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COMPLEX NUMBERS

BRIEF REVIEW: Cantor, Dedekind and Weierstrass etc, extended the concept of rational

numbers to a larger field known as real numbers which constitute rational as well as irrational

numbers. It is evident that the system of real numbers is not sufficient for all mathematical needs

e.g.

The solution of a polynomial in equation (1) below is but for equation (2), the values ofx are:

The value of is neither -2 nor +2. We, in fact cannot represent by an ordinary

number because there is no real number whose square is a negative quantity i.e. no real number

(rational or irrational) satisfies equation (2). It was therefore felt necessary by Euler Gauss,Hamilton, Cauchy, Reiman and Weierstrass etc to extend the field of real numbers to the still

larger field of complex numbers. Euler for the first time introduced the symbol j with the

property and then Gauss introduced a number of the form in equation (3) below

which satisfies every algebraic equation with real coefficients.

Such a number with and , being real is known as a complex number. is called the

real part and the imaginary part of the complex number, z and denoted by:

Modulus of a Complex Number:

If equation (4) is a complex number, then its modulus (or module) is denoted by and given

by:

Conjugate Complex Numbers:

For every complex number (4), there exists its conjugate denoted by (5):

)1.......(..........042 x

2x

)2.......(..........042 x

4x

4 4

12 j

)3.......(.......... jz

1j

(z)ror R(z) oRz Re(z)ror I(z) oI

zIm

)4.......(..........jyxz

||z

22|||| yxjyxz

)5.......(..........jyxz

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The Representation of Complex Numbers

Recall that:

There are 4 principal formats for representing complex numbers. These include:

Graphical representation Polar form Vector interpretation Spherical representation

Graphical Representation:

Equation (4) and (5) can be represented graphically. This form allows us to represent every

complex number by just one point in a plane. Also, in the reverse sense, every point in the plane

may be associated with just one complex number. Assume x = 6 and y = 2, (4) and (5) are

graphically represented thus:

Such representation as described above is called the Argand diagram.

jjjj

j

jj

jj

i

jj

jjj.j)(jjjj

.

11

11

1

1.;;1

23

3

2

2

2

1

224232

P(z )

y

6

j

-j

z

z

-y

-x x

-2

2

P(z)

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Polar Form of Complex Numbers:

From the above figure, we can transform the coordinate(s) to the polar form thus:

Note the following:

In Pictorial form:

)x

y(zandrz

rerejSinCosr

jrSinrCoszSojyxzbut

rCosxrSiny

r

xCos

r

ySin

kjj

arctanarg;||

)(

,

;

;

2

1

1

2

2

3

2

j

j

e

je

e

je

j

j

P(r,-)

y

x

j

-i

z

z

-y

-x x

-2

y

P(r,)

dians and is in ra

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ELEMENTARY FUNCTIONS OF A COMPLEX VARIABLE

These functions include:

Polynomial functions Rational functions Exponential functions Trigonometric functions Hyperbolic functions Logarithmic functions Inverse Trigonometric functions Inverse Hyperbolic functions

POLYNOMIAL FUNCTIONS

A polynomial function, P(z) can be defined as:

Where a0 is not zero, a1.an are complex constants. N is a positive integer

called the degree of the polynomial P(z) and Z is a complex variable.

RATIONAL FUNCTIONS

Are defined by:

Where P(z) and R(z) are polynomials.

EXPONENTIAL FUNCTIONS

The exponential function of a real (not complex) function is simply written as . So, for a

complex variable, z given as :

If y is used as a radian measure of the angle to define Cos y, Sin y etc, then the exponentialfunction in terms of real valued functions is defined by:

In case z is purely real i.e. y = 0, we have ez = ex and if z is purely imaginary i.e. x = 0, we have:

Recall that e is the base of natural logarithms (e = 2.718281828). Complex exponential functions

have properties similar to those of real exponential functions.

)6.......(...................)(1

2

2

1

10 nn

nnn azazazazazP

)7(..............................)(

)()(

zR

zPzF

xe

;jyxz

)8........().........sin(. yjCosyeeeee xjyxjyxz

)9(..............................sinyjCosyejy

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In polar form,

Illustrations:

1) Show that 2121. zzzz eee

RHSLHSHence

eRHS

e

ee

eeee

ee

yyjyye

yyyyjyyjyye

yjyyjyeeRHS

yjyeyjyeee

RHSLHS

zz

jyxjyx

jyxjyx

jyjyxx

yyjxx

xx

xx

xx

xxzz

,

.

...

.

)]sin().[cos(

]sin.sincos.sinsin.coscos..[cos

)]sin).(cossin[(cos.

)sin(cos).sin(cos.

21

2211

2211

2121

121

21

21

21

2121

)()(

)(

2121

21212121

2211

2211

)210(;20

)2sin()2cos(1,

0

exp

00|

arg|

;

)sin(cos

:

)sin(cos.

