’
’
I
:—
I
—
I
_ ~—
—
,—————
I
_J__—‘
____’__
f
4—;—
_—____
—
__—'
__.———
—
‘_—’
_____———
‘_’__,,
f
_4——
’
’
,,_____—
I
’
___————
J7
I.
—
"
I
—
7 7’—
’—
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_———
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—___———
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—
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’—

THESIS

Ely

 

59M MPUEMEBNS 53F WE
WHYEWS 5st A QQMPLEX NRMBLE

WmS HER WE EFE'fifl HF METER 93F RM

DALON HAL am
19:35

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MSU

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RETURNING MATERIALS:
Place in book drop to
remove this checkout from
your record. FINES will
be charged if book is
returned after the date
stamped below.

 

 

 

 

0

 

 

SOME APPLICATIO?S OF THE

FUNCTIONS OF A COHPLEX VARIABEE

A Thesis
Submitted to the Faculty
of
MICHIGAN STAT? COLLEGE
or
AGRICULTURE AID APPLIED SCITNCE
In Partial Fulfillment of the
Reguirements for the Degree
of

Master of Arts

by
Dalon Hal Ely

1935

1%??th

Tim"!
E52

 

 

 

 

 

ACKNOWLEDGMENT
To Doctor James Ellis Powell

whose suggestions and help are

responsible for this thesis.

“05‘4‘32.

TABLE OF CONTENTS

INTRODUCTION

I. Hydrodynamics

II. Light

III. Electricity

IV. Heat Flow

V. Potential Theory

VI. Algebra

VII. Evaluation of Definite Integrals
VIII. Differential Equations
IX. Plane Geometry

X. Statistics

XI. Cartography

Bibliography

10
18
23
26
29
31
33

h2
M8

53

INTRODUCTION

A student is first introduced to complex variable when
he studies elementary algebra. At that time he feels that the word
"imaginary" expressed very well the usefulness of such numbers.
This feeling is usually retained until he takes a course in the
functions of a complex variable, and even after a first c urseiue
seldom appreciate the applications of the theory to other branches
of mathematics and to physics. It is the purpose of this paper
to point out some of these apilications. We hope that it will add
interest to the study of complex functions and that it will induce
more students of mathematics, physics, and engineering to make such
a study.

No attempt has been made to give an extensive treatment
of the applications; it has been our purpose, instead, to give
enough applications in the different fields to suggest others. In
some of the problems discussed the only method of solution is that
of complex variable. While in other problems the solution is ar-
rived at more readily by this method. We propose to so arrange
the material that it will be readily available to the teacher for
presentation to a class, or to the student who feels that tixe
spent on such a subject would be w sted as far as usefulness is
concerned.

fie will discuss oroblems arisin in algebra Where the theonr
of poles is applied; problems from st tistics are discussed where

Euler's formula and the principle of inversion are used; We will show

how to evaluate certain definite integrals by the use of the theory

of residues; we will give a proof of the existence of a solution
of a linear differential equation, using some of the properties
of complex series; the existence of a relationship between trig-
onometric functions and exponential functions will be shown and
illustrations as to how this relationship is used will be given;
and for the use of the teacher of elementary algebra, we will show
how certain geometrical theories can be proved in a simple manner
by introducing complex numbers.

We will also discuss problems from the physical field.
Thus, we will apply Laplace's equation and conformal representa—
tion to problems in hydrodynamics; Euler's formula and the principle
of inversion to electrical problems, Laplace's equation and the
theory of inversion to problems arising in the theory of the potential,
the principle of inversion to problems in the theory of heat flow,
and Euler's formula to the theory of light.

We also will show how Euler's formula and conformal
representation are used by cartographers and map makers in mapping

the earth on a.plane.

 

HYDRODYTANICS

The first problem we shall discuss is tiat of a two-di-

O Q
I

mansional, non-rotat-on:l motion that often appears in the theory of
hydrodynamics. The straight forward solution of this problem requires

that we find a solution of Laplace's eQuation that will also satisfy

“n

he boundary conditions.* when we have done this we determine the

pressure by means of the equation

(1) fl 3 - w- jjz-I- 3/

6? pressure,
f velocity : /(%2+(%3Qf2
w

force potential,

[0.: cross section area

and W(X,j) is a. solution of Lailace's equation.

This orocess .ay be tedious or even practically imoossible.
Another method, wLich, while it is not direct, is much more fruitful,
is to take a particular class of solutions of Lcolace's equation and
see to what class of problems they may be a plied.

If we consider the plane of motion as the co ilex plane, then
the com lex analytic function,

0
w : ulx,‘5)+ 4 V0. ,

which satisfies Laplace's equation is such that if we plot from the

z—plane to the w-plane with

uf : L4 #- 5“

 

* R. A. Eoustoun, Kathematical Physics

r:

we will have a conformal map.* Hence, since the velocity potential is
oeroendicular to the stream line, if we consider the real part of the
function w to be the velocity potential, the imaginary part will give
the stream function. On the other hand, if we consider the imaginary
part of w to be the Velocity potential, the real part will be the stream
function.

As an example, let us consider the analytic function
(2) uf: WMHJ).

Since

(Av-(V: M(X+¢j)
: MxMaJ+Mx M?
= Mxm3+ (Mi/4943:

 

we have
() u: go‘s/ml,
3 ,

v/ :',Aué¢4()f/duénq7 3
and

a2 v2
(M) “5+ ,- ,‘X—l,

.2

(5) “2 ”KM—L

4

ma “M‘J ”

Plotting (#3 and (5) on the u,v—plane we have a family of ellipses and

a family of hyperbolas so arranged that the ellipses and hyperbolas

 

* For a discussion of conformal maps see section X of this paper.

 

 

 

 

 

 

 

 

 

 

 

Fig. l

are perpendicular.

Now it is known that if a liQuid is in motion the lines
of force are peroendicular to the stream lines. Considering the
figure we see that the vertices of the family of hyperbolas all
lie along the upaxis between.the points (—1.0) and(l,O). hence
we see that if the hfnerbolas are t ken as the stream lines the
figure represents the flow of a liquid through a slit and the
lines of force set up by uch a system are elliptical. On the

other hand we might consider the family of ellipses as represent-

’\

ing the stream lines. again considering the figure we see that
the ellipses approach the straight line segment from (-l,Ol to
(1,0). In this case we have the motion of a liquid rotating about
a thin plate and we see hat as we get away from the plate the
stream lines are elliptical and the lines of force hyperbolic.
It is readily seen that by taking fj)0 we have the motion about
an elliptic cylinder.

