHY THEOREM. CAUCHY'S INTEGRAL FORMULA. TAYLOR'S
EXEANSIONS. ASSOCIATED RADII OF CONVERGENCE.

In the theory of functions of one complex variable,
the Cauchy theorem and the Cauchy integral formula are ex-
tremely important in that they enable us to obtain power
series expansions for the functions being considered. In
the case of functions of two complex variables we use some

similar theorems for the same purpose.

‘22: Cauchy Theorem. The Cauchy theorem for functilns
of a single complex variable 18“
Theorem 5.1a.
31). A region S _i_g bounded by _a_g ordinar-y closed gum 0.”
fig). f(s) _i_g regular _i_n 8.
H5). f(s) i_s_ continuous in O.
01). £f(f.)dfi= 0. [9, indicates values of s on 0‘.)

For functions of two complex variables we have a
corresponding theorem. “’
Theorem 5.1b.
31). I _i_s_ _a_ Egg-dimensional, closed, Egg-m surface.
Hg). 1 possesses‘glgggl, ana tic representation

(Put. army. 47,) = 0. i=1,2.

 

Oopson, p. 61.
By an ordinary curve is meant a curve that may be broken
up into a finite number of divisions, each of which is
either a rectilinear segment parallel to one of the co-
ordinate axes or else has the property that it is deter-
mined by a function, y (81, where (x) and its in-
verse function, x (y), are single-valued and have
first derivatives that are continuous except at most

at the end points.
**‘ Behnke, p. 41.

3*

29
33). P £9.11. contract _i_ng region S EEME 323333.
34). f(s. .81) _ig regular _i_n 8.
£5). f(s. ,sz) _i_s_ continuous 21; F.

Cl). lfl fi.,€1)d%,d%l= 0. [3,3,1 indicate values of s, , 2,,

on 3.]

We should note that the surface I is a boundary surface
of a region in which f(z. .31) is regular. Also, it is
the surface over which we integrate to obtain the con-
clusion for our theorem. 7
The converse of the Cauchy theorem, known as
Morera's theorem, is also of interest. For functions
of one complex variable it is '
Theorem 6.2a.
3]). fix) is. continuous in: region S.
38" £f(€)dfi= O_f_o_r_a_n_y_ closed gum C _i_g S, 11.1.9.5!
C incloses gn_ly_ points 93 3.
Cl). 4f(s) _i_s_ regular _i_n S.

For functions of two complex variables. Morera's theorem
reads,”
Theorem 5.2b.
3],). fix. ,s,) _i_s_ continuous _i_n _a_ region S.
' Hg). Lflfinfiddfidfi, = 0 taken over Eh: boundary

surface, 1!, $3331 dicylinder lyig completely
a 8‘

 

 

01). 1". ,'3_) 3:9. regular 33 Se

 

"‘ Townsend, p. 80.
’* Behnke, p. 41.

30
The interesting fact about Morera's theorem for functions
of two complex variables is that a special surface of in-
tegration, namely the boundary surface of a dicylinder,

may be used.

Th: Caucgy Integgal Formula. For functions of one
complex variable this important theorem is usually stated as ”
Theorem 5.3a.

El" 8 jg g finite, closed region with. _a- boundary C M

consists 93 g finite number of ordinary curves.

 

Hz). f(s) _i_g regular within S and continuous on C.

01). For 3g. inner point 3 _o_i; 3

f(s)=(-i-TJf-z—) Iii—gang,

In the case of functions of two complex variables,
we do not have a complete generalization.”I Whereas in
Cauchy'e theorem we were able to use a surface of integration,
for this theorem the integration is made over two curves in
succession.
Theorem 6.3b.
H1). 8 _i_s_ a. generalised dicylinder 11:15.13. boundaries 01
same 's-Maani 02 mm: 5:133}...
Hz). f(s.,s,)'ig regular inside S Egg_continuous on

_t_1_1_e_ boundary 32$ 8.
C1). Egg point (3.,21) _o__:_‘._ S
2.
1'1 . 2,) a 1 “fink emf...
z ’ (211.1) £4 ($.'30)(‘i"zz,

[3. and 9;, indicate values of s. on C. and 2.. on 0:.
respect ively.]

 

 

 

‘ Copson, p. 66.
” Behnke, p. 40.

31

(As we shall show, this theorem is proved by
applying twice the Cauchy integral formula for functions
of one variable.

We have given a region 3, a generalized dicylinder,
which consists of a region S.in the z.-plane and a region
8, in the s,-plane. Now if we take a; fixed at some point
si in 8;, we then have a function of one variable which is
regular in 8.. Applying theorem 5.3a, the Cauchy integral
formula for the case of one variable, and integrating about
C., we get

(1) f‘ll 93;):5‘7 ffzé'Jzigfl fit 9
C. I- ’

 

for any value of z. in 8,
Next take the function f(%,,z'1) from (1) and con-
sider 5. as fixed on.C,,with :1 now varying. Applying

theorem 5.3a once more we have

mun) a. __1_._. [Wasp

Substituting this for f( finsL) in (l) and dropping the

prime from.s;, we have that for any point (s,,s,) in 8

(2) ' flz,,z,)= fiffiélaflmL £1349...

 

-1. )( erg-8:.)
It is interesting to note that in proving the
Cauchy intergral formula for functions of two complex
variables we have not been forced to use all the con-
ditions given in the hypotheses. We might restate the
theorem, giving only such hypotheses as were used in
the proof of the formula (2).

32

Eecrem 2.4.
m. as. _____.am 2 as. 19.3.9. lanai. as me:
f(s. ,s,) _i_g regular 5511.}. points (2.,21)
.1225: )zII1é d,.
112). my}; a. may, |%.|=d.. flew-i) _i_g
analytic _i_n_ s, 321; [2,] < d, and continuous

 

for Iig' = d...
H3). 1“. .%1) E- continuoua 2 |ggl = d.’ [31' = 6,...
0;). For 3g: point (z. ,z,) _i_n_ S

f(s.,z;)= (E-ér-Ty‘é: ‘Lm d€,d€, .

A.further interesting observation can be made
concerning functions which have the same boundary values.
For the theory of functions of two variables, as in the
theory for functions of one variable, we can see that if
two functions, say f(s.,s,) and g(s.,z,), have equal
values at all points of the boundary of a region of the
type we have been considering, then the two functions

are identical throughout the region.

