5'.“

I ‘7‘. T

.9

n-sw‘"

3 i".

127
178

 

THS_

7376.93 fort

joml

A Sivaw
19(17

,A

the Degree of M r‘

 

 

MSU

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y“.

(‘7'."’\1'-f " Pi“ -\\,M
A61.» V uufia' dalel. J.

I wish to thank Dr. J. E. Powell
for his suggestions and criticisms in

the preparation of this thesis.

RESIDUES CF A POLYGEfiIC FUJCTICN

A Thesis
Submitted to the Faculty
of
XICHIGAH STATE CflLLEGE
of

GRICULTLLE AID APPLIED SCIEICE

In Partial Fulfillment of the
Requirements for the Degree
of

Aaster of Arts

by

John Artnur Straw

1937

Al'\TVfT§‘T'.“'l"fifi
VV.I $A‘dll o'- a

1. Introduction .............................
2. Polygonic functions ......................
3. Poles and zeros of poly;enic functions ...
4. Poor's definition and results ............
5. Definition of a residue ..................
6. Theorems on residues .....................
7. The function F(Z,§)=fi(z)+fi(i) ......
£3. The functions F(2,é)= fi(i)+ifi(2)

and F(2,§):£(€)+zfa(é)
9. Possible generalizations .................

"‘ ' -- 11‘71r"
lo. blOllO:le~Dlly o.0.000000000900000000000000

'1 USU)”

:1]
F1

SIDUES or A POLYorrio Furcrion

1. INTRODUCTION

In the study of analytic functions of a complex
variable the theory of residues plays a very importan
role. In the study of polygenic functions a corresponding
theory should prove valuable. Professor V. 0. Poor gave
a definition for the residue of a polygenio function but
did not carry his theory very far.*

It is the purpose of this paper to formulate a
definition for the residue of a polygenic function. The
definition used gives, except for a certain type of pole,
all the results vhich Poor obtained. It enables one to
consider a more general class of polygenic functions than
Poor considered and to arrive at a few more conclusions
than he stated. The definition holds for a fairly general
class of polygenic functions. Eany of the theorems (or

corresponding theorems) on residues of analytic functions

 

* V. 0. Poor, Residues of Polysenic Functions,
Transactions of the American nathematical Society,
Vol. 32 (1930), pp. 216-222.

can be proved, in a modified form, for polygenic functions;
for one particular type of polygenio function all the
theorems on residues of analytic functions will hold. The
definition has the fault, however, of not carrying over
theorems in which an integral about a curve enclosing more
than one pole is concerned.

First there is given a brief discussion of polygenic
functions, followed by a discussion of the types of poles
and zeros of such functions. This is followed by a
summary of Poor's definition and results. We then state
our definition and the results obtained by it. This
definition is then applied to three particular types of
polygenic functions. Other possibilities are mentioned

and some suggestions offered.

kn!
o

2. PCLYSENIC FUXCTICIS

A function of a COmplex variable is said to be
"regular" or "analytic" at a point if it is defined,
single-valued, and possesses a unique derivative at the
point and in the neighborhood of the point. If a function
which is analytic in any region or in the neighborhood of
any point (or even on the arc of a curve) is extended
analytically as far as possible, then the function and
all its function elem nts obtained by analytic contin—
uation is called a "monosenic" function. A "polygenic"
function of a complex variable is one that is not mono-
genic; that is, a polygenic function is one which is not
analytic in any region or in the neiglborhood of any point,
although it might have a derivative at several points.

Let v: be a function of the complex variable z=x+iy.

In this paper we consider

w=F(z) = u (M) +£v<x.y),

where u and v are continuous and have continuous first.
partial derivatives. If m represents the slepe of the

curve along; which A? app-roaches zero, the derivative

of w with respect to 2 can be written*

did .. AW_ 2:- owl-(Al
dz-‘kzLA " Axso Ax-HLAg
4330

 

 

__ 3' uxbli'uyfl+€,Afi+éaAy+£(\&AX+\‘/y434-63(Ix-ready)

40x: AX+LAy
.608:
2‘1! (vwb’ 4- +644 A?)
3:. “frag-Mme: 4" x 327 63
Arno Hi 4:1
42’34

glut/may +£(vx+mvy) _ u,+£vx+ Maj-r13 Vy)
l-H'xm - I+im

 

 

ll

since

and afi’"€£ =0 (L:’12)3;4)o
dx =4X30 X“-
4330A 3:0
For this derivative to be unique it must be independent
of m. It is easily shown that the necessary and sufficient
condition for this derivative to be independent of m is
that the functions u and v satisfy the Cauchy-Riemann

equations.

 

* E. R. Hedrick, Non~Analytic Functions _f_§ Complex
Variable, Bulletin of the A..erican Le thematic al Society,
XXXIX, ho. 2, Feb. (1933 , pp. 75-~5

\H

The above expression for the derivative can be
written in a more descriptive and cossibly more useful
form. Let 8 be the angle between the x axis and the

curve of approach. Then ”41:54.9 and

d)! _ u,+£3+(uy+ivy)tm~.9 - gx+e‘vx)cooo+(uy+ivy)m£
dz l+£m9 W5+£Ao§~5

 

€69 -€¢9 ’

-c'€
éwwxflVflEe—é—fi— +(Ky+(\g)(e; 618-#)‘J

 

 

Therefore vs have

‘1) 5” DWMPWEHE’”.

,
VIP. 8 I‘ G

(2) D [7(9)] ... 2L[“x+‘5*"‘%‘“yfl
PUG-‘9] '-'* 2" [“x‘vy+"‘vx+“y)] .

It will be noticed that if the Cauchy-Riemann equations

“PW; “5"W

dw
are satisfied, then 2"; is unique, and

5%: = 0M,

If we let

i = x+iy, iax-Zy

then

 

 

-21“?- _z-§
X'— 2 2 y- 2‘. )

and we have

 

W«'-'1C(2)=u(x,3)+£vogy):u(_i£ L)+‘-____V 2L3)?

f2) Hz 2).