Another problem in this field of an entirely different

type is of interest in that we employ the same function
u]: 4W},

It is known that when a liquid seeps through a porous
soil, the component of the velocity, in any direction, is pro-
portional to the negztive cressure gradient in that san direction.
Thus, if

(P : pressure gradient,

J52: a constant of proportionality,

(6)

If we insert these values in the continuity eQ‘*tion,

2:5 2.”-

 

+ a,
Z»! 3;;
We find that,
251° 9%
2 ‘____. .-
(7) _ VJ“ 3,2“ 33" 0’

Now suppose we consider the problem of the seepage flow
under a gravity dam resting on a material that permits seepage.*
Tere we seek a function, P, which satisfies Laplace's equation
and also certain boun‘ary conditions wnich depend on the nature of
the surface of the ground. That is, the pressure must be uniform
on the surface of the ground upstream from the heel of the dam,

and zero on the surface of the ground down stream.from the toe of

If we nmw choose our coordinate system with the origin
at the midpoint of the bsse of the dam (Fig.2),

(8) 03(ulv) : £g(uly)

77, I

 

 

* Warren fle:ver, Conformal Rgpresentation, with Applications to

 

 

Problems g§_Applied tathenatics. American Yathematical Vonthly,

 

October 1932, p. M65.

where .304, V) is determined by
W: (Ag-[V : a W(X+‘:j).

One may now easily find the distribution of unlift pressure across
the base of the L8H. In fact, the base of the L8“ is the reoresent—
ation, in the (u,v) plane, of the line

u.= 65

osjé 77’,
of the (x,y) plane. Hence on the base of the dam the eqaations

u = a (JO/414.943,

y' : 62.4%;VAJZ—Aifln3)
reduce to

(9) 04:: 62¢QMA:I,

Vro,
so t‘.t

(10) 0°( k, 0) =

D

'3
03

-I

4&4

sits
N:

The total uolift or ssure per unit length of dag is found to be
1° " I 4 612‘
- a - _. _
(11) 7°- 95,— m a — (ta,
.4

which is the oressure that would be oroduced if the entire base
of the den were subjected to a.head of water just one—half the
head aLove the ham, or if the oressure decreased linearly from
the static head P. at the head of the ;am.to a Vhlue zero at the

toe.

I V

 

 

 

 

 

 

 

 

 

 

 

 

LIGHT

One of the most interesting nhysicel ohenomenon that we
.encounter in daily life is that of light. “en's idees of the na—
tire of ligh' have changed from time to tine through the centuries
but little was done in a scientific wzy to determine just what
light was until the time of air Isa c Kenton. fewton perfonned

noerimcnts along this line and arr'ved at the conclusion that

light WLS in the nature of bullets hurtling through soace at an
exceedingly high velocitV. At the same time a rival school, headed
by Euyfiens, arose on the continent, which hela that light was energy
transmitted as a wave. Togey physicists have incoroorated both
ideas along with modern Quantum olysics into an exceedingly com-
plicated theor . For our ouroose we will follow Huygens and con—
sider light in the nature of a wave motion.

This motion is conceived to be of tie transverse'tyne,
as a water wove, consisting of crests and troughs. The distance
from one crest to the next succeeiing crest is called the wave
length and is designated.Ly'fi.; the time taken for one comnlete
wave to ease a given woint is called the oeriod and denoted by'7r;
the velocity, of the wave, is equal t0:%? ; the a ulitude, which is
the height of the crest above the normal, is designated by,b ;

- . 3
the i tensity of light is equal to.6 .

It is nos ible to show exnerimentally that if two waves
of equal lengths and equal an litndes are traveling in the same
direction through a medium, such that the crest of one falls on
the crest of the other, the resultant wave has the same length and

an anolituee of va . If, however, the crest of one falls on the

10

11

.

trough of the other, the resultant a alitdde is zero. In the

latter caie tfie w;ves are se tid to nest we ivelg interfere.

To or duce interference between light weves it is nec- '
essary to take light from the same source, solit it into two oarts,
lead the carts over different maths and reunite them at a small
angle. If tne cifi erence in w)f th is an odd nunter of half wave
lengths the waves will inte: fere and dar_:ne s will result.

A “e.ve *fliose amulitnde is one,m{y be exoressed Jet- metic-
ally b‘ the function

Mat—- 7 Jzt-é)

The orotlem we are considering here is that of interfer—
ence produced by a.medium whose sides are oerfectly olane end oar-
allel.* Let B be such a medium and consider a plane wave front,

ude one,incident upon it in the direction AP. This will

6+

of amili
give rise to a reilected wave P, R , and e.refrected wave in the
direction PIC’. Tliis refracted wave will give rise to a reflected
save in the direction 0,733 and a t ans: .ittec wave in the direction

C

T The reilected Have, 0,3k: give s rise to a reilec ted \MQY

II'

in the direction EzQz and a refracted wave in the direction 2232,
end so on; the original iticid.ent wave gives rise to a series of
reflected waves slid a series of transmitted waves of reoidly de-

creasing intensity. What we are alter is an expression for the

intensities of the resultant reflected and ti :nsmitted waves.

 

*-
tr!
F
n:
O
(,1;
U1
cf‘
L‘
5
lb
F3
} J
("r
(‘7
C"
H
2
(D
F"
’g,
*6
H
3

 

12

Letm be the angle of incidence, 9 the angle of refrac-
o . l . .
tion, e the thickness of the plate. Let/Cfl be amplitudes of see.
I
ceeding; reflected m.ves,d,c’ be amplitude of succeeding; transmitted

waves, and /" the index of refraction.

to
to
h

 

 

”I C, ca.
7; 7;.
Fig. 3

Draw P, N perpendicular to the surfaces and P‘H perpendicular to ”31°

Tmien the first and second reflected mves pass- tl'e plane
Pg}: they are in different phases, for the}; were in the Smile phase
at P, and have traveled different Optical distances in coming; from

Pl. Let their relative phase difference be denoted by d; then d

denotes the phase difference between any two successive Waves.

13

 

We have
e'
(12) ficlz—JQQ/
and
73H= 73730~2H737f§
: .ZIVC; £JR_¢7
(13) : sea/«64144449
sulfa
; ‘2 (at an 6 .
Hence
0’: 61(f‘863r‘cng-75‘H)
(1‘4) ._ .2 £135... '53—'59
"’ 71(aoa ‘2‘“ ‘3'“

If the incident wave is denoted by

34;“ $7M- 15—)

we can represent the resultant wave by the function
a 217 X _
RM[?('¢-v) ”Q
in which the amplitude, R, and the phase, d: are to be determined.