Taylor’s Series Expansions. Once we have discussed

 

the Cauchy integral theorem, the next logical step is to
investigate the possibility of using the theorem, as we
do for functions of one complex variable, in developing
some kind of a Taylor's series expansion for a function
we know to be regular in a given region.

We new state the theorem concerning Taylor's series

for functions of one complex variable. ’

 

‘ Copson, p. 73.

33
Theorem 6.5a.
H1). fl!) _i_g regular _i_n the neighborhood ls - a) < R
2;. 2.1.1.2 223m. . = ..
Hg). f(s) 321E mesented _i_n that neighborhood _a_!

g convergen_t_ power series 93 the form

 

f(2) = f(&)+f'(&)(3-C)+ fu(?2(z_a)‘+'°°'+£‘:)(—§l(z")h+...
. n!

We might recall that this series is uniformly convergent when

|soa|éRH where R.<R.

We state the corresponding theorem for functions of
two complex variables. "'
Theorem 6.3b.
31-). f(s. ,zl) _i_g regular _ip g dicylinder 8;
Is. - a.|<d. , Is; - a;|<d.,.
Hz). f(il| ,sz) _i_s_ continuous 32 3113 boundary £3 8.

Cl). f(s. ,2;) can 333 represented _i_g S _a_g _a_ convergent

double power series 2.3 the form,

«-
f(s. .21.) = Z 8...; (Z. - G.)m(3a ' 9...)! e

 

"5.30
where
3“ = _L. ah"f(&|Lag .
' m! l! as? as}

To prove this theorem, let us suppose that we have,
as stated in the hypotheses, a function f(z.,z,) which is
regular in a dicylinder S, and continuous on the boundary
of that dicylinder.

 

‘ Behnke, p. 40.

34
First, let us fix 5, at 3;, some value such that
Is; - a;(<2d,. We then have f(z,,21), a function of 2.
alone, which is regular for (z. - a.l<d.. Now, using
Taylor's theorem for functions of one complex variable,

theorem 5.5a, we can represent f(s.,zi) by

1". .31) = f“.’z;)+ Mew-+- e e e
as“ l!

a" mu ), (s. -a. Y“

I
3:.
J

32:“ n!

(3)

eeee+ +eee

ConSider the general term

.8"f(a.Lg{_). (20'3”,m
313' - m3

of the series. Take f(a.,z;) and let 2; vary. We again

(4)

have a function of one complex variable. Applying theorem

5.5a once more we get an expansion

f“. 93;.) = 1(aleal)+ afta'L‘aLL-‘Le M)+ e e e

 

 

311 1!
(6) i , 1
eee+ a 1(“laL2-OM+OOO
bid) 1!

By a well known theorem of complex variable this series may be
differentiated term by term.“ Substituting (5) in (4) and
performing the differentiation, we obtain a general term of

(4), and thereby a general term of (3) also,

(a) __|____, 3""fhu‘al... (z,-a.)~(s.,-a;)1 .

m! 1! at." at.‘

The s; has been replaced by 2,, since it was adopted as a

 

means of notation, only. We see that in (6) we have ob-

1
tained the general term a.“ (z. -a. )"(zt-az) as given in

 

’ Townsend, p. 237.

35
the statement of the theorem.

‘ We can obtain an equivalent, although less useful
form for the coefficient amfi_. .dpplying the theorem
concerning derivatives of functions of one complex vari-
able on the general term, (4), of the series (3) we ob-

tain for (4)’

‘7) (”5% [Mpadm' “g“.Laq-fii °

Fixing a. on C| and applying the same theorem on the
general term of the Taylor's expansion of f(fi,,sz)

we obtain

.. o . . y m) f“ en) 1'81 Q .
(8) fit. .81) + 21” cmcfiu _.;7_L +

Substituting (8) in (7) we find for the coefficient of the

general term of the expansion for f(z.,zl)

.“Q=('§_L— w‘E—‘z— )ff fit (‘2: “)1“ agad'g' ‘

This is an equivalent form for the coefficient in (6).

 

We shall now discuss several examples of Taylor's
series expansions for functions of two complex variables.
Example 1. Expand f(z.,z,) = 23.+'3z§ about (l+i,2).
This function is regular for all finite values; hence,
the power series expansion will be convergent for any di-
cylinder about the point (lfi,2). Finding the derivatives
and substituting values as required in the conclusion of

theorem 5.5b, we obtain a finite power series,

f(s,,z,) = (l4+2i)+2(s,- 1 - i)+12(z,,- 2)+3(z.- 2)‘.

 

' Townsend, p.77.

36
Example 2. Expand f(s,,z,) : ‘ErgLr’ about the
point (1,1). This function has a singular point at
s,= 1. Hence our expansion will be convergent only
for dicylinders such that (z, - l)<L(2t By taking

derivatives and using an induction we find that

 

 

 

 

a; __ a, a": _ a‘c t-n‘u 2.
al. - fl... ’ Bi.“ -o’(m>1)9 6%: = (1”. 1,1*.

8”"? = (4)91! , and 9m“): : O for (m>l).
alibi: (31‘1,1+' 61:“ 6%:

Substituting these in the expressions given in the con-

 

clusion of theorem 5.5b, we obtain for the expansion about

the point (1,1).

 

 

I e4 1
f , l = +LELL1J lL-l, (IL-1)! +('|1.1) J11 (’L'l)
(.I B ) 1.1 " 1 l - 1 + 2:. (1 - 1,1“ )2“ 1 - 131...

jégsociated Radii pf Convergengg. In dealing with

 

 

power series expansions of functions of one complex vari-
able we define the radius of convergence of the power
series. If we describe a circle having a radius r about
the point 3 a such that the power series being con-
sidered converges for all values of 2 within the circle
and diverges for all values of 2 outside the circle, then

we define r to be the radius pf convergence of the power

 

series‘. Moreover, the power series converges uniformly
and absolutely for all values of s such that Is - a|<:r,,
where r,<: r, and ordinary convergence may occur only on
the circle I: - a)=

In investigating the possibility of radii of con-

 

’ Townsend, p. 230.