,,nnand yx

njneNow

where rthe originexcluding

plae complexthe entirunction isonential fnge of theSo, the ra

e of zevery valufori.e. eAlso, |e

yeander|e

yerwhere

rejre

tten ascan be wri

yjyeeeee

z

zz

zxz

x

jz

xjyxjyxz

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2) Show that 2121/ zzzz eee

3) Find all values of z such that 1z

e Solution:

.....),,(nn)(z

o orx

eger.n is an; wheren; yx

e; eee

orm; i.e.onential fnumbers inextwo complquality ofciple of eg the prinate uWe can equ

.ee.ee

RHSLHS

LHS.eeeSo, e

radianinangle iswhere the

RHSjecall that

jjyx

jjyx

jyxjyxz

j

210210

01log

int20

expsin

1sincosRe

0

0

RHSLHSHence

eLHS

e

e

e

eeee

e

eLHS

RHSLHS

zz

jyxjyx

jyx

jyx

jyx

jyx

yjyx

yjyx

,

.

.

21

2211

22

11

22

11

222

111

)()(

)sin.(cos

)sin.(cos

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4) Find the values of z for which 14 ze Solution:

n

z

....),,nn

; yx

y;x

e; eee

jecall that

eeeeee

jyjx

j

jyjxjyxz

22

210(22

0

2404

12sin2cosRe

..

2404

2

2044)(44

5) Find the values of z for which 13 ze Solution:

n

z

....),,nn

; yx

y;x

e; eee

eeeeeejyjx

jyjxjyxz

23

2

210(23

20

2303

..2303

2033)(33

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TRIGONOMETRIC FUNCTIONS

Recall that trigonometric functions of real variable x include sinx, cosx, tanx etc. For complex

variables, trigonometric functions are defined in terms of exponential functions as follows:

jzjz

jzjz

jzjz

jzjz

jzjz

jzjz

jzjzjzjz

jzjzjzjz

jyjy

jyjy

jy

jy

ee

eej

zz

eej

ee

eej

ee

z

zz

ee

j

zz

eezz

j

ee

z

ee

z

e manner:In the sam

j

eey

*** fromSubtract

eey

*** &Adding

**...............y.........jyand e

*...........y.........jycall:e

)(

tan

1cot

)(

2

2cos

sintan

2

sin

1csc;

2

cos

1sec

2sin;2cos

2sin

:11

2cos

:11

1sincos

1sincosRe

Many of the properties satisfied by real trigonometric functions are also satisfied by complex

trigonometric functions: e.g.

21

21

21

212121

212121

22

2222

tantan1

tantan)tan(

sinsincoscos)cos(

sincoscossin)sin(

tantancos)cos(

sinsin;csccot1

sectan1;1cossin

zz

zzzz

zzzzzz

zzzzzz

z-(-z)z;z

z-(-z)zz

zzzz

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Illustration:

6) Prove that 1cossin 22 zz Solution: Recall that:

14

4

4

224

2

4

2

4

2.

4

...

22

1cossin,

2sin;

2cos

2222

2222

22

22

22

zjzjzjzj

zjzjzjzj

zjjzjzzjjzjzjzjzjzjz

jzjzjzjz

jzjzjzjz

eeee

eeee

eeeeeeeeee

ee

j

eeLHS

RHSLHS

zzSo

j

eez

eez

HYPERBOLIC FUNCTIONS

Are defined as follows:

zz

zz

zz

zzzz

zz

zzzz

ee

ee

z

zz

eezechz

eezhzee

ee

z

eez

eez

sinh

coshcoth;

2

sinh

1cos

2

cosh

1

sec;tanh

2cosh;

2sinh

The following properties hold:

21

21

21

212121

212121

22

2222

tanhtanh1

tanhtanh)tanh(

sinhsinhcoshcosh)cosh(

sinhcoshcoshsinh)sinh(

tanhtanhcosh)cosh(

sinhsinh;csc1coth

sectanh1;1sinhcosh

zz

zzzz

zzzzzz

zzzzzz

z-(-z)z;z

z-(-z)zhz

zhzzz

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These properties can easily be proved from the definitions.

Illustration:

7) Show that 1sinhcosh 22 zz

14

4

4

22

4

2

4

2

4

2

4

..

22sinhcosh,

2sinh;

2cosh

2222

2222

2222

22

22

zzzz

zzzz

zzzzzzzz

zzzz

zzzz

eeee

eeee

eeeeeeee

eeeezzSo

eez

eez

Exercise: The proofs of others are left as exercise

Try the following:

14

j)(1TanhthatShow(3.)

cos3z(b)

cos2z(a)

find2,zcosIf2.)

jtanztanhjzcosz;coshjzjsinz;sinhjz

jtanhztanjzcoshz;cosjzjsinhz;sinjz

csc1cothz;2sechz2tanh11.) 22

zhz

LOGARITHMIC FUNCTIONS

zarithm oflthe naturaz, calledwrite w, then wee If zw logln

Thus the natural logarithmic function is the reverse of the exponential function and can be defined as

follows:

)2(

...2,1,0),2(lnln

kjj

rerezwhere

kkjrzw

Note that z is a multiple valued (in this case, infinitely many valued) function with the principal

value or branch. The principal value of

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quivalent. or its ewherejred asimes definz is somet 20lnln

However, any other interval of length like etc can be used.