The first reflected wave is represented by

' 2
/?'£44x-i;r(f“'1%&

114.

the second by
I . 1277' ._J; _
fl’CC %/?(t V) ad)
the third by

n’lc’LA[£g(t-%)'ZJ],

and so on. rl‘he resultant is the sum of these waves. Hence
I 2” a 0 2”
fl M[?(t'é)‘ {/3 fl 3AM igrt- %)+A’cc{?tcm{’f‘(f-%)-dj
+x1‘cc’1m[’J(t-l)-2dj+m~

M nag—- <t-%)-a~-Iuj+~~

To evaluate R and Jwe must sun this series. To do this as the

series stands we would have to aohly the principles of Fourier;

if, however, we int odice com lex terms by the use of Euler's

formula we will have a series of complexterms which can be easily

sunged. Using the ex1c‘nsion for .B‘Lxla- b) , We have
RLAE—“(r- -£)m/- 794.4%.— ”(r-é)»;

+/I’(c:3:fl’(t-£)g¢a( 9/166 MW. :{f‘fjmd

(1
5) +flicczdmflzrr(fiz)m‘2x_flw(;1”(f_%)/dfi:zy
+....... .. ..........
4’” CCMg(T‘-E)Coa((fl 'UJ)- fl ccm——(t-3)M(h 041
'f"" ’ .

mp 17 1
F0 all; eq .Lat ng tr -e coefficients (3sz (ti/end, (14.5.5: “Elt‘ v)
We get
2i “J3 ”1“”,(6 (Md-f4 mZd-ffl’maal-I-HH4-fl m ”obi-“)l

(16) 2h-Z
1 a ,7 . I 0
WAAJ: fl’cflflakd +A’J4~Zd+/Z,AA~3¢(+-"'+/l ,AmnoH-“X

15

. I“
The series of (15) and (lo) are power serie es in/fl with the coef-
ficient of each term less then one and hence are c1nvergent. Now

by the use of Euler's foraula
K - '
e X: m,” 4’4”"

we can combine tiese equations into the followina form

J 9' zn-.z
ae‘fi maze/(e 02‘ was” ~61-.-)

(17) e“
: ,4:+,e1:c 7:;F3EZK '

inis canoe written in the form
dd 2"€7
./ xzflccc (l-fl’c
71763 2 4+

(l-fl’ c‘vO-flflc"?
xz’c't’(coqa(+ (Mal-A1?
/— 2/:"mA4-A’7

 

(18)

 

=/z+

2 .
”ICC ’{doqol-d’) .fl’fC Md
“ /?+ I- /2’ ‘md—r/Z’T +‘ l- 3/!"aqal'f'A

 

But if two complex numbers are equal their aiolitudes are egual.

Hence

 

4 .
7(2_ a,“ d’c’d’t’Con-A’zj A’cc’A—«X y
z -.z/:' Mali-AV” "' 7- WA’ .(+/t’

This may be sin1lifiea to

add

7? — 1-2/2’ coed-r)!”

(19) yflz’dé‘zzé

 

H

 

(I'V'Z'z/‘f'ijmzfié

If T is the amnlitude of the resultant transwitted w ve, We have

by the nrinciple of energy

7?’+T¢=I.

Hence

(20) T

 

2_ ("Ag/2

:- 2A’2m4 +4”

We are interested in the maximum and minimum values of

 

2 z
R and T . We have
2
fi’ = 0
when
,4JA€£¥::¢7 0
Therefore

X

or
2ߢmd= mt

where N is a positive integer. Thus when the path difference is

an integral number of wave lengths we have no reflected light.

 

 

Also
2
fl: _ 4/!
(22) " we)“ 2
.1 + 971
La
.Z
i s a. maxi mum when
SLLCZ§§L2 I.
Therefore
anafi )7
551' yr” # = 62M”?

01‘

2 at (44% = (WM/)3L

Where N 18 a oOSitive interger. Thus R 18 a maximum wnen the oatn

J"

difference is an 0;; number oi half wave lengths. Th minimum

value of R is zero and the maximum value is
t
472
("47‘

 

17

..y— 2 o I o L - o
anen R 18 a mlnlnun,T gas a maxlnum valve eQual to one,

- - 2 . . 3. . .
an' w;en R is mwx:mum.T mas a ulnlnum valve eQu( to
a 2
[-21
a

 

1+4

III ELECTRICITY

the comwlex variable to
In electri—

.4-
‘ uS.

The apolications of

problems in electricity are divided into two on.

notentiel theory we cogsider oroble s of the srme tyo as
For exam tie t edsfor—

cal

those discussed under hydrodgnzmics.

nation
(Al’z (Lfifia‘iZ;
‘his DEDCT, might be interpre-

which has been discussed before in t
ted as reoresenting the electrostatic field due to a source in
or the electrostatic field due

1e sheoe of an elliotic cylinder,

‘eu plate from which a stria has been removed.
the oroblem of an electrical

-.
t.
A
in?"
Cilks; L
\J

to a
co )lex variable in

The use of
exilcined the ?cll Televhone Lab-

q

circuit he?

*1

Deer
nwnner.‘

since the ohysical

oratories in the follow ng
"From the physical stahddoint it (electrical Circuit
theory) is naturally a study in real quantities,
and mathematical methods
lgost exclusively

C}

forces and currents are thetselves reel,

originally used in connection with it dealt
ext Cunturg, howef r, a Dro—

Luring the pres

with real variables.
taken place, until now the use of comjlex

.818

and real numbers are

gresvive

transition 1
quantities is by all odds the usual

thing,

rather rare.
3 n ation for Electrical

’JJ

 

“'30.“ , f" *1. .1h4. ._
uiiierenticl solutiois

* T. C. Fry,
Circiit Theory. American Biathezicticcl fionthly, 1-}2'), P. hag.

18

 

19

"---- To reduce to mathematical terms, then, the problem
of elementary circuit theory is that of solving the system of dif-

ferential eQuations:

mm + 5MP)?" "* ’7” ””3"
+ 5,. (4°) a = 95%?)

:61?)

ll

‘511’757*“427“Ttk* ......

(23)

I
l
l
l
l
I
I
l
I

_—--_-——~
.0....-.

l
I
1
51,(pjjl+51(fljzf """" +5A(P),74:fa (u!

Where the F's are linear differential operators with constant coef—
ficients, the f's are simple harmonic functions or sums of

such functions, and the boundary conditions require that the system
shell be either at rest or in a steady state at the time zero.

“ I need scarcely remark that the stuny of such systems
is a recognized o'rt of first courses in differential equations.
The suggestion that the founcations for circuit theory be laid in
sudh courses is, therefore, not a.very revolutionary one; it re-
quires at most a slight change of emphasis to accomplish it. Per—
hans this will become even clearer as we mention the four facts
upon which the use of comwlex quantities rests, none of which is
unusual or non-mathematical in chzracter.

"I The sum of any solution of (23) due to a set ff ,
and any solution due to a set gJ , is a solution due to a set fJ+gJ

"II This is an inverse to I. Of course, there is no
exact converse to tie irinciple of superoosition, for even if we

know a solution of(23) ins to ff+ gj we cannot ordinarily separate

20

the part which is due to 3:!" from that due tojj. There is one im—
portant exceptional case however.

" If the functions fj are real while the functions 1] .rre
pure ianaginaries, then the real part of the solution is due toé’
and the imaginary part of the solution is due to j J .

“III. The derivatives of an exnonential function are
proportkmal to the function itself.

"IV. Euler's equation
Co‘uz: 62k1,(f‘£;4*;vz4

“Hence if(23) can be easily solved Wheanfiflis an expon—
ential, and if it is true that whenffi’jis a complex we can sep-
arate out the real solution frmm the imaginary solution, it fol—
lows at once that we can obtain the solution due to the simole
harmonic function, doow‘fl €))by first finding the solution due to

the comlex exponential, edfl’g , and then discarding the imagi-

nary part.