37
vergence when considering functions of two complex
variables, mathematicians have introduced the idea of
associated radii of convergence." .A pair of positive

numbers, rf, r{, are called a pair of associated radii

 

 

‘2; convergence of the power series expanded about the
point (a,,a;) if this series converges for

Is, - a.l<r.’ , ‘3; - a,‘<r{
and diverges for

)s. - a.)7r.’ , (2‘ - a..)>r.', .

It can be seen that the values of r( and r; depend
on each other; that is, the power series might still
converge if rf were larger and I; were smaller, and vice-
versa. Hence, if we set

)2. - a.l= r, and '2; - ‘g'gr;’
then the pair of associated radii of convergence of the
power series p(z.-a.,z,-a;) describe a curve ‘P(r,,r,) = O
in an. r,r,-plane, and this in turn would mean a three-
dimensional manifold in the s.s;, four-space. This
three-dimensional manifold is the boundary of the region
of convergence of the power series. We shall designate
this region of convergence as a region K3

.Analogous to the theory for one complex variable,
the power series converges absolutely and uniformly in-
side.K, and ordinary convergence can occur only on the
boundary of K. However, ordinary convergence may also
occur on the planes 2.: 0, z,=-O, protruding out from
K. *Such protruberances from the region.K.are referred

to as spines of the convergent space.

 

** Behnke, pp. 36-39.

38
Example. consider the series £25.22 . This series
possesses a region of convergence (”1|< 1. It also
possesses the spine
5' = 0 v (31' 3 1 -

There are several interesting properties of the
associated radii of convergence.
Property 1. Monotone property. If r.’ < r,’ and r,’,
corresponds to r! while r: corresponds to rf, then
r; ) rf. That is, if the radius in the z.-p1ane in-
creases, the radius in the zt-plane either remains the
same or decreases.
Property 2. If r{,r{ is a pair of associated radii
of convergence of the power series p(z.-a.,z,-ar),

then there exists at least one point (2:,21) where

'23. " rt 0 |.a " I“

at which the regular function represented by P(s.-a.,s;-a,)
becomes singular. This corresponds to the statement in

the theory of functions of one complex variable that

the power series expansion of an analytic function holds

out to the nearest singular point.

CHAPTER VI
SINGULARITIES.

.18 in the theory of functions of one complex
variable, we say that a function of two complex.vari-
ables, f(z.,z,), has a singular 22123 at (a.,a;) if
1(I.,lz) is not regular at (a.,a;) but has points in
any neighborhood of (a.,a;) at which it is regular.

As in the case of one complex variable, these
singular points may be classified as non-essential
singular points and essential singular points. How-
ever, as we shall see, in the theory of functions of
two complex variables the non-essential singular points
themselves are of two types, poles or points of in-
determinacy.

We say that a function f(z.,zn) has 3.2227

essential singularin at the point (a.,an) if there

 

exists a neighborhood U(a.,a;) of the point and two
functions g(l.,z;) and h(z.,ze). which have no common

factors and are regular in U(a.,ai). such that

f‘B. ’21) = w

h". .32.)
in U“! o‘a) and 11“: 932.) = Do If 8“. e‘a’ ¢ 0, than

(a. ,a.) is called a £o_l_e_, or non-essential singularity
of $2333.33; _k_131_<_1_. If g(a.,a,_) = 0, then (a. ,a1) is
called a point of indeterminagy, or a non-essential
singularggz of the second kind. All singularities

 

which are not non-essential singularities we define

as essential singularities. Thus, for example, the

39

40
3n
31-.
of the type (2., a), and a point of indeterminacy at

the point (O,a), while the function f(z.,z;) == e L“

 

function, f(z.,z;)== has poles at all points

has essential singularities at all points of the type (2.,a).
In discussing functions of two complex variables,

reference is often made to functions which are mero-

morphic in a region. Essentially, we say that a function

is meromorphic in a region if it possesses only regular

points and poles in that region. More accurately, we

define a function f(z.,z;) to be meromorphic in a region

 

8 of the four-space if *

(a) there is an exceptional point set E not decom-
posing S; [that is, B - E is connected 1,

(b) f(z,,zz) is regular in S - E,

(a) corresponding to any point of E there exists a
neighborhood and two functions g(z,,zl) and
h(s.,zl), which are regular in that neighbor-
hood, such that f: _g__ [the fraction being
in its lowest terms] in the points of 3 - E
lying in the neighborhood, and

(d) the set E must be the minimum set having this
property.

In studying the singularities of functions of
two complex variables, much of our work will be de-
pendent on a theorem that is generally referred to as

the continuity theorem. We shall here state the theorem

without proof.”

 

‘ Bochner, p. 198.
** Behnke, p. 49. ‘3 proof for this theorem may be found

in Bochner. pp. 199-201.

41
Theorem 6.1.
3],). fh, ,zz) _i_s_ regular 213.3 circle:
[2,-8. =1“, 21:: 8;.
This circle lies on a two-dimensional plane:
I. varying , 2; = a; .
1: 31131;]; designate 3.112% _a_g 8. .
Hg). 29.9.3.2. exists a sequence 93 two-dimensional
planes, designated {8,} , M converges to
8. - 229.2 .12

(P) (v)
a; = e such that lim 6 = a;.
y—boo

 

 

I13). f(z. ,st) _i_s_ regular 22 these planes for |z, - a.‘$r.

01% mun) EWEP. 5. £23; I?" - MK“

If we wish to consider single-valued functions
only, the continuity theorem may be stated in somewhat
more useful form.’

Theorem 6.2.
31). fits. ,zd _i_s regular on 2 circle:
‘2. - a.I=r, 3;: a; .

32)e f‘ z. .31) E Bizgular _a_»: (Q. ’83,)e

 

01). There always exists _a_ d > 0, such that on each

 

plane, 3;: 6‘, where 'a; - 64(d, there _i_ggt
least one sigular point for values _o_:_f_ z. _i_n-

side_g_f_ ‘2, - a.|<r .

 

This theorem may be proved directly from theorem 6.1.
We shall illustrate theorem 6.2 with examples of

several different functions.

 

" Behnke, p. 49.

a, 42
Example 1. Consider f( 3, ,st) e‘wa' . This function
has essential singularities at the points where z,== a..
The hypotheses of theorem 6.2 are satisfied since the
function is regular on the circle: [2, - a.‘= r, zl== at,
and since the function is singular at (a.,a;). Hence, as
we would expect, the conclusion of the theorem holds for
this function. Moreover, d may have any positive, real
value, since no matter what complex value we take for ,
we have a singular point (a. ,6.) on the plane 3,: as.
This singular point (a.,em) completes the requirements
of the theorem by having its z,-value inside ‘5. ~ a.‘<£ r.