1w
LOW

“ (“X*"VX)25%+(“:+5‘9) 52%

= (a,+£v,)(—2’-_-)+(uy+ixy)(z'7)

: "f ux+vy HM-“yfl = DEG-9]}

u-Omx41V&)(%)+(Ky+4\60(%—)

" é’[“x’h+i(“y+"x)]‘ FERN]

Therefore

(3) "EMDE‘Rifl‘I'PL-‘FGU -1l9_ 3w+___ 2w éztfl

bi

_. 9F(z;?____) 3_f___(z,é) 6-259.
92 25

If the polygenic function

We {(2) = My) +qu>

is such that u and v are continuous and zzave con inucus

first partial derivatives, then the function

W=F(z.i))

obtained cs above, will be continuous and have continuous

first partial derivatives in some regions A and A.*

‘45.. —

 

* by the re ions A and A ve sh»
regions"; that is, regions in the

plane such tlat if z is in a then 2
conversely

1 mean "conjugate
lane and in +‘6

da~ x.. (.1

Such a function of two complex variables is called an

analytic function of the two variables.‘ We shall here-

H)

ter spea LC of such functions as being analytic in z and

’1
U.

‘
I

E in the regions A and A and shall consider only this

type of polvuenic fur ction. fie will also use the terminolOgy
"a function is anelytic in z and z in the neighborhoods of

a and a", meaning that the regions A and K contain a and

E respectively

5. POL ES ALD ZZRCS CF POLYGEIIC FUKJTICUS

In analytic function theory ire consi ler residues at
poles of our function. In connection with residues of
the lOgarithmic derivative we are interested in both poles
and zeros

To aid in the definition and discussion of poles and
zeros of a polygenic function, we list six types of factor

which are zero when 2: a and z :5. They are

 

* E. Goursat and E. R. Hedrick, Functions gi'a
Com lex Variable, Vol. 2, part 1, Ginn and Company,
(Iéf 5 page 219.

(a) (Z'afer
(b) (5%)”.

(c) (z-a)I(i-’),

(a) [ac(z-a)+¢3(5- ~50]:
(e) [(1 ale-a)£(e, i)+(i- wt) “{“(2 i)]F

at
(f) ..., (2 a) ‘(5-4) {(M5)
In all the above types the exponents are considered to be
sitive integers; in type (e) 2'18 shall assume that £(a,d)¢0
and F1(d,i)$0; in type (f) we assume that «FdeH-o
for all i's. We note that these six types are, for the
most part, in the order of increasino generality. Types

( ) and (b) are readily seen to be special cases of the

9.)

other types. Type (c) is seen to be a special case of
(f) while (d) is a special case of (e). Type (e),
turn, is a spe ci: 1 case of type (f), since wren ve expand

type (8)

(“3.

at just a sun of terms of the type given in
(f). Type (f) then is the most general type of factor of
the polygrenic function F‘zié) which is zero when 2: a and
5:5 since it includes all the others and also any corr—
bination of sums and products of the others. The most
general factor which could occur in a numerator and which
is zero when z:a and 5:5. is of type (c) if we agree that
when we have a factor of type (d), (e), or (f) we will

eyennd and gemsrate into factors of tYDe (C)

 

1...!
C)

We now define a pole and zero for the function

iz-a)£(i-i)“f(eé)
a (z-afi'a m'qfioaé)

‘2’

(4) Rafi):

where all the exponents are positive-integers, #1)],
$(d;5)4'0 , and fi'Mdfl-O for all i's. This function
has the most general types of factors, in both numerator
and denominator, which are zero when zza and 2:5.

Let m be the least of all the nunhers 1; 4-K". Then if

M71440

we will say that Fm?) has a pole of order ”rt-(1+ K)

WI

at 2:3. and '2': . If

M<1+K I

we will say that [7235) has a zero of order (1+K)-/M-

at 2:8. and 2:5.

A particular case of (4) is

Fa’i ) Z (a-a)’((i-d)K1C(?;é) .
13.-.)"gcz,:)+e-aޣaz>]7"

he shall consider this function as having more than one

 

term in the denoninator, the terms being obtained by ex-
panding the denominator. Let m be the smaller of 1,?

and {If . Then for this function the definition above

11.

gives,
if mad-m, a pole of order mt- -(1+K) at 2:8. andz =5;
and if mull-K. a zero of order (Xi-VP!“ at z: a and E-

For the still more special function )

(z— -a)!(i- 5)“ “173) J
Ed!- -a)+F<z- c'i):l’p '

 

Rafi) =

we have,
if P>£+K, a pole of order P- -(Ie+K) at 2:8. and 5:5;
and if P<X+K , a zero of order (£41) ,0 at zsa an (1’ 2:5.
In section 5 we define the residue (at z: a) of the
function
(5) - {(2.5%)
(z-a) (3—1)

“here 4(¢:z)'¢’0 and where k and l are any integers,

 

positive, negative, or zero. From our definition above
we see that, L

if K70 and !> 0, there is a pole of order K4”! at zza
and E: 5;

if K70 and £40 , there is either a pole of order [(-1
or a zero of order .Q-k at z: a and ‘53-: a. We note from
the discussion above that the function (5) includes all
types of zeros, since k and 1 can be negative, but in-

cludes only poles of type (c).

12.

Type (d), which is a special case of type (e), is
of special interest and presents a new type of difficulty.
Such a pole arises from the factor («ETPE‘WV . This
factor is zero at 2:3. and ‘z'=a then and only when
Y=—°<a-Fi , so that the factor reduces to [«(z-a)+ “1‘48
If MI¢W| , there is an isolated. pole at zza. How-
ever, if lad-"Ir. so that P=cce", then «(z-«HHE-Z)
vanishes at every point on the line through the point zsa

whose slepe is 64‘. £123 . To show this we set Z-a,:/be‘9;

then

5"5- '-'-' X‘t'y-(a,-¢'a,,) ‘-"-' X" a. ”l. (y “‘1’

 

 

. ‘ r . T" "_' ~59
= X‘d,+‘(y'ag) = X+Ly—(a.+‘-la) = 2"“. = A8 I

where 2- a. denotes th- -e conjugate of t- n.