"This is the entire argument in favor of the use of com.
plex imaginary numbers in electrical circuit theory. It is hard
to imagine any explanation bread on vectorial ideas, which would
approach it either in conciseness, simplicity, or generality.“

As an example of this method of solution, consider the differen—

tial equation . 1'

 

 

* Page, Introduction to Mathematical Pmrsics.

21

é!“ a!“ __ .
(2h) 5‘22“?" 422;; 44, é/t/+tc(z')

where a,, a1, 8‘, are real constants, and b(t), c(t) real functions
of the real variable t.

If we assume a solution of (21+) to be
(25) M = X/t/f {7/27,

we have, after substituting (25) in (21+).

4/51 613' 611 . 6/
dt‘ ’dt‘fdlzt? *‘4’7;

Equating the real and imaginary parts of this equation we get

 

(26) a, + ('4, +4317“ ('43:; : é/r/H'c'fd.

2
,r 54/?
a :2]; *4 21”” 1: 5/“

I t .4!
(27)
~———-—--~ —’ ( J
4/ '1‘: 1‘ 42 b; + 4, 7 _.. C I

which are of the sz= me form as ( 27).

Now since we know that the impressed electromotive force
(EL-CF) is a harmonic function we may consider it to be the real part
of the function

€(th+zLaakthE 58““;

where Ea is not necessarily real. We may now write

15' = [0,65% (é’agafj‘

0

be impressed E M I is then

(28) £‘= MMY/é’eawt’w = [.324 (MHW.
Now if we denote the current by I, the resistance by R, and
the inductance of the circuit in henries by L , we may write
the equation for Ohm's law in the following form

(29) 4 gi— 7PI+ 24/194? = [.8
Comparing this with (214) we see that if we consider

('0: t-

1:1, A —- a ”—4“ and 21-: 6tJ then (29‘, and (2h) are

I’ -

identical.

To solve equation (29) let us assume a trial solution
LT== J;,c"at' (J; nzq‘nzaaaad/uéér/1444).
Substituting this value in (2“) we find it is a solution if

10/7f+4'(w1~5’2) _— [0/

 

[a -‘-' I/fiz-r-(wA—dijz .I‘e‘tf
.6“, (VJ

Pita 71f

 

 

r0.

/
where ZZQA¢J' = “ll-'Z;E
fl

Therefore E and I represent vectors rotating in the
x,y-plane with an angular velocity427, the latter lagging behind
the former by an angle d’. The actual electromotive force and
current at any instant are represented by the projections of these
vectors on the x—axis.

If we put

2 : fi+"(w£_ _/_.)= IfE2+(Q/~_fij‘fe"f

(426 J

 

..(OF

I

/ I
fi*"("”"a'12) M2+(wL-'jgz
it follows that

£7:= £:_Z;

I -‘- X75

H:

 

IV HEAT FLOW

Many problems arise in the field of engineering
to which the theories of heat flow can be applied. For example,
it is of very practical importance to know the conductivity of a
boiler plate, or a bearing peaking. It is of theoretical interest
to know that the length of time that the earth has sustained life
may be determined by considering the conductivity of the earth's
crust.

For the purpose of this discussion let us suppose that :7
all space on the oositive side of the y,z-p1ane is filled with.a
homogeneous solid of diffusivitylf . Let the temperature on the
y,z-p1ane be given as a harmonic function of the time, that is,
the temperature at all ooints of the y,z-plane is the same but
this temperature has a periodic variation. For example, if

T: temperature, and t==time, then
T: 4t% mt-r- 61""mm’2‘,

where a and b may be comalex'but n, m, m"are real. Also let um
suppose that this function is the same fer all values of y and 2.
It is required to find the temperature throughout the solid when
the periodic state is fixed.

Clearky T is independent of y and 2. One of the condi-

tions T must satisfy is

37’ 3’7“
0 "'__' - —————. o ,_
(3.) at .. at} _\

That is, the change of the temperature with respect to the time
is prooortional to the velocity of the change of the temperature

with respect to the distance from the y,z-plane and the constant

of preportionality is equal to the; diffusivity of the solid.*

 

*‘R . A. Houstoun, Introduction 32 Mathematical thsics.

23

 

21+

That is to se , T must also satisfy the conditions

Tr-axzy'amt, web/«1:0, (m: M) )

(31) 7': 00,,W2': 00.
To solve equation (30) let us assume a trial solution
.(32) T: edfrfif’
When this is substituted in (30) we have
‘t’p’: Eflzc‘fl’ef

or
(33) a =Kfi€
It is obvious from (33) that A cannot be real and negative. There-

fore let us assumed is imaginary and
0K 2 i ‘0 J 0
Substituting this value in (32) we have as a solution

:tl'd't‘ £1
(31*) T: e '1'): I ’1"
Now we know that

(H 02: .2 z:

and
. 2 '
(/~(/ : -Z(,
Hence
I .
V? = 7;: Ni"),
and

p z E, (l‘ ().

Substituting these values in (31+) we have
T : an": (1% (/1: c);

we x i am i a I)

Now the condition for x = 00 demands that the sign of the

exponent be negative. We have, therefore

‘ _ J: ' +¢'/II"- 2 X _ ' - 1.

where A and B are constants.

substituting x::o in this equation we find
[It - (It

But to satisfy boundary conditions, when x::o we must have

'T': 62,4Jn.kn't.
Therefore

'H- x Jar-1x
F7€(( g38€(( [2:]

must take the form
a ”“6 t—g X);

with
[:M.

Hence

- in ,
T: as gfaMJMt-gx)

is the solution to the problem.

V. POTFITTUJJ 'IVY'IORY

, In discussing potential theory, we are interested, not
only in the develOpment by complex variable, but also in comparing
this development with that in which co;.1)lex variable is not used."I

In two dimensional problems, the equation to be satisfied

by the potential is
1- l
31‘ 33‘ ’

from which we get

 

**
(36) '7' = {(2-1 {7) +F0v/J).
If Vis to be real, then F must be the function obtained fromf

by changing Z to - 4' . Now let

f(x+l:j) '-'- (4+5 V, u and v both real,
Then
rung) = u-I'K
and
'V' : 24!.
If we let
17 =-2K
we have

7+ (V = —2V+2¢'u
(37) 3 dam”)
: +3"j-(X+/j): (PAH/j),

where JOY-NJ) is a cor;r_31e’cel;..r general function of the single

 

* For an example of the development without complex variable see
Ifaclfillan, T4 e051 2i: the Potential.

** Jeans, Electricity and lfagnetism. P. 261

36

27

complex variable (1* {j} .

Thus the most general function of the potential which is
wholly real can be derived from the most general arbitrary function
of the single comlex variable (KHZ!) by taking the potential to

be the imaginary part of this function.

Now, if

0mg): 774/17-
and

Z 2- X+<' ,
(38) J

W : 17+ ('7‘
and if we let the element in the Z-plene be
(39) ’4‘": 421
and the corresponding element in the w-plane
I of
(1‘0) 44:4”: 5-2-42,
we can get any element QQ ’from PP ’ by multiplying PP ’ by 4%];

that is, by
at”) 02 MW).

