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TH .

ANALYTIC FLNCTECIRS
OVER A
CERTAIN HYPERCCMFLEX ALCEBRA

Thesis fcr the chrec
of

Master of Science

Camus} chdard Stewart

1 937'

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MSU RETURNING MATERIALS:
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be charged if book is
returned, after the date
Stafllge'mfi‘b. ‘ '

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To Professor J. E. Powell,
for whose instruction, advice,
and encouragement I am deeply

grateful.

ANALYTIC FUNCTIONS
OVER A: .CHIEEJRTAIN

H Y P E R C O m P L E X ~ A L G

t1)
U3
SJ

A Thesis
Submitted to the Faculty
of

MICHIGAN STATE COLLEGE
of
AGRICULTURE AND APPLIED SCIENCE

In Partial Fulfillment of the
Requirements for the Degree
of

Master of Science

by

Samuel Woodard Stewart

,~-

1937

Section

OWCD'QONUI-PUNl—J

1

ll.
12.
13.
14.

15.
16.

CONTENTS

Introduction

Chapter I

THE HYPERCOMPLEX ALGEBRA
The variable
Definitions
Inequalities
Product
Conjugate
Quotient
Determination of a nilfactor
Solution for the conjugate nilfactor
Summary of nilfactors

Relation of conjugate and nilfactor

Chapter II
DIFFERENTIATION AND INTEGRATION

Limits; continuity
The derivative
Stokes' theorem in S4
Definition and prOperties of the

integral
Cauchy's first integral theorem
Cauchy's integral formula

11

iiCSid

Page

mfimkflUI-kkfl

ll
13
13

15
20

23
25
27

17.
18.
19.
20.
21.

iii

Chapter III
ELEMENTARY FUNCTIONS
Definitions
The exponential function
The circular functions
The logarithm
Comparison with functions of a

complex variable

Bibliography

31
31
32
35

37

39

ANALYTIC FUNCTIONS
over a

CERTAIN HYPERCOMPLEX ALGEBRA
INTRODUCTION

This paper is a study of the algebra of a four-
component hypercomplex variable which is commutative
and associative, and of analytic functions of that
variable. After a discussion of the algebra involved,
the 'Cauchy-Riemann' equations are derived; Stokes’
theorem is extended to four dimensions; Cauchy's
integral theorem and integral formula are demon-
strated; and prOperties of certain elementary func-
tions are considered.

That the present variable was studied rather
than the quaternion is due to the fact that analytic
functions of a quaternion are limited to linear func-
tions. Even the simple function w = Z2 is not analy-
tic when 2 is a quaternion. Quaternion algebra has
the advantage, however, of having no nilfactors.

The study made here follows to a certain extent
a paper on general hypercomplex variables by P. W.

1

Ketchum. The present paper, however, gives explicit

 

l Ketchum, P. W., "Analytic Functions of Hypercomplex
Variables", Transactions of the American Mathematical
Societ , Volume 30, 1928, pages 641-667.

-1-

statements and proofs of results for the four-com-
ponent variable, where Ketchum's paper is restricted,
by its consideration of the n-component variable, to
more general statements and indications of proofs.

In the introduction to his article, Ketchum
lists several references to original papers by those
who began the study of functions of hypercomplex
variables. Four of these references, the ones most
closely related to the present work, are given in

the bibliography.

CHAPTER I
THE HYPERCOMPLEX ALGEBRA

l. The variable. The four-component variable
used will be denoted by
z = x1 + 1x2 + 3x3 + kx4 .
The units are 1 (one), i, J, and k, with the multi-

plication table

 

 

k k ’J -1 1 o

It is to be especially noted that

i=J=-l, 12=k2=l.

The x's will be restricted to be real numbers.

It is easily shown that addition and multipli-
cation are associative and commutative, and that
multiplication is distributive. The factor law
does’not always hold true, for (i + j)(i - j)
=12-32z-1+1=o, althoughi+JfiOand
i - 3 fi 0. Such quantities as i + j and i - J,
for which a second factor can be found which makes
the product zero though neither factor equals zero,
are called nilfactors. We shall find that it is

-3-

possible to Specify all nilfactor values of the vari-
able. In case a product is zero and it can be shown
that no factor is a nilfactor, then the factor law
may be applied.

2. Definitions. A determinant which occurs in

the discussion of the algebra is

x x
X = l 2
x3 X4 .
The absolute value of z is defined to be
1
_ 2 2 2 2 2

In the discussion of functions, w will be used
to represent the dependent variable. That is,
w = f(z) = ul + iu2 + ju3 + ku4 ,
where up = up(xl, x2, x3, x4). Similar to the deter-
minant X, Ulwill represent the determinant
ul u2
u} 1.14 O

U :

The absolute value of a function will be denoted by
. _ 2 2 2 2 1/2
The product of z by a real number, a scalar,
a, is defined to be

az = ax1 + iax2 + jax3 + kax4.

The quotient of z by a real number different from

zero is defined to be the product of z and l/a,

X X X X
"L" 1 +1: “‘5‘” 4

a ' a a

 

 

 

3. Inequalities. Beginning with the well-
known inequality of real variables,

a.2 + b2 ; 2ab ,

it can be shown that:

(1) I121! - Izzll é Izl + .21 é Izll + lzel ,
<2) x é IX! é mm + Ix2x3| é-g—Izle.
(3) Izl é 1x1! + 1le + 1x}! + 1x41 éelzl .