Hence

.5 cd’ «:9
«(z—n)+,8(§-a’.)=cc~e +ace Joe

'5 LI (-5 n, «'5 4'4
zen/(e 6+8 6 )=:-—'3 6 +9)

This expression vanishes whenever

2&9 (4’ hr (I (Hr-+4)
G =-—e :e e 2.2 )

or when

. . 77‘4-
2L5=L(17’+4’) ; a: 2’ '

 

q

13.

Therefore, when lad-slfi] , there is not just a pole at
2:3. and 2:51 but a line of poles, the line passing

through z=a of course.

4. POCR'S ssrrrrrion ArD RESULTS‘

It is natural, and in most cases probably desir-
able, to model definitions for polygenic functions after
those for analytic functions. Poor does this to some
extent in defining the residue of a polygenic function.

The residue at the point z==a of the analytic function

‘¥(29 is usually defined by the expression

R“: fijc‘flzhlz J

where C is a closed curve lying in a region in which {(3)
is analytic except at a; C encloses the point z==a but
encloses no singular point of {(3) other than zza.

his definition is not a satisfactory one for polygenic

functions because he integral is dependent on the curve C.

 

* V. C. Poor, Residues of Polygenic Functions,

Transactions of the KEerican Mathematical §ociety,
Vol. 32, (1930), pp. 216—222.

 

14.

Let C be taken as a circle of radius I about 2:8.
(or an arbitrary curve within such a circle) such that
the only singularity of the polygenic function ‘F’(?)

enclosed by C is the point zza. Poor defines
. , .
Rest“): le -—--.f (2)42
2:“ ”'50 21M C‘F

Using this definition Pobr states and proves the following
two theorems:
Theorem u: If {(1') is a polygenic function in a region
A and has no singularity at 2:8. or in the neighborhood of
zsa, then the residue of ‘Hil at zza is zero.
Theorem 3__2_: If the polygenic function F63) has no
singularity at z: a or inFthf neighborhood of zza, the

at

residue of the function fi' at 2:8. is equal to F(1).

I
For the function mi he shows that if

[L|>la| , Resflz): j- ).

acne

- .. ..L
mum, §55F<2>- at)
151:1«1, ggasanzeaa-‘go

The results in the first two cases seen somewhat unsatis-
factory since they involve only b or only a. Referring
to page 12, we see that in the first two cases there is
an isolated pole at z=§=o , whereas in the third case

there is a line of poles.

15.

Poor also shows that if {:(t) is a polygenic function,

 

then
' I Fault
and
9._——*‘*>— - .1— a as .. ..L... Ht)
2- “'lié’gzm‘Lfi( apt-£13.02,” (t ) 0”“.
Similarly
- ., . 1 -F(t) -
Tahiti-327775 {-3 ”Lt
and

 

343(2),”, ..qu 9 __(=.f;_-‘§-’)4€=Lim—'-.fl ~63 ...—1.12

We arrive at similar results in terms of partial derivatives

of the polygenic function 0

5.D221::iricr CF A R2SIDL 2

Let us write our polygenic function as a function
of the two complex variables 2 and.E. As mentioned
. above we will consider only functions of the type
which are analytic in z and E except for poles. We
shall limit our study to poles of types (a), (b), and

(c) on page 9.

Vie shall consider a function FG—flé) which can be
written as the sum of a finite number of terms of the type
duh?)
‘ K
(2-4)” (2-81)

where k and l are any integers, positive, negative, or

 

zero, nd wher Cfpkflyé) is analytic in z and E in the
neighborhoods of a and E and -£h(di)¢0 . We shall
define the residue of the sum of a finite number of terms
to be the sum of the residues f the terms. It will then
be necessary to consider only the residue of a single
term. We consider the general term by examining the
function

{(2,2)
(ii—a)" (5—52)“

d
where i: and l are any integers and 'F(?,Z) satisfies the

 

(5) F0155) =

" d
conditions given acove for fixfzfifl . Let C and C be two
closed curves lying entirely in A and A and enclosing a

- . - - .
and a respectively. A and A are congugate regions,

- ~ -.- - h F<zi)- . .1 +- -
containing a and a, in LUlC : is ane YolC in z and
E except at a and E. C and C enclose no other singular
points of Far!) except a and a. Our definition of the
residue at 2:51 of the polygenic function (5) is

, . = anvil-l 4,- ~
(0) Res Fate) .. m3hw-Kififi. dz

in» 3-61

akfl‘lfa i) A _

+0)! “*‘?"C"£'W ' (341),, 4-. - - .. '

 

[-1
,5) Ai 3“ (1,") .21.?—
+(K‘1)!k+I-ICk-I£WW (.5- ay+OIK+p-I Cx-f/‘IL—ékf-Z'I ' z-a,

21(44sz (3 6i) 9‘!!! ') “4L.
“v'lkd-Iqwaei‘a‘g'afré’ g??? 4' ”MINUC .H a)?" “3"“) J

I ‘ " . (Mi) Je
:: Tfi‘tYKbQ-I)! [$3, (‘-I)l"*‘e"c"'€ D.” «4-7-6 (e-Z)‘

 

HA _
4" 2105- ”1*...144'. f2 Rb“) Az—

H, a; "Ml-t (a-a)‘

 

where he shall agree that,

if K5 0, the first sun: nation will be Zero;

if 19.0 , the second su;n...ation will be zero;

any term containing a partial derivative of negative
order will be regarded as being zero. A partial derivative
of zero order is just the function itself.

For the particular function

e. .. £6: 3)
U”) F(Z,Z)= W )

our definition gives

12—.(6- -.). [-16 -. 314““). 43’”

£95 F617;):37’7W'l) (a, 157% (111): o

For the function

- ¥(2;
F (m) -(———._af£ ,

(5b)

we have

2‘. {(099). 45 .
-‘)z"" (z- -a‘

 

Res F(z,§)= 277.20“), 2.0-1). HQ”

Isa.