01442
If we express J; as

(In) if: (NM-{1) = Hm“ I'M 6’) 4w“
we can find the element dw by multiplying dz by (O or lag/and

 

turning it through the angle 9 .
7:10?! let us examine (a , we have

(Wm 49 + (Mafiff’: 49’tu

(142) ax ax
: 2125117
o, a: 1

av .37 .. 2372 21:23
gm - {emu/o

28

Thus we see that if'Vr is the potential, the modulus,(f

measures the intensity R. And since
R= 9770“

we have a simple means of finding 0‘, the surface density at any

point of the surface.

VI ALGEBRA

Let us now turn to the realm of mathematics. It was
here that the ideas of imaginary numbers and hence complex vari-
able first arose. We first reCOgnize the need of imaginary num—

bers when solving quadratic equations of the type

2
X +1 */= 0..
Since the use of complex numbers in the solution of

eQuations is familiar to anyone having had a course in college
algebra, we will, instead, in this section, show how the funda-
mental theorem of algebra can be easily proved by the use of some
ideas from functions of a complex variable.‘

Theorem: E. eguation 9_f_'_ daggeem complex coefficients

f0?) = z”+'a,z"’$ ----- +42, = 0 ,

has a complex (real 9}; imaginary) root.

 

For, from our knowledge of singular points we know that 2‘“) has
no singular point in the finite plane and has a pole at infinity.

Putting
(M4) 2 = 21-,-
it follows that
(15) 4912') = H?)
has a pole at
z’: 0.
It can be shown that if
2 = 20
is a pole of order K of the analytic function J-(z), then gig) is

analytic in the neighborhood of Z, and has a zero point of order

 

* E. J. Townsend, Functions 2.2 Complex Variable, P. 291

29

\JJ
0

Hence??? is analytic in the neir'I-uborhood of the origin.

Consequent y, since
I - .1... ‘

fog») ‘ cmz’) . . _
is z—*-nalytic in the neighborhood of the origin,?—(/}-) is analyticin 4
the ne‘ghborhood of 1:00 . But, as is easily proved, every analytic
function. not 7. constant, :zzust hat-'e at least one sitv'ulrr point
eitlmr in the finite region 0':- at infinity. Sinceféa cannot have a
Sinf”-)‘-l£“l‘ noint at infinity, there mus-t be at least one point in the
finite rr~-_.,_rfion, say Z a , at which the functionjyl—i) has a singular

point. This ..Ol‘;'it cannot £36 {.23 esr'.e1'1tial singular point, for in that

5

case 20 would not be a. r-gu ar point offa’). Hence 2,, must be a. pole of
.1. . .
j‘!) ans. consequently a zero point of the {given funtionf/Z). This

estrblishes the theorm.*

 

* For a proof vrith'iut the use of the theories of a com'vlex variable

see Dickson, First Course in the Emory o 773118tl0319 r). 5g

 

 

 

VII .TEYE EVELVATIC‘FT O'f‘ BETTY-YET? I‘ETE‘T’TALS

All.

Certain integrals
that are difficult to evaluate by ordinary means can be evaluated
by the use of the theory of residues.‘

Consider for example,
277 de
M
f 2+m6 .

0
To evaluate this, let

 

 

 

 

_ (79
Z - 8 ,
r _ 61:?
(140) 63(6 - (z- I
_ elfif 6- (9- 2+?
Hence
:7
(147) J J? : f 71;!" ——-p(£‘ .: .3. 6(2‘
2*009 x. .l <' z
a 024(24-91‘2) t Z +‘Yi4'l
where the path. C, is determined by the substitution
Z: 8/9

As 9 varies from a to 277', z describes the unit circle about the

origin. The integrand has poles at
it 2-2 1 V3)

but the only one to fall within C is
2 = - 2+ VF.

Hence if we expand 7‘13) about
2 —- ~2+ V37

by considering I
I - __._.—--
(M8) 237;? _ (2+2- V3)(Z+2+V37

 

 

and then expanding the factor
1

3921-5
about the point by Taylor's series and multiplying each term

 

 

¢ E. J. Townsend, Functions 9; g Comolex Variable. P. 2814

 

31

l
by 2+2~VZT ,weget

 

___L.._— -/‘ + ------- .
(’49) gaff/{fl - +5(2+Z )

Since the theory requires only the coefficient of the term with

the exponent (-1) this is all of the expansion we need.

The residue at

i: -.2+IT?‘
l
1325 . Therefore we have
277' 0(9 277/3
______.,.——_-_- ( "(—— ‘ v”
(50) j 2-1-on 4 277 2/37 3
0

as the value of the integral.
As a second example let us consider a function with

infinite limits, for example
‘0 all ,
3 J
co(X +I)
Substituting
x = if

we see that the function
I

(z‘W’
has a pole at

Z: (.0
Expanding 3H2) in powers ofO‘c' U we have

1 - ---"' - -—-——-3‘ - - - -- ~
(5 ) 5'7”!) in; ’6)" “(z t)2 Mlz- ¢'+) '
The residue at 2‘: 4 is
he:
" Ié '

Hence we have
377’

00M ii. I: ---,
(52) jmz'w '27! i
-00

VIII. DIFFZQEYTIAL EQUATIOHS

To discuss comoletely the many uses of the functions of
a comolex variable in the theory of differential equations is im—
noseible in this paper. We will illustrate the use of some of the
theory by giving a proof of an existence theorem of linear differ-
ential munitions."I To simplify the proof we will consider an
equation of the first order.

Theorem. Given §_linear differential equation 9f_the

 

Knunmr

first order

(A) g: 530,3)

rah-

siere a and y_ are corrole}: and {(53) _'1__«.;_ analvtic in some finite

region
D: [X‘Xoléa
lam/$5,
for al g” y'in the region. ithollovs that there exists a solution

 

 

a: (A) missing through the noint (x.,j.),
For, since fix, is analytic in L‘- it follorrs that ffxyhas an unper
bound, K, in D. **

Expanding fayabout the points x yowe have

of

H293) = m. 3.) + :5, (21.4.) (1-1.) +230.» gag-3o)

) + ’u (1.. 11.) ( H41“ ’33 “IJ‘JU‘U‘S'BJ * ’13"°J')(3‘3Jz
(53 + .2 .1 _

.f_-—-—-—--—--

 

* Voulton, Differential Ecuations

 

** E. J. Townsend, Functions f §_Comnle§;Vzriable, P. 39.

 

3h

Define the function F(X; y

 

(5”) Fftj} = ___#_'_j__i“ x-x. “ J
lgs-N’ ‘7)

 

 

 

 

 

where (I‘
a’m,
b'< b.
Expanding
I
M M 1-1 , Mid
/‘ 3.3‘ /"*--———«-o
b’ 4”
. ¢ n
M ~ M[’+Lg'a+ (flj+-—-+(.L3j+~]
/- 3.3“ - 6’ 6’ bl )
and b, 2 X'A’ h
’ -- 4.15.? X‘XO --.. (“L—3 4<o-
/-£.:£¢' /+ 4”¢{”Z7)* + 4’ ‘

These sgiies converge in a r gion
DI: LPN/(6L
Ig-m 5
since they are geometric progressions with the ratio less than one.