The second part of (3) is demonstrated by the same
method used to prove the correSponding inequality
of complex variables.

4. Product. The product of two numbers is,

w = ul + iu2 + ju3 + ku4 = zlz2

= (xllx2l ’ x12x22 ' X13X23 + x14x24)

(4) + 1L("11"22 + xl2x2l ' x13x24 ' X14X23)

+ 3(x11x23 ' x12x24 + X13x21 ' xl4x22)

+ k(x + +

11x24 x12x23 x13X22 + x14x21) ‘

This product is symmetrical; that is, if xlp and x2p
are interchanged the result remains the same. The

absolute value of the product is

‘2122' = (Izllzlz2l2 + AXIXE‘yL/e
irE'lzliizzl .

IWI

(5)

IM

The inequality follows from the inequality (2).
5. Conjugate. The conjugate, Z, of a number z
is a number such that the product, w = 23, has only

real terms. That is,
u2 = u3 = u4 = 0.

Given 2, in order to find 3 let 21 of (4) equal 2

and 22 equal 2. Then we have three equations to

solve for the components x1, x2, x3, x4 of 2;

they are,

u2 = x2xl + xlx2 - x4x3 - x3x4 = O ,

(6) u = XBXI - x4x2 + xlxj - x2x4 = O ,

- x4xl + x3x2 + x2x3 + xlx4 = O .

c:
.p
I

These equations are sufficient to determine
the ratios of the x's. The values of the x's are
proportional to the three—rowed determinants, alter-
nately plus and minus, obtained by dropping the first,
second, etc., columns successively from the matrix
of the coefficients? On simplifying these determi-
nants, we find that

- 2 _ o
cxl — xllzl 2x4X ,

2
-x2IzI - 2x X ,

0x2 2 3

c; = -x [zl2 - 2x X
3 3 2 ’

c; - x Izl2 - 2x X
4-4 1:

 

2 BScher, Maxims, Introduction 39 higher Algebra,
page 47, theorem IV.

where c is a real proportionality factor different
from zero. Taking c equal to one, we shall mean by
the conjugate of z,

E = (xllzl2 - 2x4X) - i(x2lzl2 + 2x3X)

(7)
- J(x3lzl2 + 2x2X) + k(x4lz(2 - 2x1X) ,

which may also be written as

- _ 2
z _ Izl (xl - ix2 - jx3 + kx4)
(7')
- 2x(x4 + ix} + jx2 + kxl) -
The product of a number and its conjugate

reduces to

(8) w = z; = Izl4 - 4X2.

In case 2 is a nilfactor, the rank of the matrix
of coefficients of equations (6) is two and the solu-
tion for the i's given above is not valid. The com-
ponents of 2 then are properly the same as those of
the conjugate nilfactor?

6. Quotient. The quotient of two numbers, Z1
and z2, is defined to be a number w such that

z1 : wzeo
Multiplying each member of this equation by the con-
jugate of 22, we get

zlzz = wzézz.

Dividing each member by (2232), a real number, we

 

3 For a demonstration of the statements of this
paragraph, see section I0.

find for w, using (8),

 

Z Z Z Z Z
(9) W =—--Zl :: -————1_2 :: 3i; 2 .
2 z2Z2 [22! ' 4X2
Let

C = (xllx2l + Xiexce + X13X23 + Xl4x24) '
D = (xllx22 ' X12x21 I x13x24 ' x14x23) ’
E = (X11x23 + x12X24 ’ X13X21 ' X14X22) '
F = +

(x11x24 ’ x12X23 ' x13x22 X14x21) '

Then, after simplification,

Z
(10) w: 1 = 1
Z2 (lz2l4 - 4x3)

 

 

[(Cl22l2 - 2FX2)

- i(Dlz2I2 + 2EX2) - j(Elz2|2 + 2DX2)

+ k(F122I2 - 2CX2)] .

We also find that the absolute value of the quotient

 

 

is
2 2 1/2
m = 21 = '21) ”2‘ “:X1X2 .
Z
2 Izal - 4x2

 

 

7. Determination g: g nilfactor. A number z1
different from zero is called a nilfactor if there
exists a second number 22 different from zero such

that their product is zero. The number z2 will then
be called the conjugate nilfactor to zl.

If w = 2122, then for a known number zl to be
a nilfactor we must be able to find a Z2 such that

w = O; that is,
u1 = u2 = u3 : u4 = O .
Taking the values of the u's from equation (4), we

get the four equations:

u1 = X11X21 ’ X12X22 ' X13X23 + X11324 =

u2 = Xl2x2l * xllx22 ' X14X23 ' X13X24 =
(12)

“3 = X13x21 ' X14X22 + Xllx23 ' X12X22+ =

”4 = xl4x2l + x13x22 I X12X23 + X11X24 =

In order for there to exist solutions not all
zero for the components of Z2, the determinant of
the coefficients (the known xl's) must equal zero.
The necessary and sufficient condition, then, thgt

Zl_______________________

x11 ”x12 'X13 X14
x12 X11 ’x14 'X13
X13 ’X14 X11 'X12
x14 X13 x12 X11 .