. For the function

_ (2" ‘5)K'F(i';é)__ {'(z’z) 4—
(5c) ' F(3,2)= (y “)7- (g.a)£(i-a)'“ 2

we have, if 1" K2

DI.“ -.“F({15')L_ .
@195 H255)“ zT—"uex-mé“ '0' Mn (”£757“— -""TT<"'"-. (2-0" J
if :‘K(', ReSF({)z)g 0
Zng

"9

ror the still more particular function

(561) Hz ,5) e if?

 

)

Res Fas- .- ,1," ——-.fz i—f’f’ .12..— 1C (4.5)
586%

Similarly for

- We 5)
(59) Hash-55L;

we have

* - l ( ’)
Eff Rafi-23;? 2’ iii?- ola: 34(4)”.

n

I‘OI'

_ (2,3)
(9f) F(z)i) " 2;“): I

 

we have

RES F(z,z)- _, In ___[J'_;_:_ BREE). J_g_ +JCHE ii) 41-2]

é-aL

_'_ Mani) + 97%;)
" as az-

For

we have

 

$250.3 H155): 37‘ [f9 3-293. %+ J: gfidé']

_, 2_____16(a);) +91C63/4)
" 22 +35

_ ‘F(2/'§)
(5h) F‘z/z) = (2-d)(i-fl) I

 

55; ream U?” W) .2: W $5.1...» 5%]

3-5.

_9_____ 7:69;) +2‘F(a__£__2)
" 2 2 “7"" i

rum

b-UO

These examples suggest the following theorem.

THECRE; 5.1: If, in the golygenic function (5),

lw

l are gositive integers and «F(?,§) _i_§ analytic LE
.2 in the neighborhoods 2£_§.and'§, then

m»? - amt-r
.., —— (Jz)
(I) £88 F(E)i )- (K+ -—7—-_)! Effie-Id 3 {:4

 

'1‘]

or, by (b),

‘) 2K+1."F(4?)
Res Ft” 3“ ++flyEZU -l‘) K+1.,Q-l(gT‘J:W

 

)H'h-Ril, 91') deg
4' ; ("Dix-+1- :Ci-I (2-72 *I‘W (z-a)’

' K 4-1-1 ..
.1... -.,I I 9" {(4:40
= cum-m [251" J'IrI-I-ICi-I ((6-1)!°W)

9H!-

9 I
4' Z“.")!k+1"lc¢'-l (479 ”a” )]

(at ("0- )ixu’". 9?".

 

' ”1 9"!"{61‘150
= (K-I-I-I)! ;K+1—.C¢-,W

=I

 

-I Azk‘ffl-(‘aii'l '

(::.:T t')

)3

“”*D’"
U \
'33. flag

0\
. a
..h
[Ii
(3
n!
.9!
b3
0
P4
:11
t

Using our definition we now state and prove several
theorems corresponding to the common'theorens of analytic

function theory. Our first theorem is

 

THEOREH 6.1: If, in the gclyaenic function
42(3)?)

F‘eiwm’

 

 

the function 'FG'yi) is analvtic in _z_ and

IN!

__I_‘; the neighbor-
3222s 2:. 2 iii 21. £1.21. if. fats)”, m 212. _residue 2:. Fats)
at _z_:giggeggigg values _o_f;l<___ and l _s___uch th__:_i_t K-f/ego,
For, since K'hQ $0 , all partial derivatives thich could
occur in definition (a) are of negative orders. we have
agreed that any ttrn containin; partial derivati es of
negative orders will be zero. Therefore our conclusion
follows. .
THECRELI 6.2: if, in BE golzpenig jungti on
, 'F(29§D
Fez) : W I
MW “213) amirzmzinmw-
Mainesdfi $1253: 1:. “main-a, ism mama. Rafi)
_iszsisswimmwgrminf' 'en 2:.

(2—4)? ..
75-5.)” ’ (F‘°"I””‘ K") 2

 

I“
TO
0

olus the sun or; the coefficients _O_i;
(5-3)3

s exp_:-2_.nsion _O_f_ F6355) in goners gr; z-a and 35-5.

1:.

)(3=0)[1...../Z-l)2

22

 

Since 43(3)?) is analytic in z and E in the neighborhoods

of a and '5, we have

24(4)) ‘))( .. .
‘F(z/§)='F(al¢)+- a? “(Z '4 4+) if; a. "4“)4'

 

 

I 2““’"£a.z) “+9" 9"*‘""fé,z) - -
“‘WEW‘z" we 3212.2“ “W”;

)H-F-I flag) 1.. __ _ K gxfhfar - M:
+K+1~lCK garage (2") (2-0.) +m.k+9-I (4.0-1 33mg"; (2-3) 1+»;

And

Fem): {(245) _ Feta)

(av-a )1 (5-3)? " (z-aPfi-a)“

 

2 ‘F(‘/Z-) . I a f(4132 I

+ )z (z-a)‘9"(é-a.’)7‘+ aé’ ' (a-c.)7(2-a)*" +

+ (afl’hicéfi) ' (“f-aw WW“). I
(Hue-I)! MAC 0 9254-35-1 (3-391: [WI-1 .,W ‘57:"

I .. W4 .. 1'!
23+}. [(4 4) I a“ 'Ffl ") (vi-3.)
+K-I-3-14 322?: 257? 'U. ——'"+ "M 1.14%: a§K+!-ta ' (5.4) ]+'

 

 

 

23.

The coefficients named in the theorem are

 

1 ..
(1‘ .......'..___.. 3+ . 234-2 {Wad}
“’ (mg-1;; “I m)... m jiw_05‘._, .

By (6)

 

- K «+14 - _
es F652 = ...—L.— ._ / 9 Hart) J;
R ) (HI-I)! E; a 1) m1 ’4‘ (-1 (37¢ “I 3,344.? 7-5757")

Zea:

 

1 ¢
+ Z. (‘ o'IK-rI-I Ci (-1 (2277' -f :ZK-‘gif ) (::)() 1

C3]

 

 

K 3"!“
_.__ l . I ‘Rflbi’
_ (K+.€-/)T [a“""’*+1"C‘"(<""”A éakuflaifl)

,? )deLGFfi"'
o )4 )
+ Z. ((-1)! K+!.'('-( (Tr-()1) 22Kf‘e'l a?" )]

 

 

 

 

(=1
- .1... 3“” WI") 2"“"F6ua
.. “+1-,” [g “+1.. ‘ . ) “my—g.» :57 . ,--. gnaw]
1L
._ I 5i a" 7693)
."' («+1.01 “lief-z MW)

and this is expression (8).

r')A
L+o

In the two sgecial cases,
(5-3)

 

f%%5)=

...—fl
(2 -¢)(é- 4-)
our definition gives a residue of zero, and these results
agree vith the conclusion of the above theorem.