= meter—(~71
s _ - O _ a
= M, ev(a7+(~;¢%s)+(e¥-N

is convergent in D Since it is the proeuct of convergent power

 

M
Then — _ ' 4.
(55) ("-33.1)("%7)

series.
Text we will show thrt the series (55) douinates the

series (53)f

Y"

* For a discussion of doninant functions see F. S. woods, Advanced

Calcalus, p. 57.

That is, we will show that

 

 

 

 

 

 

M %1f(xolj°) ’)
~23 % [£(X9130)ll
M
7% ”30.4.”,
33.x} ’5’;~7‘“~3:{/
H! ’
ah
‘41..) b:d(x..3.)l
b' )4:
. aha-sh I
(x0) 0
m “a aman‘ 31
d’mb' {Wu-M)!

Let us consider first the two series _

<5o) 1m: f(X,)+f’/X.)(z-x,)+--

 

 

‘ - _ __
(57) F (X): 1-1. - ll”. a’ + d"
- 7

 

By Cauchy's integral theorem we have
:HX ) ._ I {(04%
a r 277.! c 25-10 .

Sudstituting '6
c
(#1,) : x: e ,
[(t'Xo)/ A ’ .
d: Z/ze‘fl/gl

W 6 find that

lf‘hJ/é 5%, %-277/2: M.

35

x-x. (1—1.): _ - ,_ +_ (£32) * J

36

I ' 1
n a simi ar 'ff(r)o(:)é __l__._/_l 274’: 1:7,
|f(X,)I= ”rt/fa x.)

{(t)df 4 +1. = :K '
) x 277“ In a
szlwxdl ‘27” lj—-———'— (1" ‘0) a

This shows that (57) dominates (56). To show that a similar reason-

ing holds for two variables, let

 

__L_‘ *(t’je)i(:
(58) f(x,,3,)-2m C (t-x.) "
J71 3) - J... {(t v.17;
mf. -————(T_ ,)

 

H. Wu: .L 4,«6’77§l'12’=/‘1,
If (Xej. )/ (.277! [1: f (1'40” T‘J') $97744 b

)J J .51
I 21:15:19.7: kmfi’f/LL ZHxEIaLSI S 4’ ’

z .eztvmf 4 21-,
Ia§(x..&Io:)’ (2777).; )MJ (7: x,)(r—3.) b

I

I

I

l

I

I

I

 

 

I

I

I I
I I
' I
I I
I I
l I
I I

Dmfuzflv) f(§T)_—_-d’__7a(t A %M
IT )I: 277% I i}; (1‘ 1o)m(7’fl°)/ ’

 

‘Z ff! 17111.0“ /( A
n ““""“ In,
[a f(x..3.)/=(27’n,) Hf (cant-34"" b

 

9‘] "
)4(Zb(t /q
am+'\;(xog° I 2/ f(tT 91/ s ,m bl"
la QXTQJ" /: (.27!) IJ’T‘J‘ (1" Xe) T’jo)“ a

Now assume a solution of (53) in the form
(59) 3—3.: ak,(x~x.)+o<.(x-X.)‘+----+fi,, (II-Io)" ' ‘
Substituting (59) in (53), we get

AI+2*‘(X.X.)f '1' K‘K(X’x.)n.{!
= C” +- 6“, (x- x0)

 

(60) +6.! L4C (X‘XO)+44(X'X.)‘. f“‘ +dn(X-X,) "f-r]
”‘on X0) +61: (X- x0109 (X I.) +4301.) +. .]+-- .
aha-Oh
Where C“m : 3X "‘93" fifhuj.)
(huh)!

Equating coefficients, we find that
0M = Coo,

‘ clo+cmdfl: clef-(0160012

3‘: " caldz+coz*cu"‘ +c0142"

A:
h9~
l

Hence it is possible tosvaluate all the ok’: in terms of the C's,
and hence we have a solution to (6k) if we can prove the series
(59) convergent in some region.

To do this consider

<61) 34: ..
dz " FOUL

with
j (X.) = 3,

By use of the same method as above we can solve this equation in

a series, to get 2 n
(62) 3-3.mix-inset).--”MW-..

38

The values of the Ads here are of the same form as in (59). Further-
more. they are all positive. Hence, since the series (55) dominates

series (53) it follows that the Ads dominate the A3 . That is
saw, figs/«.4 ..... figs/M

However, we can solve (61) directly without the use of series,

 

since
3(1-/L— ,
‘ - . “25.1.:
d; (I- 2171')“ a’)
and
Md)!

-3, _. ________,—— .
ago-17)- I7-
Therefore z 1‘);

.‘1" (37%,): ~N4’/&L.T(I- 4’)+ 6'
Since
1:“)! '3sz)
we have
0': :10.
Hence
~Xo
3.. j,- Vig‘)+ma’.47(I-I—r"= .
If
3-3» = 37,
then 13? 1—0’
a]

          

        

y— :b’Y—Z/‘M'b'jifl

and

I :: 5’: W‘+2Mn’bilaj(l- €755;—

 

39

Now I is complex and therefore the series for the expansion of
(53) is convergent in a circular region having xa.as a center and
a radius equal to the distance from x0 to the nearest singular point.

There are singular points at

(63) x—X. _ /
and ‘2’ -} I

1“ I I _ X‘Xo ..
(61*) b +245M«4f(/ “wj-a
Solving (61+) for (X- X.) , we have

J&:f(/l- X~X:)=_ gATROV

"i—IX—E: eia’M

(1" X0): a-- ”a C

Hence the radius of the circle of convergence is finite and dif-

I

447%

ferent from zero. Since series (62) and (6h) must be the same,
series (b2) must be convergent. Furthermore, since series (62)
dominates (59), it follows that (59) must be convergent in some
region.

We therefore have the existence of a solution of (53).

IX . PIu'aN'E G720? TTQY

It is often desirable to have a rather simple
application of complex numbers to bring home to a group of college
freehmen the idea that complex numbers have other uses than for'
solving algebraic equations of certain types. The apolications
that have so far been discussed are in the most part beyond the
range of a freshman group and have little meaning to them. With
this in mind, Lloyd L. Smail of Lehigh University has worked
some theorems of plane geometry which he has proved with the aid ;
of complex variabled' To do this he used only two theorems of
complex variable which are easily proved to an elementary group.
These are:

l. The distance between two points 7,319,} and 7.;{22} is

given by 7373 3 [El-E2, .

2. lbs point of division of a line segment joining

points EVA} and 672‘} which divides the segment into the ratio

221?“ 2 4:0»
(65) 7’3 ' ’
Z = Ml— .
If'A

For the special points where P, is the bisector and P”

is trisector, we have

I_ _L _
(66) P " 2 (Eli-23)) A-//

I/
P : 3L(£,+234,)J )zz.