 

 

On eXpanding the determinant, we get

_ 4 _ 2
(14) D — lzll 4X1 .
From D = O, we find

2 — i
Izl! — 2X1.

-10-

That is,

2 2 2 2 _ t -
x11 I X12 I x13 I X14 ' 2X11x14 I 2X12x13 '
2 - 2 2 + 2 _
X11 + 2xllxl4 + X14 + x12 - 2x12x13 + x1} — O,
or

O .

Therefore,

- _ i. _
(15) x11 + xl4 — O and x12 x13 0 ,

2 _ 2 2 _ 2

X11 ' X14 x12 ‘ X13
Adding,

2 2 _ 2 2 2 2 _

x11 I x12 ‘ X13 I X14 ‘ X11 I X13 ‘ X12 I X14'
Let

_ 2 2

m — X11 + X12 0
Then
(16) 2m = 12112, m = $42112 ,e o ,.
since 21 cannot be zero.
Also from (15),

c- + .—

(17) xllxlj I x12X14 ‘ I X11X12 ' x12x11 ‘ 0 °

Taking the signs in (15) one at a time, we have
two possibilities:

Case I,

(18) X14 = X11 ' X13 ” 'x12 '
_ 2 2 _
X1 ‘ X11 I x12 “ m '
2 _ 2 2 _ _
lzll — 2(xll + x12) — 2m — 2Xl ,

(19) z1 = r + is - js + kr .

-11-

Case 11,
(20) X14 = ’xii ' x13 = x12 '
- -x2 -x2 - -m
X1 _ ll 12 - I '
2 _ 2 2 _ _ _
lzll — 2(xll + x12) — 2m — 2Xl ,
(21) zl = r + is + js - kr ,

where r and s are any real parameters not both zero.
In either case,
(22) m = lel .

We are now able to state the theorem: The neces-

 

sary and sufficient condition that z1 pg g nilfactor
Lg that its components satigfy either (18) 9; (20).
This is true because (18) and (20) follow directly
from D = O, and each gives D = O. I

8. Solution for the conjugate nilfactor.

When D = O, that is, when 2 is a nilfactor, then D

l
is of rank two. This can be shown by consideration
of the second- and third-order minors of D. The
two-by-two determinant in the upper left-hand corner
of D equals m, which is never gero, (16). The three-

by-three determinant in the upper left-hand corner,

for example, reduces to

2
XllIle ' 2x14X1 '
Considering the equations given under either case I
or case II, we find that this determinant is zero.

The other three third-order determinants having m

-12-

as first minor reduce to similar values and are like-

wise equal to zero. Therefore, all three-rowed deter-

minants of D are zero, and D is of rank two?

As D is of rank two, we can solve equations (12)
for two of the x2's in terms of the other two.

Solving for x23 and x24, we get
x = -;- x (x x + x x )
23 m 21 ll 13 12 14
I x22(x11X14 ' X12X13)]‘
In this expression, the first quantity in parentheses

is zero by (17), the second is X1. By (22), m = IXlIl

Therefore,

(23) x23 :_T;iT— x22 .

Similarly,

X
_ l
(24) X24 " lxll X21 °
Referring to the equations given under cases

I and II, we have, under case I,

(25) x23 2 x22 , and x24 = -x21 ,
(26) 22 = p + iq + jq - kp ;

and under case II,

(27) x2} = ~x22 , and x24 2 x21 ,
(28) 22 = p + iq - Jq + kp .

where p and q are real parameters, not both zero.

 

4 Bacher, Theorem I, page 54.

-13-

9. Summary prnilfagtors. Let a, b, c, d be
any real numbers such that

2

a + b2‘# 0 , and c2

+ d2‘# 0 .
Then, by equations (19) and (21),

zl = a + lb 1 jb : ka

is a nilfactor, and, by equations (26) and (28),

z2 = c + id I jd 1 kc
is a conjugate nilfactor to zl, regardless of the
relative values of a, b, c, and d. This can be ver-
ified by direct multiplication of Z1 and 32, the pro-
duct being zero.
If zl is a nilfactor, a conjugate nilfactor is
most simply found by changing the signs of x13 and x14.
It follows from the associative law of multi-
plication that the product of a nilfactor by any
hypercomplex number, not zero and not a conjugate
nilfactor of the given number, is itself a nilfactor.
It should be noted that if X is zero, 2 cannot be a
nilfactor.

10. Relation g; conjugate and nilfactor. If
2 is a nilfactor, then D is of rank two as shown above.
The matrix of coefficients of equations (6) is the
same, when x in (6) equals x

p 1p
three rows of D, and is therefore of rank two.

in D, as the last

This renders the solutions for the components

of 3 given in (7) invalid. The last two of equations

-14-

(6) are of rank two and may be solved for £3 and E4
in terms of i1, Eé, and the x's. The components of
the conjugate then are found to be the same as those
of the conjugate nilfactor. This is evident by com-
parison of the last two of equations (6) with the
last two of (12). The name "conjugate nilfactor”
was chosen for this reason.