The notation used in our definition of a residue is
such that an interpretation is needed for poles at the
point at infinity. To avoid difficulties of notation we
shall define the residue of F(Z,i) at {=00 to be esual
to the residue of t :e trar sforrr ed fun ction 6(i, 2) at :30)

vi iere he aiply tre rec iprocal transformations

UsinL. this definition as have

1?”"E? ) §§=-EF')
to Fan) to get @235).

THEOREJ 6.3: if, i. he ooly:er ic function

F6515) :- 312" {(3,5)}

the function 'F(2,§) _i__s_ analytic i z and

 

IN!

i the neighbor-

 

hood'gf the goint §£.infinitv and.;§

Lint-[YiyED gޢ9 )

1.49
:09
then the residue of. [(1,5) 37.13. 2:90 is equal to he

 

'neaative of the sum of the coefficients of

 

 

F
37;, ,(Pgo, I).....1))

r0
‘0 I

31313 t_h_3 negative 31: 133 _u_:11 3f; the coefficients 33

.gg. (W)
_13 1:33 expansion 3: F633) 13133 neighborhood 33 {3“,
Applying the transformations

5? = "" I if = ‘27 I
z.
to F595) we have
HELUT)_ 3935’)

GIG-5%): H497)" W = W
where 3(i)5) is analytic in 2' and E'in the neighborhood
of the origin and where 3(60)¢0 . he must also apply the
above transformations to the differentials of z and El

Thus
Xian-3342' , dis-rigixi’

in the in teg rals of (c). Keeping this in mind, theorem (6.2
1 fl - '| ‘-') ‘ .
says tnat tne residue of 6(3):? at i=0 is equal to the

negative of the sum of the coefficients of
awffl

...-{P ) (Pg-@111) )

plus the negative of the sum of the coefficients of
«H
I

2 f ( :0 I‘ . . . K)

‘7'}— ) ? ’ , )

.z

O!

in the expansion of 6633;) in the neighborhood of the

origin. But these terms are respectively the terms

and
5*

3-517 )(ggo)l,.....l<) ’

in the expansion of Fat-i) in the neighborhood of the
point at infinity.

The expression for the derivative of a polygenio
function, equation (3), suggests a definition for the
“logarithmic derivative" of a polygenic function. In
order to make this derivative unique (independent ofB)

we shall say that

DU J+FU ] 2f(z,'2') +3F®E3
C ,. __ (t) (a) __, 2a a?
(J) Log. 08% F621;)- fa!) F '- F(Z)

Then we have

 

 

different from zero, the 103arit‘mnic derivative of Fax?)

has 3 simole Eole gt _z_-:3 and 3 simgle Bole at 2:51;;

furthermore, for any inteorel values of k and l, the

 

 

residue of the loiterithmic derivative of F612) is

equal to K1,].

 

 

 

 

 

 

 

 

 

 

 

 

 

Since
9/7132): (. - ..
a z %)K[-(z a)!” ”(5.52) +1(z-a)£ ’f(zli)])
then 91%?) 274(2)?)
92' _ 2;:
F655) " fax?) + *-
3F(z,§ , .. -
than Masai) -QLQJ)
52 2% +2'K
f6?) 2) Hflaé)
By (9) 2_f_____(95) I _9_____1:(£,é)
: )2 2K;
L05.Der. HZ?) H! ‘T +-—- _a + {(izi)+ .
This proves our first conclusion. Also we have
L 2/(25) ' 2H5?)
. . 2,; 2%
Res 0.7 Der H 3:) Eff—L. {(2 (:12) +£5.55 f(s,i)

é=¢t

          

isa.

The first two terms are zero because the functions

 

”(E/é) )fél'zé)
22- 25

 

Rafi) Hz, 5)

are analytic in z and E in the neighborhoods of a and a.

'11

or, by hypot‘z’iesis, ‘F(£,Z) satisfies this condition and
so its first partial derivatives will also. Also, by
hypothesis, #(a,£}#0 . The functions,

2f(2)§) 946313.) -
*2: W 35 - )
Hz, 2') 1‘ (*2 e)

 

 

thus satistr the hypotheses of theorem 6.1 where K=Z=o
end so the residues of these terms are zero. Applying

(6) to the term -—-- ”we get

 

 

 

'0-

’? I
es 3-“ .W(. fife. 4%: 1 O
:05 c,

For the term e get
3-4.
K _ I K' “..

Res - 0‘]:- _-_ din—K .
234' 3. Z- arc ¢ z a

Therefore

58‘s Log. Der. Ffzé): n+1.

This theorem includes both of the following theorems on
analytic functions:

Tl—LEOEEL'; A: If, in the analjtic function

F(£)= (bare-[3(2), (1)0)

the function {(2) _i_§_

9‘

nalytic in the neighborhood of e:a_

and 2.1:. {(¢)¢o, then the longrithmic derivative of; [7(a)

 

 

F0
\0

rs
0
fr)
H
i .
U1

I_f.: £2. 32.11—31- analytic function

4(2)
F(2 )=""——2 (1":7) (170))

the function {(11) _i__s_ analvtic _i; the neighborhood of £30.,

 

and i}: ‘F(a)$o , then he locerithmicderivative of. F6!)

m

T,

res a simole pole at z=a with residue epual to :_l_.

y

 

The definition given here for the residue of a poly-
genic function has one serious objection. The theorems on
resid es in analytic function theory which involve the
integration around a curve enclosing more than one pole
do not seen to carry over in any suitable form. The
reason for this is seen in the definition since it in—
volves the substitution of “the” point at which there is
a pole in the function 'FCE,Z) or its oedrt al derivatives.
Among the th eorens fro:n analytic ft notion theor3 wlich do
not carry over for this reason are:

TI-ILCRELI : I_f_ {(2) is holornorghio 1.3 a region _s_

 

exce t for a,finite number of noise and if _ism mg curve

 

1 ins entirely in _S_, and enclosing all the “p oles g: -F(z)
then the integral, J;«F(2)J% , _i__s_ esual _t_c_) 111% times the
sum _g_f_'_ the residues _c_>_I_“_ {(2) _i_n_§.