Note that this function has a two-dimensional

manifold, or more specifically a plane, of singularities;
that is, for z.- a., any value in the zt-plane gives an
essential singular point.
Example 2. Consider f(z.,z.)=:._;r_%_§;:. This function,
we see, has poles at all points (z.,z,) for which 2.: 23..
How, applying theorem 6.2, we find that all the hypotheses
are satisfied, since the function is regular for values
on the circle: Is. - 2a.I= r, z;=s a., and since the
function is singular at (Baked. Our next step is to de-
termine in what way the conclusion of the theorem applies.
On any plane ate-.6, we have a singular point when a. = 26;.
Oonceivably, our choice or the plane a,=.61 might be such
that the value z.= 26., might lie outside the circle
I:.- 2a.|= r. Hence, for a singular point (2 51.62.)

satisfying the conditions of the theorem, the z.-value,

43
25,, must be such that I26. - 2a.| < r. Simplified,

this requires that ‘€;- a.‘ < .2, meaning that the choice

at: e, must be made within a distance .12: of a.. In

short, for this function, d = r .

2
Observe that in this example, also, the function

has a two-dimensional manifold of singularities, since
for each value of I; there is a corresponding value of
a. giving a pole.

W. Consider fh. ,z;) = ”"1: . It is in-

(‘g " '2.)
teresting to note that this function has poles at all

 

points (a,a) except at the point (1,1) where it has
a point of indeterminacy since the function takes on
a value .8. there.

Seeking to apply theorem 6.2, we again find the
hypotheses satisfied, the function being regular on the
circle: '8. - 1|: r, 3,8 1 and having a singularity
at the point (1,1). The conclusion of the theorem
naturally follows. On any plane 2, =£, we have a
singularity when z.=€t . In order for the value of
8. giving a singular point to be inside the circle
'1. - 1|: r, we see that our choice of 6.; must be such
that ‘6..- 1! < r. This means that the value 2., = e,
must be chosen inside a circle of radius r about the
point 5‘: 1 in the zt-plane. In other words, for this

example , d 2 r.

There are several interesting consequences of
the continuity theorem which we shall now proceed to
state and discuss.

Consequence 1.

H1). f(z.,z,) ég_regular in a region S of the four-

 

_s_p_a_c_e_ except £25 2 Eng-dimens ional manifold _a_._t_
in: Easi-

Cl). Then f(z. ,zt) _i_s_ regular _i_g 1133 m: interior
23’s.

Before proving this consequence, let us investigate
its meaning. In simpler form, this consequence states
that for functions of two complex variables, if there
exist any singularities of the function in a given four-
dimensional region, then there must be at least a two-
dimensional manifold, or a double infinity, so to speak,
of singularities of the function in that region. For
example, a function cannot have one point of singularity,
such as (a.,ai). Neither can a function have a one-
dhmensional manifold of singularities. Referring back
to the examples already discussed in this chapter, we
see that in each case, the singularities formed a two-
dimensional manifold.

This is quite in contrast with functions of one
complex variable. For functions of one complex variable,
singular points are for the most part isolated. This is
always true in the case of poles.‘ An essential singular
point is in most cases isolated. However, an essential

singular point may be the limit point of an infinite

 

‘ Townsend, p. 270.

45
sequence of poles or of an infinite sequence of essential
singular points.‘

We shall now indicate a proof of consequence l
for the case where S is a generalized dicylinder. By
hypothesis our function f(z.,z;) is known to be reg-
ular everywhere in 8 except for a one-dimensional mani-
fold. Suppose we consider the case where z. is fixed
and 2‘ describes a curve in the zg-plane.

By hypothesis we know f(z,,z;) to be regular

s.-plane st-plane

\/ K/

’1‘”. 6e1e
in 8 except on the one-dimensional manifold: z,

 

 

an e

2‘ on arc LB. We will take any point a, on LB and

show that the function is regular at (1.,at). In 3

on the zit-plane take a sequence of points, {CM , none

of which lie on LB, with a; as the limit point. Con-
struct a circle in 8 about the point z.= a.. Let r
designate the radius of the circle. These constructions
will be possible for any at chosen in 3. Our next step
is to apply theorem 6.1, the continuity theorem. Our
f(z.,z;) is regular on the circle: ‘3, - a.|= r, s;=-ai,

and it is regular for '2, - adér on the sequence of

 

' Townsend, p. 271.

46
(v) ..
planes 5.58 . Hence, by the theorem, f(z.,z;) is
regular for is. - a..< r, zl= a1, and so the point
(a.,a;) is a regular point. Since this is true for any
choice of a;, there cannot be any such manifold of

singularities. .L proof for the general case would be

similar.

Consequence 2.

El). ‘QEIEQIQ dicylinder S:
lz'“a||<d'u ‘32.’ 34(51-
H2). 93 _e_a_c_h_ two-dimensional £13.22 7.; = b“ m
signings. izz-adenMEE
Eo_s_t_ one singular 2.2.1.22. 23 3.1.1.9. regglar function,
f(z.,zi).
H3). f(z.,z,)_h_a_s_gt__1_e_a_§_t__o_n_g singular w, P, _i_n_ 8.

01). There exists exactly one two-dimensional surface,

 

F, passigg through P such that all singularities
‘3; f(z.,z;) which lie 32 8 lie 2g F, and each

point 23 F _i_sg singular point 2; f(z. ,s;).
03). F satisfies §g_equation 2.x g(z;) for '31- ail<1d1.