The product of a nilfactor and its conjugate is
necessarily zero, by the argument of the preceding
paragraph. Therefore, it is necessary to bar divi-
sion by nilfactors as well as division by zero. A
nilfactor in the denominator of a fraction has the

same effect as a zero.

CHAPTER II

DIFFERENTIATION AND INTEGRATION

ll. Limits; continuity. he definition of a
limit is the same as that used in complex variable.
The necessary and sufficient condition that a sequence
or a function approach a limit is that each component
of the general term of the sequence, or each conju-
gate function of the function, approach the corres-
ponding component of the limit. The usual theorems
on the sum, product, and quotient of limits hold
true, except that nilfactor, as well as zero, values
of the limit of the divisor are barred.

The definitions of continuity and uniform con-
tinuity likewise remain the same. The necessary and
sufficient condition for a function to be continuous
is that the conjugate functions be continuous. Other
theorems on continuity and on limits also carry over
from complex variable?

12. The derivative. As in functions of a com-
plex variable, the derivative of a function is defined

to be the following limit, if it exists:

L f(z +Qzl - NZ)
(:2 = O .Az

(29) g: = f'(z) =

 

 

5 Ketchum, page 643, footnote, states that in Town-
send, Functions 93 g Complex Variable, every theorem
of Chapter II, pages 20-41, is true for hypercomplex
variables.

-15-

-16..

A function which has a derivative at z = a is
said to be regular at a. If it is regular at every
point of a region, it is said to be holomorphic in
that region. It is called an analytic function if
it is holomorphic in some region.

If a function has a derivative, that derivative
must be independent of the method of approach of
(z + Az) to 2.. By setting 42 equal, in turn, to
Axl, 14x2, JAXB’ and kAxa, and taking the limit
as Az approaches zero, we find the following values
for the derivative, if the partial derivatives of the

conjugate functions exist:

 

dw __ 3,“. all; 3/“3 2}“;
dz I am. +1 245, +3 9%, +k 2M. ’
_ 122. _ 1.31"; J, .2511. _, kids.
- 24x, 32;; ‘3 ébfla .QAZg ’
(30)
_ .2312. + 1.3.41. - J14; - 1,2512.
- 3%, 3/143 29%: 9A4; ’
_ 3/44 - 1 3/43 - 3/01 + k 3/”I
‘ am. am. 3 .92.. ‘7. °

On equating the corresponding components of these

four values of the derivative, we find that the neces-

 

sary_conditions for a function 33 have a_derivative

 

 

are that the partial derivatives 9: the conjugate

 

 

functions exist and satisfy the equations,

-17..

 

 

 

 

2’“: _ all; _ 9/43 __ 9/44!
our, - Jar. _ Q/X, - 29%,, ’
.2531. _ , 9’”: _ .9212. _ _ 9""
9%: — 2%, — 2A3; _ 3/433 ’
(31)
3/4, _ 3,412 _ _ 3A; _ _ 3/44
3453 57x), 345! 07/35:. ,
3,“, _ - 3/42 __ - 9/43 _ afly
aw... _ 07/26: _ 3/». - 3/26. '

These are the equations corresponding to the Cauchy-
Riemann equations in functions of a complex variable.
We shall call them the Cauchy-Riemann equations here
also.

By substitution from these equalities in the
values of the derivative given above, we can write
other expressions for the derivative, each involving
the partial derivatives of only one of the u's.

They are,

(1W _ 2,“, - 1 9/1 _ J 3/4, + k 9/“,
dz - 2“! 2%: 3/233 2%,}. ’
_ .3132. , 1.2.42. _ 3.3.4:. - bide.
- 3441 29/26, 32:4,, 9433 '
(32)
2%.? 9/43 2/‘(3 3/41
=m-im+Jm-kM.
241:3 343,. 9/», 29/2151
:_?_:‘L+,.§fl.+J.f_<”_L,k.€/£z_.
346,. 24/3 34:, 34v,

-18-

If the conjugate functions up have continuous
first partial derivatives satisfying the Cauchy-
Biemann gguatigggiig’g region, then w = f(z) hgg g
derivative f'(z) lg that region.

Since up has continuous partial derivatives,

it is totally differentiable, and

2"! 9,4
1

u
A p 3/”, 9”?

3»!
+ #Aflg‘f éflAfll + éizAflz

where the 6's converge uniformly to zero as 43;

approaches zero? That is,

‘64.; l‘ 5 for IAXiI é lAzI< 5 .

 

 

Then
2" _ .Aul+i.<iu2-i-j.¢iu3+k.l.iu4
132 I 432
- I 2 abeIAAx + i 2 97g; [ix
4 3/4 4 3A -—
+3): 3 Ax k2 ” Ax 6,
1 ¢9Afif P 1 .9/2} p

for lAzl< 5 . The absolute value of .6. is less than

16 6 because

I--’-‘”‘
A 7

 

él, and Iéfji<€ o

 

1s

6 Pierpont, J., Theory 9; Functions 9; heal Variables,
Volume I, Article 426, pages 271, 2.

-19-

Substituting in the above difference quotient from

equations (31), we can reduce it to

 

15w _ [ 5b“, abut .3913

AZ 3/». + 1 3/». + J 9/24.

+ki’f’L]£Z_+Z.
49A”: ‘4}?