HECREE b: The sum of the residues of‘a rations l

 

function 1 zero.

 

 

K»!
C)

TE’CREJ c: The integral )

1 Fti)
27:? C. Read; )

 

taken 'n‘a oositive sense around the boundary g Cf.§

 

closed region in which the "rational" function -F(2)
is holomorp hic exceot at a finite number of ncles is

e ual _t_9_ the number 2;: zero Eoints g: FOE) in this

F

reoion diminished by_ the number of E 16 s, each zero

F

 

point and each 90 ole being counted a number of times so al

 

33 it 3 order .

 

7. Tz—Li mrcricz: F(z,z)= {(2)445 Hi)

In this and the next section we examine the three

3 articular types of polyger ic functions

1.

(a) F632) = £(2>+{,(é)
(b) Fa?) =£(2)+i+',(z) ,
(c) F(z,i)=£(z')+zfi(§)

Keener, in his geometric regresentaticn of the recti—
linear second derivative of a polygenic function, mentions
seven special types of functions, the tires list eda bove,

and four others which are Special cases of these three.*

* Edward Ka sner, The Second Derivative of a Poly—
:enic Function, Transactions of the American“ ”ethematical
Society, Vol. 30, (1928), cage $00.

 

\-rJ

It is seen that for the three types of functions listed

above we have
(a) &§(zji);0 I (b)%;(%/§)=0 ) (c) g2,i)=0a

For type (a) we shall write
- , 42(2) 491(5)
. F(Z,£) = £(2)+£(z) - (in-:52 + (5,1)K I
where k and l are any integers; and where m ('3) and 41(5)
are analytic in z and 2 respectively in the neighborhoods
of a and E, and where 42(4)=¢-’0 , 4102).;90 . Applying

the definition to each term separ ately :e set

 

iisaL

ResF(z,i)=W a71[(1—I)!1-C,,JW4M ale]

 

+2”: (K'IN [FK 0‘ K- _,CK 'fg (‘L—i-ffi‘Aé-J

n F 42/504 I 44') _ -
2—— %+-—-— -——- de:Resf(e)+Kes~H%)
' , ' _-K I _ _ a
firm c (2 a. 311*! £64: a) 5.34 a" )
w‘n ere, in the last line, £62) and {3(5) are con-
sidered as functions of the single variables 2 and E
respectively. Thus for this function our definition

gives a reasonable result.

Since the residue of fi(i)+fi(5) turns out to be
the sum of the residues of '5‘?) and 4313) considered.
as functions of the single variables 2 and‘E respectively,
all of the theorems concerning residues of analytic
functions will hold for this function. The proofs of the
following theorems consist of first the statement,

Res F0125): fies {Jen-fies 4-;(1) ,
as“. 330. -

and then a statement or two from analytic function theory

:1)

(usually statement of the corresponding theorem for
analytic functions).

TEECT131; 7.1: If, in the oolygenic function

.. _ (2 e65)
F(z,z)-— manger): fi-fi— (-375),

the functions @(g) and «(5) are analytic 3333 and

43(4):}! 0 , £(Z)¢0 , 33922.1 ii

150) 100 , ResF(z,2)='-Ee_.s£,(i),

3:“

Res Wei): 1565 Hi) I
«2:4 53:“-

Kso, 150, KeSFiz,§)=O ;

law:

“1:, ) Egg, Fans): (yawn/a.) .

1.3
O {l
p - 1
H

n
’11
H
O
-\I
O
H:
b
H

the Eglysenic function

_ , 49/2) +415?)
F(z,z)= £(2)+f,(a)= (z --;-;¢ +317)“;

the functions 4(2) and @(5) are analytic in;

 

and z rescectively _i_n the EGiPhOOIhOOdS of a anda and

_ @(4)¢0, @0049 0, __‘1__t1en the residue of F(z’i)_ at
2:3. i egual Lo the coefficient 9_f_ («z—a.)-' _j__1_1_ the 23$.

 

oen sic of {(2) about s=a plus the coefficient 2_f_ («E-a.)
the expansion 31; {1(5) about 75:51.

THEQFE; 7.3: If the golysenic function

F(z,§)= fi(z>+w‘1(€)

is analytic in _z_ and :2: in recrions _afg and £3 exceot for a

 

 

finite number 2;: Eoles and _13 g and g are curves, lyinv

11.1.1.5 532913.: enclosingng singilarities 31: £63») 33 fig)

exceot the Eoles _c_>_f_ F652) _i_ng and 21:, then

 

f6 fifzuz +f, HEM};

_i_s ecual 1:2 2771' times the sure of the residues of F(Z)é)

 

0.

an A.

I;
lb

1

.1.

TE GEEK .4: If, in the rolygenic function

E

 

flag) : £‘(a)+¥,(€)= z’flani’Vfif), (10") I”) 1

the functions «(2) a__r__1d (9,62) are analytic _i_; z and

 

: res Jcectively _i_n t___h_e neighborhood c__f_ the point a__t_ infinity

 

 

' , $0, L' " :éo
%;2¢(2) ’ 3!: We) ,

then the residue _o_f_ FOEE) £31; 2-300 is

 

. . . ‘l . t .
nergatlve 3;: the coeff1C1ent 2f; .2 3.}; 1:118 6X2 ensign“ 0

 

 

_1_5‘_._ the neighborhood of the Eoint at infiritx glue the
-I

:3

emtive 2; the coefficient 3; ‘z‘.’ _i_:g _t_h_e_ exggnsion 2.1:. 43.05)

IS

_the neighborhood _o_:§_ the goint _a_i_:_ infinitv.