 

* L. L. Small, Some Geometric-:11 Applications 31; Complex Numbers,

American Mathematical Monthly, 1929, P, 5014,

13.0

Lil

One of the theorems that he proves is the following:

I. The mid-point 2;: the hypotenuse 9;; _a_ right triangle is s 111..

 

distant from the vertices.
For, consider the right triangle with vertices at
0M, fife}. 3164.
N

M

 
   

c 3’441‘6‘7}

 

 

0 5/6?
Then the point C which is the mid-point of the hypotenuse is
given by Ci} {44'6”}. Now
0C = [2L(a+b¢')~0/= ZL/a+6¢'/=%W§
A78=Ia+bil= W. -

Hence
0c=j~/76’= BC= 6/7.

By similar methods Professor Small proves eight other

theorems .

X. STATISTICS

One of the fields where one would least eXpect to
find a use for complex variable is the field of statistics. How-
ever, in some of the developments of statistics, a.know1edge of
complex variable is very useful. Consider, for example, Laplace's
deveIOpment of Bernouillis' theorem.*

Let us consider the probability expansion

(mg/f

where

34 = probability a thingwill not happen,

00:: probability this same thing will hanpen, and /z is a
positive integer.

We have
(67) (flog/if; {77/ 7*fl’7’w (1/7/92--m”
where (f/ : fl (#5,)(fl‘22' . . .[fl- (XL/217 .

x!

The problem is to obtain a simple expression for the general term,

which we will designate by‘jx .

[(-22

5:(o°+;/§- (fif’ifiM %+(.z/,9f *"”
(68) +::./)iflf my! +63% ;¢*(X+1M ;l+/+-"'

 

* From notes taken from Lectures by Professor Carver of the
university of Midhigan, by Professor Crows of Hichigan State

College.

L*3

and let . fl JL‘J‘ fr fl" Mae)!“
3: (is We 7: (was
MM; 52’- (:on W"“’"“"
(59) (w/zuxlflleuz-21-1-Jil' +:/¢e-/N"
+ 1+} "U '

Solving for the term containing (1) , we get

(”ff/1‘;€~(fl-wm’[(f€w+d°€'w)‘nffl lit-1‘-” _ ’
1(1); ft 34/on plepz—zn-Jjwz'

(70) /Z i/ZvP/fla /‘z+/ (/2-21—2) 40"
1H; 6 —- - - — - *
/z _/za)c'
... f e

Tn+9“l"ctl‘l" botn me: were of the above equation with respect to

we have

<7an:11 (17,0 fine I: ""“W‘ee We”? 1’1”“

To show that the other terms are zero consider any term as

(7JJ“”"1/f 6 5w

1?

we!
(72) 2/7}? (mzw-uwz ) a)

gem___-Z_&7~ ( 176.042
.2

F7
h qu/fl)f/Z-:flo H/ H
0/" 7rd; Lemmy 8 #8“):le

’

LLB.

We now have the general term expressed as a definite

We now wish to evaluate this integral. It is easily

integral.
seen that
4.01:)
-(Jz-szw‘ a; -(/r-2wa""’*/l/’j§?€ we
(73) e (f: +fe ”/2 e .
Therefore

(xJ/‘f‘ " 9%] c“”‘“’“’"*“"7Wiifswiza

, .4
Expanding "(i7 94803798 “yin terms of X; we get

1::mw
C = 0, Q = as flff/I-é/fl
4 mm”, 6.; -‘- ' ”Vise/III-Wfl.
9: Vflff’ c6 .-.- éV/Z/bf/l- iof/s/za/QZ
arm/{4w -~

Hence we have

(75) (if); #-

Uaking the substitution
dd.)

77 . 2
3' 2(x~/z,e/ca¢—2fl/°X““ 5/4).

M5

(75) becomes $01
X/ / =2}? e 4’
-77

z. . 3
I 7/- (x-fl/aj 1.9/— 1/23? (I (at) 5'75“”):

" C ‘2 1,23 + / .77

_ .457 . (S 3' 9' :a(&)

or,

21-177 - aw (fl—L420
(76) (Qt/ff GYfl/e (Ir/Wang //+ 3123 1“ mfg/a)

”cf (PA/140‘” ‘fig—Q‘J

3,277
< > ’77 if may"
77 c (-27 zJ'- .8
+5337;- 6” ” ‘ -= wad/w";
' Z7

Since the series in brackets in (76) converges to the value
wz' ~49!
ext/Q7978 +f€ j)

the integration performed in (77) is valid and the resultant

series converges.

Letting
t
)7, (,p/zltwa- 221‘.”
WIFE}; (3 ’2 4/40.
~77
we have, on differentiating, .q 40‘
I 77 ~ I“ i
(00'): *— 5” ”Na” ,2 («Jib/a),
-77 1
,7 - l- 4&1!) .
(My: 8” Wm 2 (£00309;

(78) ' ~77

I
l I
I
' I
X ' I l
' l
' l I
I
l

I. 77 - ,_ file—9‘60: ' n
m"(z)=2'7"7‘/c"”’”"” 4 (4") ”m

_.—.. _“-I-"‘
'5 . t ‘ .—

 

vfll K -o v no

1+6

Comparing (77) and (78) we see that
('y (6’)

(/x-g)//ZX 1:0(X)+3/23 (fl( 309+;Z7y4) (19"
This is analogous to MacLaurin's expansion-

Using Euler's formula
8": (24. 14- («444413

(WXJtakes the form, fl, X a?" g
070%,, —" W/[JQ(X‘/’/ya7+tldw'“ (x zz/waje ,2 fla)
_/r 1“

0“le ‘l 2

fl- —
:2;7/C ”2 Zd(X-/I/bjwa/a)+7;7 ’76 My fl/jwa/a)
”Z7

The integral

7 ӣ40: ,1
16‘ z/AM» (x-fl/DJMJ‘J
'5’
may be shown to be zero.

Using bz

W? ""9.
f0; ~41"; WWW— 2/84 Lenard": 228 y:
, ~00
With
: a?) é:(X—fl/Z

we have

flzmpza)‘
‘2z7;/:fl 62 ”‘2 ic;M2(X“/§Q)4Jaéflzfi I <3 ‘ZCQH1(1“A%94061&3
77—0

oo_ -fiz3249 W524)
:377:: C? Z43?Q (1"/€;V6J0Aa7‘rz;11£9262}Q(QX—/7Q)d)0/UJ

 

LFl

It can be shown that whendfifvz is large, the remainder

i 8 small .

We find the value of the general term to be 1
W; -&’—’£‘2 1/ 7w; Lie?
(79) (Ir/yi/ J/L— 8 ‘20 + g/ZadjfZ—e 20“ 77.- - - -

 

where

4/77.

As an example of the use of this formula let us suppose

that of 100,000 men of a certain age, 800 die within the year.

It is of importance to insurance companies to know the orobability

that say 1000 men of the same age will die in another year.