 

Several examples will serve to illustrate this con-

sequence e

5a
z - b
"g- .||<d.| , ‘z‘- &;‘<dl. Take 3.. 8112311 that d|>'a|- b.‘ ,

that is, such that b. is in the dicylinder. The hypotheses

 

Example 1. Consider f(z.,zz):= in a dicylinder:

of consequence 2 are all satisfied since on each two-dim-
ensional plane 2,: b.., where b... is a value in \x,- szl< :11,

we have only one singular point (b.,b;), and since the

47
function has at least one singularity in the dicylinder;
for example, the point (b., ar+%§J. Now, the question
is, can.we find the surface F of the type described in
the conclusions of consequence 2. Consider the plane
2. = b. , 2., varying. This plane passes through (b. ,a;+-d§-).
All the singular points of f(2.,zt) lie on this plane, for
in order that a point he a singularity, it must have a
2.-coordinate, 2.== b.. Furthermore, each point of this
plane is a singular point of f(2.,2;) for the same reason.
Finally, this plane has an analytic representation, 2. = b. .
Eunmple 2. Consider f(2.,2,):= (2,-2i%(2.+2;) in a di-
cylinder about the origin: l2.| < d. , [s,|<d:. Im-
mediately we see that the second hypothesis of consequence
2 is not satisfied, since for any plane 21:; at, where
e... is a value in ‘2,|<d;, there exist 3.29. singular
points (ai,a.) and (-a.,a;). Hence it is impossible

to find any two-dimensional surface I on which all the
singular points of fin. .32.) lie.

Consequence 3.
El). The point set 3‘ lies _i_._n_ _a. region S and _i_s_ cut _‘qy
_a_ny two-dimensional plane 2,: b; once at the most.

 

Hz). f(2. ,2.,) _i_s_ regular 31:. mm _9_f S which does
£9.33. belong 19 I".

Hg). f(2. ,2.) _i_g siggular flflkflmflsfi F.

'01). f(2. ,2i) _i_g singular EEMME F.

 

 

The meaning of this consequence may be somewhat

48

clarified by the discussion of an example.

 

Example 1. Consider f(z.,2;) =' sin ( 1 ). Let

S be a generalized dicylinder about the :rig'i‘n, and let
I be the set of points (2.,2;) lying in S such that
2.: 2;. We see that F is cut by any plans 2,: b; in
one point at the most, namely, the point (b;,b;). Also
f(2.,2l) is regular at every point (2.,2‘) of S where
2.#=2,§ and f(2,,2,) possesses an essential singularity
at the point, (1,1). Thus, all the hypotheses of con-
sequence 5 are satisfied, and f(2.,2.) should be
singular at every point of F. Upon examining our function,
we see that at every point (2.,2g) for which 2.: 2,,

f(2,,2;) possesses an essential singularity. These are

the points of F.

Consgquence 4.
H1). f(2.,2;) Ag regular 23 all boundary points 23.3

region 3.
H2). S'igwg closed, finite region with 2 connected

boundary.
Cl). f(2.,2.) can 23 analytically continued 32_each

 

inner point 23 S.

This is the most important of the consequences we shall
discuss. To better show its importance we might restate
it more simply. Consequence 4 states that $3 §_function

.25 singular anywhere 33 the interior 2£.E closed, finite region

 

with g_connected boundary, then that functionngg'necessariiy

49

siggular _i_n .95. 22929.3 .933 boundary 29.3.9.3 93 21?. region.
Herein the theory of functions of two complex
variables differs greatly from the theory of functions
of one complex variable. In the case of one variable
a function may be regular on the boundary of a region,
yet still have singularities inside that region. For
example, the function f(2) = -——%?—— has a singular
point 2: a, yet on the boundary of any region enclosing
2==a as an inner point, f(2) is regular.
We might investigate a few examples of functions
of two complex variables as consequence 4 applies to them.
Example 1. Consider f(2.,2‘)'= gi-. We shall use three
different regions to illustrate three different applications.
(a). Let S be the dicylinder: ‘2.)( d. , \2..- ai‘(d;,
where Iatlj> at, as shown in figure 6.2. The function

2,-p1ane 2t-plane

fin
\%

f(2.,2,) = EL- is regular at all points of the boundary,
2

since the value 21: 0 has been excluded from the region.

 

 

 

 

 

Figure 6.2.

By the conclusion of consequence 4, there can be no singular
points inside the region. We see that this is true, since
all the singularities of the function lie on the plane 2 0.
(b). Let s be the dicylinder: |s.| < d. , )3.) < d..

The function f(2.,2g)=«%L has a pole at the point b%%,0).

L

50
Since this point is interior to the dicylinder, by the
restatement of consequence 4 there must be at least one
singular point on the boundary of the dicylinder. The
point (d.,0) is such a point. In fact, every point
(2. ,0), where |2.)= d. , is a singular point on the
boundary of S.
(c). It should be noticed that a function may have a
singular point on the boundary of the region which is
being considered without being singular at any point
interior to the region. This does not contradict the
statements of the theorem. If we consider the same
function, f(2. ,2.) = .34.. , in a dicylinder: ls.|<d.,
)2;- a;|(la;' , we havetsuch a situation. For this

2.-plane zgoplane

m  
K / 5:15.:

Figure 6.3.

 

 

dicylinder the function has poles at all the boundary
points (2.,0) such that l2.l==d.. Yet there are no
singularities inside the region, since in order for a
point to be a singular point, its zz-coordinate must
be zero, and only boundary points possess such a value

for their zz-coordinate.

CHAPTER VII

ZEROS. NON-ESSENTIAL SINGULAR POINTS. MITTAG-
LEFFLER'S THEOREM. THE WEIERSTRASS PRODUCT THEOREM.

53522. For functions of two complex variables
we can define zeros in the same way that they are de-
fined for functions of one complex variable. If f(2.,2z)
vanishes at the point (a.,a;) and is regular in a neigh-
borhood of (a.,a;), we say that (a.,a1) is a 5253 of f(2.,2;).

From our discussion of singularities, where those
singularities might be confined to poles, we would expect
that if there exist any zeros of a function in some region
there will be at least a two—dimensional manifold of such
zeros. Suppose we wish to consider a function f(z.,2l)

for its zeros in a region S. We will accomplish the same

 

end by considering the function 1‘ 1' ) for its poles
‘8 9'1
in 3. By consequence 1 of theorem 6.1, if -——L——- has
f(3|,21,

any poles, it will possess at least one two-dimensional
manifold of poles. Hence, f(2.,2;), if it has any zeros
in 8, will possess at least one two-dimensional manifold
of zeros in S. In a similar way the other consequences

of the continuity theorem may be interpreted in terms of

l
f(zle'l’
In order to determine these zero manifolds as we

 

zeros by using the function .
shall call them, the preparation theorem of Weierstrass
is used.’

Theorem 7.1.