F(Z',i) 3 £(£)+‘F2.(5) )

THEJREI 7.5: If, 'n he oolzsenic function )

go

the functions 16"?) and 41(5) are r tional functions

_c_>_f_ g and _z_ respectively, then thew; the residues 931
he shall say that the logarithmic derivative of
F(Z,f)=£(€)+é(i) is equal to the sum of the logarithmic
derivatives of £03) and f;(i=) . Using this definition

we have the following two theorems:

TLiECRELl 7.6: f "n the golygenic function

~’ ——

F (2,2) = f, (a) + mi) :: (an-4V4; (2)4- (é-a‘LMfi) ;

the functions (flat) and fljé) are analytic gang; and

Z reeiectivelg: in the neighborhoods 2;; a and E and if
gauge) fi(d)¢fl) then if; K=fi'0,,?;¢0, the loEarithmic
derivative _o_f_. F652) has _8; sim‘ole Eole El z=a and _a_._

single Role at 3:5; furthermore for _15 and i an}: integers,

 

the residue 3_f_ the logarithmic derivative of F(z)§) 513

2:8. i_s_ ecual to [(4-1,

if;
2;;
{.4

m fir/"211' . '7 o m‘n ~112r 4‘
J..-.:a visual. " o 7 0 4.1.13 bwu .L t;
“-1- -

Elfi£L03 DerflaMe-I-f— [Lo fl(é')al€ ;

 

 

 

taken _i_n t 1e bosit; ve so rise around _t__‘1‘1e boundaries C and g

_c_>_f closed regions A and E _i_n which th__e_ "rot1 onal functions

{(2’) 2312 {3(5) egg holoznorphic _i_n g and _i_: respectively

 

 

of _z___ero Roints_ of «6(a) in A glue t__‘:__1e number 2;: zero
ooints o__f_ {(5) in A diminished 51 the number 35001135 3:;

43(2) _1_Il A olus the nmn‘oer of; holes of; {1(5) 335, each

zero and each Role being counted 53 number of times eoual

 

to its order.

 

 

s. .2 1131111101. F(z,é)=fi(z)+£f,(a)
111:3 F(vb5)=fi(i)+a£(i)

For the particular function, type (b), page 30, we

shall trite

- (
F6512): £(2)+z£(%>= g5} fi‘gfiau

where @730 , where 4(2) and £62) are analytic in the
neighborhood of z-:.a, and where (3 (A)$0, 43,(a.)¢01

Equation (5) gives as residue)

(12) fies F(z1é')= mu. ), [31,- )1 1., 9""1 4(a) 70,71]

2:4.

 

\M
O\

 

' £12) flex»)
+ fir/(13;?! E643! 15191-2: c7"? WfiHA ‘1)Q,Q,I£EI;)I At]

_ 4m) 1 4m 1 4___(__2)
”fif‘jjzl'é "TA?+7_m‘—__f J2

cam) . in C(i-a) a" div-«>12.

,. )
=§cs£é2> +ftes-—- :S-‘I'Tr' +43: is: £112),

ew-

where these resiiues are evaluated as for analytic functions

of one variable. This result is also obtained if 'F.(-2)+Zfi(i')
is combined. into one term, using either t .1e assumption 1.21;
or [<11 . The exoression (10) is as one would expect since

__ 41(2) :1 (5-55.)417.(i)_ 1‘3. (‘2)
2+, (a)- (2 ‘ 7"" (2-4)13_ + (21") )2

'..e note that in the soecial ca so where (1:0, equation (10)

gives as the residue

(10a) Reeflzi) = Res {(2) + K65 3355:21- I
*30 345 9:0 3
since if 0.1-0 , then I = 0.

The function '85) +2 {4(2) is similar to the function
above. Since this function is obtainable fron the above
function by merely interchanging z and E, the conclusions
for this function can be obtained by the interchange of z

3:153 :r'.’ in the results for the function. £(£)+§£¢(2)o

By interchanging z and E in (10) we get the residue of

F(z,2') =f(é)+z£(é)
to be
(11) 1353 = gfgfliHéfg 3(2)”; + (1 {12551543,
The following theorems, which we state for the function
F(z’§):-E(g)+§-Fz(2) can be easily proved. by using (10)
and statements on residues of analytic functions:
rarest; 8.1: g; a the leygenic function

¢f2> 24312)
(#3.‘) ’ +-——T-(?“) ,. ) (“#0) }

I

F (2, i) : 75, (z)+ 219(2) =

 

the functions fléz) and day) are analytic in the

neighborhood 9;; zsa and ii fi(¢)¢'0 ,/;(d):éo , then E

 

, Mt)
a6§0) 13 M, £5: F03): [:33 (+4?" +53: 4363)}

.>\\.
\I
§

‘0

1k
in

.)= t
o , if: flu @g; {62),
X; 50,14 20, éfifléi) =0;

’6’»? =.,, , Egg: Kai) =#(4)+Zf £14);

 

[i=5 1;: :1, £52 flea—7 €(a)+4;(4)+i if?) .

f!

THZCFEK c. : f, “n the golygenic function

 

I

#62) i @(z)

F(z;§)=£(z)+'é£(z)=m+ z-a) .. (4990))

the functions @(i) and %(2) are analxtic _i_n the
neigho orhood of 2:8. a_n__d if ¢(“) #0 , f (4):}:0, th
the residue of F(z,§)

 

g3; z=a is egual to the coefficient
of (2-4,)" _i._n the expansion 23: «(i-(z) about 2:8. plus the
-z.
coefficient g (5-9.) _i_r; the exgansion gt: £(z) about zza
glue E times the coefficient 23‘; (3.4)", _i_ t e expansion
of £02) about zza.
THEORELJI 8.3: __f, _'_1_i 'he golyflnic function
I
I

F(z,z)= ,(z)+'z' £52): azflzwiz ‘420-3; (”CM/6M”

the functions £62) and 41%) are analvtic $1 the

neighborhood 2;: the pgint _a_t. infinit‘g fig;

1.212% mango , $32» 4g<z>¢o ,
her 3112 _residue 9.1:. Fez) 22:. a—w is: 2:22; in 21.:
negative 2:239. coefficient 21; z" E the expansion _o_f_ 4'62)
_i_}; the neighborhood 9}; infinitz Elli... t_h_e_2_ negative 9_f_ the
coefficient o_f_ 2"ng the expe nsion 9_f_ £02,) 3.2 3132
neighborhood of; infinity.
In finding the residue of this function at 55:00, me do
as we did for the general function (see theorem 6.“).
That is, we ag>ply the transformations

.’ - I
2"?227—‘2-7)

v
\

to F(3,2) , and define the residue of F(Z)Z) at 2:00
1 . ~ ‘ . G ' ‘.)
to be the residue of the transformed function (i)?- at
I- . . . . . . "’9 I
2—0. Then in finding the reSidue of 6(3)? at 2 ===0,
I .
we note that 0.30 and so we use equation (10a).