Taking the limit as A z approaches zero, we get

 

f ‘Z’ - dz -7/;:‘*17;7‘
2x“; + 1 3/44.
+ —-—-——-—-— ——-—
J 3/)" { J/X’I ’

since we know that the partial derivatives exist and
that i? I < 166 , While 6 approaches zero with A z.
The value of f'(z) found is exactly the first of
equations (30); so the theorem is proved.

As in functions of complex variables, it can be
shown that if a function has a derivative, that deri-
vative is continuous. From this it follows easily
that the first partial derivatives of the conjugate
functions up are continuous. Accepting this as true,
we can state the following theorem: The necessary

and sufficient condition that a function w = f(z) be

 

holomorphic gg a given region is that the first par-
tial derivatives of the conjugate functions exist,

are continuous, and satisfy equations (31).

-20..

The rules for differentiation of a sum, a pro-
duct, and a quotient remain the same, as can be shown
from the definition. The derivative of z is found
from the definition to be 1. Starting with this, we

n is

can show by induction that the derivative of z
nén - la and that the derivative of a rational func—
tion may be found as usual, except at nilfactor
values of the denominator. V

13. Stokes' theorem is S4. By an argument
similar to the proof of Green's theorem in the plane?
it can be shown that on a curved surface with curvi-
linear coordinates u and v, we have the Lemma:

.9/9 .962

i: P(u,v), Q(u,v), EUV" and a»: are continuous

 

is s region S' and C is ss ordinarg curve gs the

surface is S' bounding s region S s: the surface,

 

 

then
(33) J! ( 43-3- - 315—) du'dv
5 ,a .QA’

:II (P du + Q dv) .
With the use of thgs lemma we get Stokes'
Theorem: 12 Pi(xl, x2, x3, x4), 1 = l. 2. 3. 4.
are continuous functions with continuous first par-

tial derivatives is s region R si 54, C is an ordi-

 

nary closed curve is R, S is s ssrface is_R,‘boundsd

 

7 Townsend, E. J., FUnctions si s Complex Variable,
pages 54—56.

-21-

p1 C, with parametric equations x1 : xi(u,v),

i = l, 2, 3, 4, where the first partial derivatives

 

 

 

 

c9 - .
3:?" £123 “—437, exist and are continuous and the
3’7!’ 3‘4-
7. a _‘I Q - _______4__ _ A
second paitiai derivatives 9/49” .. ——-——-)”3A exist,
then
£91Xm + P26”? + P3“; '* P261999
+ 9P; 3P,
= ' “_—“— ' ————- d d
J£[ ( 47/», 3/», ) X1 X2
P a
:(__?__Z_____f£__)dxedxj
8/14,, 346,
~ + 33; 373
(34) " ( 94¢: ' 2%”. ) dxjdxz}
3F
1 ( __3__F_,__ ' —""?'— ) dx dxl
3/)65 3%, 3
8P 9?
: ('—_"JL' ’ _‘_‘£; ) dx4dx
2A4... 3/252 2
+ a P1]. 3p, 1
- _____. _ ______ , d .

The 1 signs used above do not indicate ambi-
guity, but a difficulty in determining the proper
signs of the Jacobians which occur in the proof.

The result as stated is sufficient for the purpose
desired in the proof of Cauchy's first integral theo-

rem given in section 15.

Let
2
x :31 x =_€_/f.£_ etc,
pu 2A ' pu" 3,143” ’

Then, by the lemma, (33),
JcPlXm '2 JC Pl(x1udu + xlvdv)

:J L(Plxlu)du + (Plxlv)dv]
C

 

' a .9
._ JIS.-E“_(P1X1V) - 7(Plxlu)]du dv .
By writing out the partial derivatives indicated

and performing the subtraction, we find that

9” xl,x2
J6 Pldxl : - JJs-‘a—m-JI— J(—JT')C1U. (1V

3%.? u, V
5
x
- 11—43;» J( u} v4 )du dv
5
.. .9? + 3F
: + JJWL: XmdX2 - J} 3/); C1 3(1X1
s, 5.. 3

where 81’ S2, . . . 86 are the projections of S on
the xlxe, x2x3, x3x4, X3Xl’ x4x2, and X1X4 planes,

respectively. Similarly,

- P
[ P2dx2 : +J %— dx2dx3
c 5 3

2
3P 9P

+ .__:£__ 1 .___£_.

.. L 3m, dx4dx2 IL 34/, dxldx2 ,
5

I

3 __§;__ + 0" Pa
3

Adding these four integrals, we get the equation of
the theorem, with integration over S, since x1 and x2
take on the same values over S as over the projection
51’ etc.

14. Definition and pronerties s: the integral.
The integral is defined to be
(35) Jf(Z)dz= L 2f<$pp)AZ.

C ND=O p=l

where

w = f(z) = ul + iu2 + ju3 + ku4 ,

bl
N
ll

Apxl *1AnX2 +3Apx3 +kapx4

= + i + + k
c€13 spl 6p? ‘3ng g‘p‘i
is a value of z in the interval ,Apz on C, the path
of integration.
The product to be summed in (35), expanded,

consists of sixteen terms, a typical one being

(- J u4( t:‘p)Apx2 ) .