In this problem
f: .0fléf
/: .?Z&
/Z = Mama,
,2: 1M-

Substituting in (77) t
(we a
~x , I 33;; yam/(m2

(XXX/0 -m€ . + 6.3.(7936).r2;7

 

pus-a

XI. CARTOGRAPHY

The problem of mapping is the problem of find-
ing a one to one (1.1) correspondence between the surface to be
mapped and the surface upon which the man is to be made. For the
most part we are interested in mapping the earth on a plane. Now
we know that it is imoossible to apply a sphere to a plane; that
is, a Sphere cannot be mapped on a plane in such e.way that cor-
responding arcs will have equal lengths on the sphere and the
plane. Therefore, when a map of the earth is made there is
necessarily some distortion.

The maps with which we are most familiar are conformal
maps. These have the listinction that infinitesmal areas cor—
respond and that the angle between two arcs on the earth is equal
to the angle between corresponding arcs on the map. The most
common projections which have this prOperty are Lambert's conformal
conic projection in which the earth is considered an ellipsoid of
revolution and Mercator's projection in which the earth is con—
sidered a sphere.

If we have a surface,
F(6(,VJ=&,
on which the linear element is

(80) d: '3 Vala‘f- 7/1/2

in whichV' is a function of I" alone and U'a function of u. alone,

the element can be expressed in the form

d3 z-.-.- M2014 171-402))

us

n9

2
where E! is a function of u and v. To see this write d3 in the

form
,1, z_ 4/4; 0/1.!"
(01) 4/3 -2717 --.0_ +7-7— ’

By changing the parameters so that
.31:

fry- ’
dr

J’s—l

9

'l

(32)

 

the linear element becomes

p542: F‘VY/e‘M/‘IJ
When the linear element is in this form, the surface is said to be
expressed in terms of an isothermal orthogonal system of parameters,
and the net of u, v curves is said to form an isothermal conjugate
net.

After the surface has been expressed in this manner in
terms of isothermal orthogonal coordinates, the general conformal
representation of the surface upon a plane can be determined.

This general representation is given by the equation
1+1}: f(u+¢'v)
where fflui V) is an analytic function. The element of length

”(3%.- m2<azu=+m=l
is represented in the plane by
d: ‘= 47! 96/72?
If we let the curve C be represented by the curve C, in the plane

and if 9 represents the angle which the curve C makes with the

cum e

-v= Wit,

we will hare

 

 

 

 

Hence .
4 . ééh’L“{V; .a§¢+¢afv
C2 ='(2%Q67+'5«4**¢7 : %e#7‘ fighr-l
or ,
2/Q_ £ZEEZL_K'
62 ' ¢a£2-cé/V'
Therefore
Adm-63) dzu-zQ/V’ ééh/éfi
‘3 " fl-M' aerfléfl’
or

(an) 8'2‘7‘9'W: “44*"4/5 ”“29

Since
X74111: )((Z(+4IV)
so will

1'13! : {(a—{W s

50

If we differentiate with respect to the complex variable ,Kil;7

we have i I

544 4'45 3 04+ (1959-)! (“7“,"),
or , . 61;'+l'

(85) f (”NV/'7 @4420,

In the same manner éAQW-‘l

(8b) f’(”("’v) : fl—zZ/V .

Substituting (85)" and (86) in equation (314) we obtain
I I '
1€{9'3/_ 11.5.3.5.) .
— f/(“+"’/J

If B and B, are another pair of correSponding curves starting

ALL?

‘1‘? WF‘T'“-"."

from the same point and if their angles are denoted by<17 and

(D, we will find by the method used above that
«"74" 4") W

C f/(a+¢.V) .
Hence- .
2‘l(€_§’) 62((tp—fll)
’- l
01‘
€‘g; = w- 0')
and

awe—e. I
This shows that the angles between curves on the surface is pre-

served in the representation on the plane and the map is conformal.
' Ion if we consid r the earth to be a sphere its eguations
are
X: Mamz
(37) J:/d.43-ued.u'-lg
53: Ma,
where v is the longitude and u.the latitude. Here the element of

length is

_ 3.. 2

d5 - .99: fééz-xdzje
with
Mawm’a—MaAA WIV,
Mawvala-I- Aaluocml/‘(VJ
: —-,d»;zé(cxéc.
in terms of u.and q we get

dz
. . n
Expressing as
I 1
413 2:: (Qfinazétdbangb(gaata,4»«‘vdcza: .
4- (M344 M‘vq-Aak‘u 444214de
+ ai~§z&(¢aéx£:
The transformation (82) takes the form

, .9;
a Jet—f wa= 47%)

V

<88)
V

If we solve the first equation of (88) for u.in terms of
Hand then find the values of x, y, z in terms of H and be sub—
stituting the values of u, v in (87), we can substitute these
values in equations (8h), (85), (86), and show that the transforma-

tion actually does give a conformal map.*

 

* For a discussion of this transformation treating the earth an an
ellipisoid of revolution and of the errors introduced by the trans—
formation, see the phamphlet by O. S. Adams, General Theory 921 the
Lambert Conformal Conic Projection, U. 5. Coast and Geodetic Sur-

vey, Special Publication :70. 53— 1918 r

7'23? (1

is

‘L. 'r'.

“"‘-g-\unx- .
1 _ __ _

51
KN

BIBLIOGRAPHY

1. E. J. Townsend, Functions of a Complex Variable.

2. Pierpont, Functions of a Complex Variable.

3. J. H. Jeans, Electricity and magnetism.

h. Maxwell, Electricity and Hagnetism, Vol. I.

5. S. G. Starling, Electricity and Hagnetism.

6. W. Weaver, Conformal Representation, with Applica-
tions to Problems of Applied Hathematics, American Hathematical 1
Monthly, October, 1932. p. h65 L"

7. O. S. Adams, General Theory of the Lambert Conformal
Conic Projection, U. 5. Coast and Geodetic Survey, Special Pub-
lication, Number 53.

8. R. L. Peek Jr., Flow of Heat in a Disk Heated by a
Gas Stream, American Mathematical Monthly, May, 1932.

9. Page , Introduction to Mathematical Physics.

10. Ingersol and Fobel, mathematical Theory of Heat
Conduction.

11. R. A" Houstoun, Introduction to mathematical Physics.

12. D. M. Bateman,‘ Hydrodynamics.

13, G. Gray and J. G. Gray, Treatise on Dynamics.

1“. Lamb, Hydrodynamics.

15. R..A. Houstoun, G. Treatise on Light.

16. Max Planck, Theory of Light.

17. T. 0. Fry, Differential Equations as a Foundation
for Electricity Circuit Theory, American Hathematical Monthly,
1929. p. M99

18. houlton, Differential EQuations.

19. L. L. Smail, Some Geometrical Apalicaticns of Complex
Numbers, American fiathematical Monthly, 1929. 0. 50M

20. J. J. Thompson, Recent Researches in Electricity and

Vagnitism.

'

lift. .2313)

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.. I ‘..I...

 

Iii I‘ll .I l“.Il’lu‘I-',.

c
illul‘nv 1|}.vv II! (II I:‘:..

illijdsluflfil‘u: dial-”6‘, u .I i. A! I .V . I 'n r . I
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. . . . . . . . . , . . . . .. 3 . a... .. . .

 

 

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