4“

TEiEC‘EELL 0.4: if, 'n the golygenic function ’
.. -' I ..
F ( 5!; 2)='5( 1) 1572(1): (z-a) ’4(£>+z-(z-a>"¢;(£), (use),

the functions 3(1) and 4(5) are anal‘gtic in the
neighborhood 2: z:a and if ¢(4)¢0 , ¢>(4)¢a , then
if {$0 , .610 , the logarithmic derivative _o_f_ F(£,E)
has _a_ simgle Eole _ai z=a; furthermore for ,6 and [a any
integers, the residue 23 the logarithmic derivative is
eoual _t_o_ {4.} ,

before, we say

U‘DT‘ l
.Lbfie’ Q.

[.0

1.09.0” Fag): Log. Den 1%!) 1" L03. Der. 3&(2).

Due to the fact that the residue of Rafi): £(i)+§'fa,(?)

involves the term)
67. [£85 £9!) ,
{hr-a.

which contains "the" hole 238., the theorems on residues
of analytic functions which depend on an integral along a
closed curve enclosing more than just one pole do not hold.
Thus theorems (a), (b), and (c) of section 6 do not carry

over as they are stated.

4:7
Q)
o

It would seem sossible, however, to prove theorems
sanenhat similar to the theorems of analytic functions.
In particular, it probably is possible to show that if'fifii)
and +£(é) zxmaretionel functions, then the sum of the

residues is zero.

’TJ

9. 'OSSIELE GEFLRALIZATIOKS
Our definition for the residue of a polygenic function,

although limited to a few types of poles, need not be so

restricted in the types of zeros included. In fact, he

can include all types of zeros. F0

"3

example, we can find

the residue of the function.) .
.. _ P

Ea-«nmz-afl tag]

i )
(3-4)}?i-J)“

F512) =-

9"

where [:an-aH'fiG-fiuf is zero for 2:8. and :3: ’
by expanding the factor [4(E-a)+fi(5‘5)JP and Separgt-
ing our function into a sum of terns of the type given in
ecuation (E) on page 15. Possibly a still more general

definition can be found which will include all types of

poles. Such a definition night he stated for the function

(2-421 (z-az)“ F (2,5)

i ze-a)’¢a-z>"‘a-<z.2) ’

in

which we have seen includes all types of zeros and poles

(see page 10). For this function we shall merely suggest

the follov;i n3 definition) In the terns of the denominator
A ‘1\ ‘.r r -. I. + —|-- 1_ . .

or (a) let an be the largest of i Kt . if there is Just

one such term, say mtg/(1,“! K5 , we shall say that

(s-a)’(i-z>" {(2.5)
(2" )2’O(i‘3)fifi(%)§)

 

I
Res F(?;Z')=R€$ -

%=A. 394
If there is more than one such tern ne will compare the
absolute values of the functions ‘F‘-(4:J) which occur in
these terms of highest total exoonent. If, of these
terms the function ‘F a a) has a larger absolute

.9 h I C.

value than any of ti e others, we shall say that

(e-a)’(s-h"7‘(si)
(MA (ii-5.)“ was)

 

IEQSjFYZ)§>==/h35
i=4 2:4.
If the terms

.Fd‘fa) 5-), .. . . ... . .‘fiAM‘a’aJ,

have the same maxinum absolute value we shall define the

residue to he

1 .. _ K ..
(2-4) (at-.1.) fag
HES F(z,z)= [ifs (2-4)“(£-Z)&’5,(£é)

 

(evaP’li-a) Hess)
fies—7 -
.354, (i “)43462 aa)KWaV1C»(z)g)

 

2»:
1C.

This definition of the residue or (Q)

the function }

(2-4.)! (5-41)“ Rafi.)

Z i): . . _'
Fc ’ 'Eda-aJ-t-flz-EUF ’

 

if Ix|>lfi|

R65 F01 2)_ _ (2-4)1(£‘4)K‘r(z)i)

‘ 2-34. dP(?’d)F
if RHIPI,

 

I

. -— .. (z-a) (2-4) Rm) .
if 13-!!!)

(2- m)! (i ~J) K-Rizz)
Eff/W55)- @345: “(PO-5 '4)?

 

+ K65 (2-4)1(5-5)KF(2,5) .. _ . + £95 (a “-4)"??? iffaé)
3:“ acP '(5(z- af’e- 5) 2:.“ Wei-vat)?

 

 

O

 

“f/ o
lo. blbL: CCE..S: ::Y

1. Heirick, 2. R., lon— —Ar 3 ic F“nctions of‘g
Conolex Variacle, Bulletin of tne Ag‘"lf,u “at' t
Society, xnzix, 30. 2, Fee. 1%: 913. 75—99.

 

 

2. Course., s. ana necricr, B. R., :incticns o a
._ -- - /’
Coucle Variable, Vo . 2, iort l, Cinn ano 60., 1915?,
page 219.

 

 

3. Poor, V. 0., nesiiues of Poly3enic FunCtions,
Transactions of the Arerican Lat ne1nctice 1 Society, Vol.
32, (193 C), pp. Elo~ta2.

 

 

A. Kasner, Edward, Tfie second Derivotive oi §_Polv~
33enic Function, Transactions of the American Lathenetical
Society, Vol. 30, (1925), pa3 s39.

 

 

 

 

1“ amt [RS :

' .u
i I“! u

 

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