-24-

If f(z) is continuous, then the u's are continuous,
and the limit of the sum of each of the sixteen terms
exists. Therefore, the limit defining the integral
exists. It can be evaluated in terms of real line
integrals by the use of the following equation, each
real integral being the limit of the sum of four of

the sixteen terms mentioned above. This equation is

Lf(z)dz : Jéuldxl - u2dx2 - u3dx:5 + u4dx4)

+ i [(u2dxl + uldx2 -'u4dx3 - u3dx4)

(36) C
+ j [C(ujdxl - u4dx2 + uldx3 - u2dx4)

+ k £5114Xm + u3dx2 + u2dx3 + uldx4) .

By argument from the definition of an integral,

it can be shown that if f(z) and g(z) are integrable

and c is a point on the path of integration, then:

1, a
I f(z) dz = - J f(z) dz ,
,5

J k f(z) dz = k I f(z) dz ,

TL . ,5 AL

J[f(z) + g(Z)] dz = I f(Z) dz +‘I g(z) dz ,
at c a. ,6 a.

J f(z) dz = {f(z) dz +-J f(z) dz .

a. a. c

Also from the definition it is easily shown that

.L
(37) I Idzl = L .

a.

-25-

where L is the length of the path of integration.

By inequalities (l) and (5),

n
)pilfi 5p) Apz

é VT 2m $p)‘°'4pzi.

Taking the limit as AZ. approaches zero, we have,

:.<- £|f( 5p) Apz‘

 

1r [f(z)] is integrable,

' j££<z) dz é L/7§_‘£flf(z)l°ldzl

a.

,1, 1
ey-z—Jmndzlel/ré‘mJIazlz-IF2 ML.

4

 

(38)

where M is the least upper bound of [f(z)] on the
path of integration.

15. Cauchy's first integral theorem. ii f(z)
_s holomorphic is s region R and the partial deri-

 

 

 

vatives 3;} (i, J = 1, 2, 3, 4) exist and are
continuous is R, and _i C _s s closed curve is R,
then
(39) Jf(2) d2. = O .

C

Since f(z) is holomorphic, then it is continuous
and the partial derivatives satisfy equations (31).
From the continuity of f(z) we know that the integral
exists. Applying Stokes' theorem four times, once
to each of the integrals in the right member of equa-

tion (36), the u's being the P functions, we get an

expression for J f(z) dz in terms of double inte-

C
grals, over a surface S bounded by C, of such expres-
sions as

“727—227“

There are twenty-four such integrals in all. On
substitution from equations (31) in these expressions,
each of them is seen to be zero. Therefore, the

value of each double integral is zero and the con-
clusion of the theorem is proved.

The hypothesis that the partial derivatives
axq
that f'(z) be continuous. By a demonstration parallel

are continuous is equivalent to requiring

to Goursat's proof of the theorem in complex variable,
it is possible to complete the proof without requir-
ing the continuity of f’(z)§

A consequence of the theorem is, Corollary 1:
The integral si f(z) has the same value along any
psis is R Joining the same two points; that is, pss
integral is independent 2:.LQ2 psps si integration.

Another consequence is, Corollary II: la.a
region is ssiss f(z) is holomorphic except sp certain
points, s psps si integration can ps deformed into
any other path without affecting the value si_iss

integral provided that is the process the path is

 

8 Townsend, pages 66-71.

-27-

cut py_ss singularities.
l6. Cauchyjs integral formula. The values of
2 which make (z - b) a nilfactor are

z : bl + r + i(b2 + s) + 3(b3 1 s) + k(b4 : r) ,

where the bp are the components of b, and r and s
are parameters, not both zero. The points repre-
senting these values of z lie in the planes A and B,
and include all points of the planes but the point

b which is their only point of intersection.

Plane A is
X1 2 X1 , X3 = " X2 + (~03 + b2) ,
X2 = X2 , X4 : X1 + (b4 " bl) °
Plane B is
X1 : X1 , X3 = x2 + (b3 - b2) !
X2 : X2 , X4 = ' X1 + (b4 + bl) 0

and

x2 2 b2 , x4 2 b4 ,
intersect planes A and B at the point b only. With
the planes A, B, E, and F specified, we can state the
following theorem, Cauchy's integral formula: ii f(z)
_s holomorphic is the simply connected region R, b

is any point is R, and C is ss ordinary curve is R

encircling, but not intersecting, the planes A and B,

-28-

0

then: (1) if can _s deforned into s curve Cl about

 

 

the point p in the plane E without cutting planes

 

 

 

 

 

 

 

 

A 92 B,

y _ l f(z) ,

(10) f(b) — 2""1 'Jc Z - b (12: ,.

(2) i_ C can ps deformed into s curve C2 about
the point b is the plane F without cutting planes
A gr 3,
_.__l__ _£ial__
(41) f(b) — 2M L Z _ b dz .

We see that

17(2) : LL

2 - b
is holomorphic in R except on planes A and B. There-
fore, it is holomorphic on planes E and F except at b.