El). f(2. .23.) _i_s_Wgt the point (a.,&1)0

 

*: Behnke, p. 57.
51

 

 

 

. 62
H2). f(a. ,a1) = 0, but f(z. .21) i o.
01). There exists _a_ neighborhood U(a, .31) 3f the

 

point (a.,a;) such that there exist:

 

a). _a_ function 1 which Ls regular
Q‘zlezl)
in U and does not vanish in U,

 

b). 3 whole number M/ 0

c). _a_ function KPH. ,21) which _i_s_ identically

 

 

equal 19 1, 25933113 form
(1) 4”?" 932.) 7' (3t’Q|)~+A|(31)(3|’3I,m-‘+”"“+Am(zx)°
where 113 A:(2,_) 933 regu_.l_ar _i_n U 9.93.

vanish 2.3 2..= at,

 

such that f(2. ,21) can be written in the f_3___rm,

(2) runs.) = (3,- a.)’“———l———wz. .z..>.

c2). g f(2. ,e,) -_=. 0, then the factor n,- e,)"’ _i_s_

omitted from representation (2).

A function of type (1) is called a distinguished
polynomial. Thus we define a distinguished polynomial
with respect to 2;: a; to be a function of the form

(23,- a.)m+l.(2.,)(2.-a.)""+ . . . + L4“).
where the A¢(2z) are regular in the neighborhood of a
point si= aIL and vanish at 2,: at.

Observing (2) we note that we can now find, in the
neighborhood U(a.,ag), the manifolds upon which f(2.,2;)
equals zero. Naturally, one such manifold will be the

plane 21: a, if the factor (2,;- azr occurs in the

53

 

representation (2). The factor 1' does not
vanish in U(a.,a1), by definitioggz.h:n;e, this factor
will give rise to no zeros of the function f(2.,2;) in
U(a.,a;). Obviously, we can obtain all other zero
manifolds of f(z.,zz) by setting the factor lP(z.,z,)
equal to zero and solving the resulting mlth degree
equation. This gives us m roots,
z.= s.(zi). 2.: gitzi), . . . , 2.: gm(2;).
These are the m, manifolds upon.which QJ(2.,2;) equals
zero, and therefore they are the m manifolds upon.which
f(2.,2;) equals zero in the neighborhood U(a.,a;).
Bochner gives a method for finding this distin-

guished polynomia1.\P(2.,zt) for a given function, f(z.,z;).*
We shall outline the method and illustrate with an example.
This method uses the origin, (0,0), as the point (a.,a;),
in a neighborhood of which we desire to investigate the
zero manifolds of f(z.,z;). This choice does not limit
the generality of the method, since our given function can
always be transformed by use of substitutions 2{=(z.- a.)
and 21=(2;- ag), enabling us to consider the transformed
function at (0,0). rather than the given function at (a.,a;).
Also, we shall consider functions such that f(2. ,ag) #- o,
permitting us by conclusion 2 of the theorem to omit the
factor (21- ath from representation (2).

‘ Thus we are considering f(2.,2;) about the point
(0,0), and by the preparation theorem, this function has

a representation,

(3) fl , )= 1 (2.,2) .
3' z; Q(8l ’zz) w I.

 

 

*‘ Bochner. DD. 183-184.

Our task is to find Q(z,,zl) and (P(z.,z;). Let
\P(2.,zz) = B(2.,21)-k H(z.,zz).

This enables us to choose B(2.,zi) arbitrarily. Then

we must find an H(z.,2;) which added to B(2,,z;)

will give a distinguished polynomial,\P(s.,z,). Sub-

stituting B(2.,2i)+'H(z.,zz) for \P(2.,z;) in (3)

and solving for H(z,,2t), we get

(4) H(2.,zt):= f(2.,zi)Q(z.,zz) - B(z,,z;).

Therefore, to find H(2.,zg) we must find Q(z.,2;),

given f(z.,z;) and having chosen B(2.,2L) arbitrarily.

1

Q‘zlezz)
must be regular in U(a.,ai). and hence may be expanded

Since must not vanish in U(a.,ai), Q(2.,2L)

in a power series

so
(6) Q(z,,zz) z. :E: q 272: .

Mmze "h"
The coefficients qwm,may be found by the use of a re-
cursion formula which involves the coefficients hm".

and fm, from the expansions

(6) . 3‘3! 931.) = a. bum €32: e
(7) fix. ,2.,) = :2 f,“ 2:“22

for B(s.,z;) and f(z.,2‘). This recursion formula is ‘

0—. Ibo!
(8) Q...“ = hm... "' i Z qr-m Lark-mm?" Z, qnm I'Mk-rao

Hut. 9:.

where k is the power of the first term in the series
.0

(9) f(2.,21) = Z 1.12,) sf"
for which 1.,(o)4=o. h"

Example. Consider

f(a. .21.) = 2i+32._z.+ z?+ z," - 32,2:~

 

’ Bochner, p. 184.

55
for its zero manifolds in a neighborhood U(0,0). In
most cases the function being investigated for its zero
manifolds would be an infinite series. The zero mani-
folds of the function we are investigating would be
most easily found by setting the function,as it stands,
equal to zero and solving the resulting equation for
its roots. However, for purposes of illustration, we
shall carry through the work for this function in the
manner described on pages 55 and 54.

Expressing f(2.,z;) as a series in the form
of (9), we obtain
(10) f(2. .21): 21+(521)2.+ (1- 32..)2T-I- z? .
Since (1452;)?é 0 when zi= 0, then k=2.
We then choose B(z.,2;) '= 2?. Considered
as a power series in the form (6), this means that all
5,“: 0, except bm=1. Considering f(z.,z._) as a
power series in the form (7), we get for the coefficients,
p.,.=0. p.,,=0, pm=l, . . .
P... = 0. P.,. = 3, p.,,,=0, . . .
p...= 1. p.,.=-5, . . .
p,'.=l, . . . and all other p...,,=0.
Next, using our recursion formula, (8), we find
q“:— l, q,” = 6, q“ = 53, qM = 567, (4M :2. 6744, .
q“: -l, q,“ = -12, q," = -l42, q”, = -l770, q,» = -23101, .
flap: 1, q”: 18. Q.» = 267. (1...; = 5825, q... = 54795, .
q» =-l, q” = -24, q,“ = -428, q” = ~6948, q”: ~108846, .
1, q,” = 30, th= 625, q..,,=-— 11355, q.“ = 193570, .

qe,e =

Substituting these coefficients in the expansion (5)

56

for Q(2.,z;) we obtain that expansion up to and in—
cluding m=4, n=4. Multiplying this by f(z. ,z;),
we get
\WI. .si) = 3(2. .3.) + H”. .z,)
(11) = z.‘+(sz,+ 175:4» 147224- 1559zf+. . . )z,
+(z;+ezf+5zz:+ . . . ).
This, we see, is a distinguished polynomial since the
coefficient of the highest power of 2. is equal to 1
and all the other coefficients of powers of 2. are
regular in the neighborhood of the origin and vanish
at the origin. To find the zero manifolds in the
neighborhood of (0,0), setting (11) equal to zero
and solving for the two roots, we obtain the zero
manifolds
z.= emu). and 2.:- sites).

for the given function in the neighborhood of (0,0).

