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Chapter 4: Elementary Functions of

Complex Variables

Ali H. M. Murid

Department of Mathematical Sciences,Faculty of Science, Universiti Teknologi Malaysia,

81310 UTM Johor Bahru, Malaysia

[email protected]

September 29, 2012

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Elementary Complex Functions

In calculus, several derivative formulas have been established

for the elementary functions of real variables, such as the

exponential, trigonometric, logarithm, hyperbolic, and the

inverse functions. In this chapter we shall extend the definitions

of the elementary functions from real variables to complexvariables and obtain derivative formulas for them. We begin

with the construction of a suitable definition for the complex

exponential function, which forms a basis for defining other

elementary functions of complex variables.

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Definition of Complex Exponential Function

Definition

Ifz=x+ iy, then the complex exponential functionez isdefined by

ez =ex(cos y+ isin y).

ez has the polar formr(cos + isin )representation.Hence

|ez| =ex, arg ez =y+ 2k,

withk integers. Sinceex >0, the functionez =0 for allz.

If in the definition of ez, we setx=0 andy=, we obtain

ei =cos + isin .

Hence

z=x+ iy=r(cos + isin ) =rei. 5


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Properties ofez

Theorem

Suppose z, w are complex numbers and n is a positive integer.Then

1. ezew =ez+w

2. ez

ew

=ezw

3. (ez)n =enz

4. |ez| =ex =eRe z

5. ez is periodic with the imaginary period2i

6. If k is an integer, then

6.1 ez =1if and only if z=2ki .6.2 ez =ew if and only if z=w+ 2ki .

7. If g(z)is analytic, then ddzeg(z) =eg(z)g(z).

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Complex Trigonometric FunctionsFrom the Eulers formula,

eix

=cos x+ isin x, eix

=cos x isin x,wherexis real. Adding and subtracting these two equations:

eix + eix =2 cos x, eix eix =2isin x

which are equivalent to

cos x=eix + eix

2 , sin x=

eix eix

2i .

This suggests

Definition

Ifz=x+ iy, then the complex cosinus and sinus functions of acomplex variablezare defined respectively by

cos z=

eiz + eiz

2 , sin z=

eiz eiz

2i .

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Properties of Complex Sine and Cosine Functions1. sin(x+ iy) =sin xcosh y+ icos xsinh y2. cos

(x

+iy

) =cos xcosh y

isin xsinh y

3. The inequalities | sin x| 1 and | cos x| 1 are no longertrue for sin zand cos z.

4. The equation sin z=0 has solutions on the complex planeonly atz=n,n=0,1,2, . . ..

5. The equation cos z=0 has solutions on the complexplane only atz= (2n+ 1)/2,n=0,1,2, . . ..

6. sin(z) = sin z cos(z) =cos z7. sin2 z+ cos2 z=18. sin(zw) =sin zcos w cos zsin w

9. cos(z w) =cos zcos w sin zsin w10. sin(2z) =2 sin zcos w, cos(2z) =cos2 z sin2 z11. sin(z+ 2 ) =cos z

12. sin(z) =sin z, cos(z) =cos z13. d

dzsin z=cos z, d

dzcos z= sin z 8


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The following other trigometric functions of a complex variable

are defined in the same way as in the real case:

tan z=

sin z

cos z, cot z=

cos z

sin z =

1

tan z, csc z=

1

sin z, sec z=

1

cos z.

Based on our calculus knowledge, we can immediately

conclude that

d

dz tan z=sec2

z,d

dzcot z= csc2 z,

d

dz

csc z= csc zcot z,

d

dzsec z=sec ztan z.

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Complex Hyperbolic Functions

The real hyperbolic functions are defined as

sinh x= ex ex

2 ,

cosh x= ex + ex

2 .

This suggests that we define the complex hyperbolic functions

as

sinh z=ez ez

2 , (1)

cosh z=ez + ez

2 . (2)

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The other complex hyperbolic functions are defined in the same

manner as in the real case:

tanh z= sinh zcosh z

, coth z= cosh zsinh z

= 1tanh z

,

csch z= 1

sinh z, sech z=

1

cosh z.

Thus it can be shown that

(tanh z) =sech 2z,

(coth z) = csch 2z,

(csch z) = csch zcoth z,

(sech z) = sech ztanh z.

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Complex Logarithmic FunctionIn calculus,

x=

ey

y

=ln x

eln x

=x

.This suggests we define the complex logarithm function ln z

such that it satisfies

eln z =z.

DefinitionForz=0, define

ln z=ln |z|+ iarg z.

WHY?

ln |z| may be computed with the help of a calculator.

Since arg zis multiple valued, then so is the function ln z.

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In calculus, ln 1= 0. In complex variables, ln 1 hasinfinitely many values.

In calculus, ln(1)is undefined since ln xis valid only forx>0. In complex variables, ln(1)is meaningful.

In calculus, the formula

ln(xy) =ln x+ ln y

is valid with the restriction thatxandyare positive reals. If

wandzare complex, can we still have

ln(zw) =ln z+ ln w?

If it is so, what are the restrictions onzandw?

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The Laws of Complex Logarithm

The following relations hold for certain specified values of the

logarithms:

ln(zw) =ln z+ ln w

ln(z/w) =ln z ln w

ln zn/m = n

mln z

ln ez =z

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Principal Value Logarithm

The function ln zcan be made single valued by suitably

restricting the range of values for arg z.

Recall that the value of arg zcan be made unique by

restricting it to the interval < . This unique valueis called the principal argument of zand is denoted by

Arg z. This suggests defining theprincipal valueof ln z, denoted

byLn z, as the value of ln zthat employs the principal

argument ofz, i.e.,

Ln z=ln |z| + iArg z.

Threfore the functionLn zalways has a unique value.

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Continuity ofLn z= ln |z|+ iArg z Ln zis a combination of two functions ln |z| andArg z. The function ln |z| is continuous on the entire complex

plane except at the pointz=0. Arg zsatisfies


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