Case (1), C deformable into C : In plane E,

1
let Kl be the circle
_ 19
x1 + 1x2 — bl + ib2 + r e ,
X3:b3, X4=b4,

where r is a constant, 0 a parameter. Then, by

corollary II on page 26,

f(z) d _ f(z)
z - dz
C z - b K z - b

I
= I _§ipi__ dz + I. f(z) - f(bl_ dg
z - b 2 - b
K, K,
Because f(z) is holomorphic in R, it is regular,

 

and hence continuous, at b. Therefore,

-29-

[f(z)-f(b)|<€ for Iz-bl< 5 .

On Kl’ lz - bl = r. Pick r < 5 . Then, by

equation (38),
I I f(z) - rib) dz
“0

 

g l/ 2 ML

= W— (_f'—.)(2TTI‘) = 23/2116

Therefore,

z - b

l L “2% : 113i =0,

and

J f(Z; : SLb) dz = O .
Kt

Setting

2 - b = r e10 ,

for z on K , we find that

1
2w

(iii-Elf;- dz = f(b) J1 d6 = 2m f(b).

Kg 0

Combining these two results, we have

J —§1%lg— dz = o + 2n1 f(b),
c
l J' f(z) d
Z 9
C z - b

 

f(b) = 2wi

as was to be shown.

Case (2), C deformable into 02: In plane F,

let K2 be the circle

x1 + 3x3 = bl + 3b3 + r e39 ,

x2 z'b2 ' X4 ='b4 '

-30-

where r is constant, 6 a parameter. By the same
argument as for the first case, replacing i by j,

Kl by K2, we can show that

“‘0’ =‘A—J‘igs‘dz
C

2nj

CHAPTER III
ELEMENTARY FUNCTIONS

17. Definitions. The exponential function,
circular functions, and hyperbolic functions are
defined by the series which represent them in com-
plex variable. The addition formulas for these func-
tions remain valid for the hypercomplex variable, as
can be verified from the expansions;

From the series we find that

e3x = cos x + 3 sin x ,

ekx = cosh x + k sinh x ,

sin Jx = J sinh x , cos jx = cosh x ,
sin kx = k sin x , cos kx = cos x .

These equations remain true when 3 is everywhere
replaced by i.

18. The exponential function. Using the addi-
tion theorem and the equations of the preceding sec-
tion, we find that

(X + ix + Jx
w = ez : e l 2

3 + kx4)

(x + ix ) (3x + kx )
e l 2 e 3 4

x
e 1(cos x

2 cos x3 cosh x4

+ sin x sin x sinh x4)

2 3

-31-

-32-

x
+ i e l(sin x2 cos x3 cosh x1+

- cos x2 sin x3 sinh x4)

(42) x

+ J e 1(cos x2 sin x3 cosh x4

- sin x2 cos x3 sinh x4)

x
+ k e l(sin x2 sin x3 cosh x4

+ cos x2 cos x3 sinh x4) .

From this value, after simplification, we have

2x
(43) lwl2 = e l cosh(2x4) ,
and
l 2x1
(44) U =‘—§- e sinn(2x4) .

Therefore,

le4 - 4U2

4x1 4x

e cosh2(2x4) - e l

(45) sinh2(2x4)

:e4xlfi0,
and we can assert that ez is never a nilfactor.
Using equation (30) we find, as would be expected,
that the derivative of eZ is again ez.

19. The circular functions. By the use of the

 

addition formula for sin(A + B), we arrive at the
following expression for sin z:

w = sin 2 = sin(xl + ix2 + 3x} + kx4)

(46)

+

+

-33-

sin(xl + ix2) cos(.jx3 + kx4)

+ cos(xl + 1x2) sin(Jx3

(sin xl cosh x2 cosh x3 cos x4

i(sin x

- COS

l

+ COS

J(cos x1

+ sin

k(cos xl

- sin

xl sinh x

cosh x2

x1 sinh
cosh x2

x1 sinh

cosh x2

x1 sinh

The values of le2 and U for

(47)

and

(48)

m2

2 sinh

sinh x3
x2 cosh
sinh x3
x2 cosh
cosh x3

x2 sinh

= sin

w
2

x3 sin x4)

sin x4

X COS

3 X4)
cos x4

x3 sin x4)
sin x4
x3 cos x4) .

z reduce to,

_ 2 2 2
— - sin xl sin x4 - cos x1 cos x4

2 2 2 1
+ sinh x2 sinh x3 + cosh x2 cosn x3
cos(2xl) cosh(2x2)
cosh(2x3) cos(2x4)

ml»;

cos xl sin xl cos x4 sin x4

- cosn x2 sinn x2 cosh x3 sinh x3

4+4

sinh(2x

sin(2xl)

3

)

sinh(2x2)

sin(2x4) .

+ kX4)

2

9

-34-

Computing the expression

le4 - 4U2
and setting it equal to zero, we find the condition

that sin z be a nilfactor to be
2 2 2 2
1 -
sin x1 + sinh x2 + cosn x3 cos x4

* 1
(49) — - 2(sin xl cosh x2 cosn x sin x4

3
- cos xl sinh x2 sinh x3 cos x4) .
Starting with the second values in (47) and (48),

the condition is found to be

cos(4xl) + cosh(4x2) + cosh(4x3) + cos(4x4)
(49') = 4[cos(2xl) cosh(2x2) cosh(2x3) cos(2x4)
— sin(2xl) sinh(2x2) sinh(2x3) sin(2x4)] .