Our discussion of zeros now leads us to *
Theorem 7.2.
31). f(2. ,21) _i_s_ regular _a_t (a,,a,_).
H2). f(a.,ai) = 0.331;. £(z.,z,)$ o.
01). in _a_ sufficiently small neighborhood U(a. ,al)
elitism M ed aliasing: mum.)

lie on a finite number of two-dimensional

~_—mmu—*——

 

02). All the points of these analytic pieces 2;

surface, if they lie ig’ U(a.,at), are zeros

9-; 1‘3. ,Ba.)e

 

’ Behnke, p. 59.

57

Thus we see that the zeros of functions of two
complex variables differ from the zeros of functions
of one complex variable in much the same way as do
singularities for the two cases. For functions of
two complex variables, if there are any zeros they
will form at least one two-dimensional manifold. In
contrast to this, functions of one complex variable
can have a zero at a point. The function f(z)= z - a,
for example, has a zero at the point 2==a.

In the theory of functions of one complex vari-
able we have a theorem dealing with the zeros of a
function which is sometimes referred to as the unicity
theorem.’
Theorem 7.3a.

Hl). f(2) i3 regular _i_g a region 8.

H2). f(2) _ig equal _t_9_ zero _a_t._a_n_ infinite sequence

 

22 points, z', z”, 2”fi . . . , which have a

 

point interior :2 S 23'; limit point.
01)e f(Z)‘-:..O age

The unicity theorem for functions of two complex

variables is "
Theoram 7e3be
H1). f(2. ,21) _i_s_ regular _i_gg region S.

H2). f(2.,zl) 12 equal 22 zero everywhere £2 St

 

where S' _i_g 2 region interior _t_9_ S.
01’ £(8| ’31) a 0 .121} so

 

* Copson, p. 74.
** Bochner, p. 174.

58

We shall omit the proof of this theorem.

It is interesting to note that in the case of
one complex variable we need know that the function
is zero at a sequence of points, only, while in the
case of two variables we must know that the function
equals zero everywhere in a subregion of the region in
which the function is regular. The reason for this
difference can be pointed out if we recall that functions
of two complex variables which are not identically zero
can still have two-dimensional manifolds upon which they
are equal to zero., Hence, when considering a function
of two complex variables, we could get a sequence of
points, lying on the zero manifolds of the function,
at which the function takes on the value zero, without
having our function identically equal to zero. However,
if we know the function to be zero at an infinite sequence
of points which do not lie on these zero manifolds, then
it necessarily follows that the function is identically
equal to zero in the region being considered.

Closely related to the unicity theorem for functions
of one complex variable we have the following theorem.’
Theorem 7.4a.

H1). 1(2) and 3(2) are two functions regular ig'g

 

region Se

H2). f(2)=g(z) 2323 infinite sequence 9.1.: points,

 

I II III

2 ,2 ,2

 

 

, . . . , which have 2 point interior
32 S 25 3 limit point.
0;). f(2)='=—g(z) _i_ns.

 

‘ Townsend, p. 248.

59
We can state a corresponding theorem for functions
of two complex variables.

Theorem 7.4b.

 

H1). f(2.,2;)Iggg g(2.,2,) 252 regular 33 2 region S.

H2). f(z.,z,) = g(z ,z ) ‘fgnggg points (z.,z,)lig
'2 region 8’ interior 32 S.

0;). f(2.,2,):—"~. g(z ,2) in S.

Igggfessential Singular Points. We have defined
non-essential singular points in chapter VI. .4 function
f(2.,zl) is said to have a gaggessential singularity at
the point (a.,a;) if there exist a neighborhood U(a.,a;)
of the point and two functions f(z.,2;) and g(2.,2;),
which have no common factors and are regular in U(a.,a;),
such that f(z, ,2,) = 595-151;)- in U(a. ,al) and h(a. ,a._)=0.

We also mentiongdzthzt)there are two types of non-

essential singular points. .L non-essential singular point

is called a pole or g_non~essential singularity 23 the

 

my if g(a. ,a1) '4: 0. In this case f(2. ,21) tends
toward infinity as (2,,21) approaches the point (a.,ai).
We say that f(z.,z,) has the value ex: at the point (a.,a;).
.a non-essential singular point is called a 22$EE.2£.EE‘
determinacy or a Eggeessential singularity gf Egg second
£3.99. if g(a. ,a1) = o.

It is possible to show that if (a.,a,) is a point
of indeterminacy of the function f(2,,z,) and if °< is
any preassigned value, that f(2,,z;) assumes the value
°< somewhere in any neighborhood of (a.,a;).* Also,

for functions of two complex variables, points of in-

 

’ Bochner, p. 199.

60
determinacy are always isolated. This follows from

the fact that g(z| ,21) = 0 h(2,,z,_) = 0 give four

equations in the four variables x,,x,,y,,y; when we
substitute 2.: x.+-iy. and zz==xL4-iy; and equate
real and imaginary parts. Solving these four equations,
we will obtain a finite number of solutions, that is,
points where g(2. ,z,) = 0 and h(2. ,z,) = o .

Also, for functions of two complex variables the
points of indeterminacy are always limit points of
non-essential singularities of the first kind. These
statements are clearly illustrated by example 3 of
theorem 6.2.

In conclusion we might state a theorem for non-
essential singular points corresponding to theorem 7.2
on zeros.‘

Theorem 7.5.

'Hl). f(z.,z;) has 3 non-essential singular point 21 (a,,a‘).