The logarithmic function for real bases other than e can be defined. Thus ifw

az , then

wz

alog . Where 0a and 1,0a . In this case, awez ln and so

a

zw

ln

ln

Illustration:

8) Evaluate (i) )4ln( ; (ii) )3ln( j Solutions:

,....)2,1,0(:

)2(4ln

)2(ln)4ln(

0tan4

0tan)arg(

4)0()4(||

04..

)4ln()(

11

22

kNote

kj

kjr

z

zr

jzei

wLeti

,....)2,1,0(:

)26

11(2ln

)26

11()2ln()3ln(

6

11

63

1tan)arg(

2)1()3(||

)1(33)(

1

22

kNote

kj

kjj

orz

zr

jjzii

Assessment Exercise

j)-((c)j(b)j-a 23ln;2

3

2

1ln;

2

3

2

1ln)(

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INVERSE TRIGONOMETRIC FUNCTIONS

To define the inverse of sine function i.e. arcsine of z, z1

sin

, we write:

wzThen

zw

sin

sin 1

Also, we define other inverse trigonometric or circular functions z1cos

, z

1tan etc.These functions are multiple-valued and can be expressed in terms of natural logarithms as

follows:

)1ln(sin,

)1ln(1

)1ln(

12

122

2

442

1*2

)1(*1*42201)(2

0)(2.2

Re

2

)1ln(1

sin)1(

21

22

2

22

2

2

2

21

zjzjzHence

zjzj

wzjzjw

zjzezjz

ezjz

e

jzjzeejze

ejzeeeeejz

us:written th

j

eezwhere

zjzj

z

jwjwjw

jwjwjw

jwjwjwjwjwjw

jwjw

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)1ln(cos,

)1ln(1

)1ln(

12

122

2

442

1*2

)1(*1*42201)(2

0)(2.2

2

)1ln(1

cos)2(

21

22

2

22

2

2

2

21

zzjzHence

zzj

wzzjw

zzezz

ezz

e

zzeeze

ezeeeeez

eezwhere

wzzj

z

jwjwjw

jwjwjw

jwjwjwjwjwjw

jwjw

jz

jz

jzwHence

jz

jz

jw

jz

jzjw

jz

jzejzjze

ejzejzejzejze

eejzejze

eeejzejz

like termscollectandExpand

eeeejzeeeejz

eej

eezBut

jz

jz

jzthatshowwzif

jwjw

jwjwjwjwjw

jwjwjwjw

jwjwjwjw

jwjwjwjwjwjwjwjw

jwjw

jwjw

1

1ln

2

1tan,

1

1ln

2

1

1

1ln2

1

1011.

0.11.011

0.

..

:

).()(

)(

)1

1ln(

2

1tan,tan)3(

1

22

2

11

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OTHERS:

jz

jz

jztw,show thaz) if(

z

z

jztw,show thaz) if(

z

zj

jztw,show thaz) if(

ln2

1cotcot6

11ln

1secsec5

1ln

1csccsc4

11

2

11

2

11

Note that they are all multiple-valued functions.

Illustration:

9) Find the values of 2sin 1 Solution:

kjHence

jkj

ej

jj

jjjjjSo

zjzjzjzj

zcall

jk

22

)32ln(2sin,

22)32ln(2sin

).32(ln2sin

)32ln(2sin

)32ln()212ln(2sin,

)1ln()1ln(1sinRe

1

1

221

1

21

221

Assessment Exercise

2csc;2)( 1(b)ca-1

os

INVERSE HYPERBOLIC FUNCTIONS

If e of zolicrse hyperbd the invez is callew then wz- sinsinhsinh 1

Other inverse hyperbolic functions are similarly defined:

z

zz;zzz;zzz

1

1ln

2

1tanh1lncosh1lnsinh 12121

In each case, the constant k2 has been omitted. They are all multiple valued functions.

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jkj

ejjj

jjjjj

jjjjj

jk

2221lncosh

.21lncosh.21lncosh

.21lncosh2lncosh

11lncosh1lncosh

1

2211

11

121

OTHERS:

1

1

ln2

1

coth6

11lnsec5

11lncsc4

1

2

1

2

1

z

z

z)(

z

zzh)(

z

zzh)(

Assessment Exercise

)]([j;(b)(a)- 1lnsinhsinh 11