We observe that two sets of values of z that
make sin z a nilfactor, a few of the values being
zeros of the function, are

_ x _ kn _ x
Xl"4' 2' X2‘3'

 

where k is zero or an integer, and
x1 = x4 , x2 = x3 = O .
The expressions for cos 2 corresponding to those
of (46) through (49) for sin z are similar and may
be found in the same way.
Finding the derivatives of sin 2 and cos z by

equation (30), we can show that

 

 

d -

dz sin z — cos z ,
and

d ~

dz cos z — - sin z .

From these equations, the derivatives of the other
circular functions may be found.

20. The logarithm. Let w = log z be defined

by the equation

(u +iu +ju +ku)
ew z e l 2 3 4

= z = Xl + 1x2 + ij + kx4 .
Then, using equation (42), we find for the xis,
ULl
x1 = e (cos u2 cos u3 cosn u4

+ sin u2 sin u} sinh u4) ,

ul
e (sin u2 cos u3 cosn u4

>4
[0
ll

- cos u2 sin u3 sinh u.) ,

u
_ 1
x3 — e (cos u2 sin u3 cosh u4

- sin u

cos
2

L13 sinh u4) ,

u
e l(sin u

1.
x4 2 sin u3 cosn u4

+ cos u2 cos u3 sinh u4) .

From these equations we compute

2u

(50) 2(xlx4 - x2x3) 2 2X 2 e l sinh(2u4) ,

-35-

2u
(51) xi + x2 + x3 + xi = Izl2 = e 1 cosh(2u4) ,
2u
(52) xi - x3 + x; - xi = e l cos(2u2) ,
2u
2 2 2 2 _ l
(53) x1 + x2 - x3 - x4 — e cos(2u3) .

Subtracting the square of (50) from that of (51),

we have, 4
u
e 1::z14-4x2,
and
(54) ul =~%;—log(lzl4 - 4X2) .

Solving (52) for u2 and (53) for u}, we findw
2 2 2 2

 

 

 

_ _l_ xl - x2 + x3 - x4
u2 — 2 are cos 2ul
e
(55) 2_ 2 2_ 2
:-fi%— arc cos x1 4x2 + :31/2X4
(Izl - 4X ) ’
and
2 2 2 2
(56) u3 =‘—%- arc cos X1 +4Xg- :31;2X4:]
(Izl -4X) °

Dividing (50) by (51) and solving for u4, we discover

its value to be

1 2X
u = —- arc tanh -————
4 2 (1212)

_i_ (zi? + 2X
4 10% IZI2 - 2X ]

(57)

 

-37-

The complete expression for log 2 is found by
use of equations (54) through (57) to be,

w = 10g z

-%— 105(lzl4 - 4X2)

 

+

.1.
i 2 arc cos

(58)
X? * X2; X? ' 392+?

”Pi. 10.“:{2 :3]

+ J -%— arc cos

 

 

It is obvious from this expression for the loga-
rithm that every nilfactor value of 2, as well as
the point z = O, is a singular point of the function.
The quantity
(59) I2)4 - 4X2 .
which is zero when 2 is zero or a nilfactor occurs
either in-the denominator or inside a real logarithm
in every conjugate function. This is true because
both factors of (59) appear in the value of u4, and
one of them is zero whenever (59) is.

21. Comparison with functions of a comglex
variable. It is easily seen, by a review of the
functions considered in this chapter, that zero and
nilfactor values of the argument tOgether take the

place in our study of the zero in the study of func-

-38-

tions of a complex variable. We have seen that cor-
responding to the non—existence, for a finite argu—

z
ment, of a zero of e in complex variable, we have

2 is never zero or a nilfactor in our function

that e
theory. Corresponding to the singularity at zero

of log 2 in complex variable, we find that log 2 is
not defined for a zero or nilfactor argument. If

we should write out the expression for tan 2 as the
quotient of sin z by cos z, it would be evident that
the function has singularities at nilfactor, as well
as zero, values of cos z.

In general, x2 and x the second and third

3:
components of our four-component variable, play the
part in this paper that the imaginary term does in
complex variable, while x4 goes with x1 in taking
over the role played by the real term. This is seen
in the absolute value of ez, where only x1 and x4
occur. It is seen again in the logarithm, where it

is the second and third conjugate functions which

give the function its multiple-valued character.

BIBLIOGRAPHY
Bacher, haxime, Introduction 39 higher Algebra,
New York, Hacmillan, 1907.

Ketchum, P. W., "Analytic FUnctions of Hyper—

complex Variables", Transactions 9; the American

 

mathematical Society, Volume 50, 1928, pages 641-667.

Pierpont, J., Theory 9: Functions of Real Vari-

ables, Volume I, Boston, Ginn, 1905.

 

Townsend, E. J., Functions 9: a Complex Vari-
able, New York, Henry Holt, 1915.
Historical references given p1 Ketchum:

Autonne, Journal dg Mathématiques, (6),

Volume 3, 1907, page 55.
Berloty, Thesis, Paris, 1886.

Scheffers, Leipziger Berichte, Volume 45,

1895, page 828.

Weierstrass, thtinger Nachrichten, 1884,

page 595.

-39-

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