 

 

01). All the singularities 3f f(2. ,21) _i_n g sufficiently

small neighborhood U(a.,a;)I2£ the point (a.,a;)

 

lie on a finite number of analytic pieces 23 surface.

C2). All the points 2g these pieces 2: surface, so far

2g they lie l3 U(a.,a;) are non-essential singular

 

points _o_f f(2. .21).

 

 

Mittag-Leffler's Theorem and the Weierstrass Product
Theorem. In the theory of functions of one complex variable
these two theorems concern the possibilities of forming a

function which has poles or zeros at an infinite number of

 

" Behnke, p. 51.

61

previously assigned points.

complex variable is ‘

Theorem 7.6a.

 

31).

Ha) e
Hz) e

01) e

02) .

OOHOBI‘DB ZQI‘OBe

Given §2_infinite set 3f points, 2., 2,, 2,, .
such that

o< lz.lélz.l<l33l 9
Lin! 2“ 2. 00 O

lit-too
Corresponding 33 each zh there 33 given fig

 

 

. g;)2~|sg . . .

 

 

 

arbitrarily chosen integral function 22 .__l;__
1 3 " 2.

namely, G.(...._.___.) .

z - In

There exists a single-valued, blytic function

which lg regular for all finite values 3; g 25-

cept z = 2k.

 

 

This function has Gh(’ 1 ) g§_the principal
3"Ik

part 33 its expansion 33 the neighborhood 2; 3h'

 

 

as

Theorem 7.7a.

31) .

H2)e
H3).
Cl)e

Given gg infinite set 2: points, 2,, 2;, 23, .

 

'22: including 322 origin, such that
|5||$lzt'§)33‘€- 0 - élzki£
le zkzw e

“9°.

 

0(2) fig 23 integral function.

There exists 3 transcendental integral function

 

’ Townsend, p. 304.
*’ Townsend, p. 313.

Mittag-Leffler's theorem for functions of a single

e 2‘9

The Weierstrass product theorem, on the other hand,

9 2&9

'2; the form

3+LG§+..H+—L-3r~‘
¢(Z)=fil-—.—-) .2k 2 Moe-t ill
Rat z“

having the points 2k and no others as zero points.

~_m~

02). The function
(5(1)
F(2) = s ‘(P(Z)
is the most general function having this property.

The difficulty in generalizing these theorems to
the case of two complex variables lies in the fact that
for functions of two complex variables, as we have already
seen, the zeros and poles are not isolated but lie on two-
dimensional manifolds. However, theorems have been de-
veloped which, although not complete generalizations, may
be called the corresponding theorems for functions of two
complex variables.

First, we require two definitions. Two functions,
f(2.,2;) and H(2.,2;), which are meromorphic in the neigh-

borhood of a point P, are called equivalent with respect

 

‘13 subtraction 23.3 in case the difference, f(z.,2&) - h(z,,2z),

 

is regular at P. Two functions, f(2.,z;) and h(z.,z;),
which are regular in the neighborhood of a point P, are

called equivalent with respect pp_division g: g in case

 

the quotient, ELELLELL , is regular and different from
h( z: 93:.)
zero at P.
The theorem concerning poles, corresponding to

Mittag-Leffler's theorem is ‘

 

* Behnke, p. 64.

65

Theorem 7.6b.

 

H1) e

01) e

2‘2 each point P 93 _a dicylinder 3 let there

 

 

33 associated g neighborhood U(P) and 3 function

 

 

fp(2. ,21) which _i_s_ meromorphic there, such that

 

for ELY. point Q, chosen from U(P) , where fQ(z. ,21)

_i_s_ the function associated with Q, the functions

 

 

fp(z. ,21) and f°(2. ,21) are equivalent with _r_e_-
spect 33.9. subtraction.

There exists _a function, N2. .21), meromorphic

 

everywhere _i_p 3, , which _i_s_ 32 each point P

 

equivalent with respect 1:3 subtraction with the

 

associated function f,(z,,zz).

The theorem for functions of two complex variables

corresponding to the Weierstrass product theorem is *

Theorem 7.7b.

31) e

01) e

_Tp each point P 933 dicylinder 3. let there

_bg associated _a_ neighborhood U(P) and _a_ function

 

 

fp(2. ,2._) which _izg regular there, such that for

 

fl point Q chosen from U(P), where fQ(z. ,21) _i;_s_
the function associated with Q, fph, ,21) and

fan. ,22) are equivalent with respect _t_9_ division.

 

There exists _a_. function G(2. .21), regular 1.3 ’3. ,
which _i_s_ 3p each point P equivalent with respect

1:2 division with the associated function fp(z. ,zz).

 

#

Behnka , Fe 650

BIBLIOGRAPHY

Behnke, H. and Thullen, P., Theorie der Funktionen
Mehrerer Komplexer VeranderIIcEen, Berlin, Verlag
von Julius Springer, 1934.

 

Bergmann, Stefan, Fonctions Orthogonales g2 Plusieurs
Variables Complexes, Cambridge, Mass., Interscience
PubliShers, Inc., I941.

 

 

 

Bochner, Salomon, Functions of Several Complex Variables,

Part III, Notes By E. Wefilexander, W. T. Martin,
W. C. Randels, S. Wylie, Princeton, 1936.

 

Bochner, S. and Martin, W. T., Singularities of Com~
posite Functions in SeveralIVariables, Annals of

emetics,VOI._38, (1937).

Copson, Edward Thomas, 5p Introduction to the Theor
of Functions pf a Complex‘Variable,-Uxford, The
ClarendonPress,_l .

 

 

Goursat, E., translated by Hedrick and Dunkel, Mathematical

Anal sis, Vol. II, Part I, Functions of a Complex
VarIaBIe, New York, Ginn and Company,-I9I6.

 

 

Jackson, Dunham, Orthogonal Polynomials in Two Complex
Variables, Annals of"Mathematics, yer? 39, (1933).

 

Martin, W. T., Special He ions p_f_ gegulapit of Functions
of Several CompIex ar a es, annaIs of Mathematics,

'VEIT‘EHT‘TIHZVTT"

 

Townsend E. J., Functions of a Complex Variable, New York
Henry Holt and Company{'19I5. '

Whitmore, W. F., Convergence Theorems for Functions pf Two
Complex Variables, American journal of Mathematics,
UOIe 62, (Ig40)e

 

64

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