1

February 5 Complex numbers

2.1 Introduction2.2 Real and imaginary parts of a complex number2.3 The complex plane2.4 Terminology and notationSolution to a quadratic equation:

Chapter 2 Complex Numbers

numbers.complex in equation lfundamentamost theis 1

.1 Especially

2

4 :Answer ?04 if happensWhat

.2

40

2

2,12

2

2,12

2

2,12

i

izz

a

bacibzacb

a

acbbzcbzaz

Example p46.

A complex number has a real part and an imaginary part. 2.)2Im(3 ,3)23Re( ii

2

Representation of a complex number on the complex plane:A complex number x + iy can be specified or represented by the following equivalent methods on the complex plane:1)The original rectangular form x + iy.2)A point with the coordinates (x, y).3)A vector that starts from the origin and ends at the point (x, y).4)The polar form rei that satisfies

).sin(cos irreiyx i

Example p48.

Modulus (magnitude, absolute value) of a complex number:Angle (argument, phase) of a complex number:

)quadrants. 3rdor 2nd in the is if ( arctan)arg(

)sin(cos

22

zx

yz

ryxz

irreiyxz i

Example p50.

3

Complex conjugate

)sin(cos)sin()cos(

)sin(cos*

irirreiyxz

irreiyxzi

i

The complex conjugate pair z=x + iy and z*=x − iy are symmetric with respect to the x-axis on the complex plane.

Problems 4.3,9,18.

4

Read: Chapter 2: 1-4Homework:2.4.1,3,5,15,18.Due: February 14

5

February 7, 10 Complex algebra

2.1 Complex algebraA. Simplifying to x + iy form

22

22

21122121

2222

2211

22

11

2

1

12212121221121

2121221121

2121221121

222111

)(

))((

))((

,

. ,

yx

yxyxiyyxx

iyxiyx

iyxiyx

iyx

iyx

z

z

iyxiyxzicycxiyxccz

yxyxiyyxxiyxiyxzz

yyixxiyxiyxzz

yyixxiyxiyxzz

iyxziyxz

Examples p51.1-4; Problems 5.3,7.

B. Complex conjugate of a complex expression

The complex conjugate of any expression is just to change all i’s into –i.

*2

*1

*

2

1*2

*1

*21

*2

*1

*21

*2

*1

*21 ,)( ,)( ,)(

z

z

z

zzzzzzzzzzzzz

Example p53.1

6

C. Finding the absolute value of z

zzz

z

z

z

zzzz

zzyxrz

irreiyxz i

11 ,

)sin(cos

2

1

2

1

2121

*22

Example p53.2; Problems 5.26,28.

D. Complex equation

. and ifonly and if

. ,

212121

222111

yyxxzz

iyxziyxz

Example p54; Problems 5.36,43.

E. Graphs

Complex equations or inequalities have geometrical meanings.

Example p55.1-4; Problems 5.52,53,59.

. of radiusa with),(at centered circlea is |)(| ,particular In rbaribaz

7

F. Physical applications

The position of a moving particle is represented by a vector. This vector also represents a complex number. Addition and subtraction of complex numbers is analogous to the addition and subtraction of vectors. Therefore the position, speed and acceleration of a particle can be well represented by complex numbers.However, because the multiplication of complex numbers is not in analogy with the multiplication of the vectors, physical principles involving multiplication of vectors can not be represented by complex algebra. Example: W=F·s.

Example p56.

8

Read: Chapter 2: 5Homework:2.5.2,7,23,26,33,36,47,59,68.Due: February 21

9

February 14 Complex infinite series

2.6 Complex infinite series Convergence of a complex infinite series:

. and ifonly and if toconverge tosaid is )(111

bn

nan

nban

nn SbSaiSSiba

Theorem: An absolutely convergent complex series converges.

converges. )( that means This

.convergent absolutely are they since converge, and both Then

test.comparison the toaccording converge and so

, , Because converges.

:Proof

1

11

11

2222

1

22

1

nnn

nn

nn

nn

nn

nnnnnnn

nnn

nn

iba

ba

ba

babbaabaiba

10

Example p57.1-2; Problems 6.2,6,7.

Theorem: Ration test for a complex series an:

diverge.must .0lim,1 , allfor Then .1 that so take,1 If 2)

converges, thusand converge,must converges, Since

. then,for Form

. , allfor Then .1 that so take,1 If 1)

:Proof

needed. isst further te 1, If 3)

diverges. ,1 If 2)

converges, ,1 If )1

then,lim If

1

1

111

1

1

1

1

nnn

nn

n

nn

nn

nn

nnNNn

n

n

n

nn

nn

n

n

n

aaa

aNn

aab

baNnab

a

aNn

a

a

a

a

11

2.7 Complex power series; disk of convergenceDisk of convergence: An area on which the series is convergent.Radius of convergence: The radius of the disk of convergence.

Example p58.7.2a-c; Problems 7.5,8,15.

Disk of convergence for the quotient of two power series:

).,,min( inleast withat coverges )(

)( seriesquotient theThen

.0)( tomagnitude) (in solutionsmallest theis Suppose

. withinconverges )( and , withinconverges )( Suppose

1210

1

20

10

zrrrzczg

zf

zgz

rzzgzbrzzfza

n

nn

n

nn

n

nn

Example p59.

12

Read: Chapter 2: 6-7Homework:2.6.3,5,6,13;2.7.8,11,15.Due: February 21

13

February 17 Elementary functions

2.8 Elementary functions of complex numbers Elementary functions: powers and roots; trigonometric and inverse trigonometric functions; logarithmic and exponential functions; and the combination of them.Functions of a complex variable can be defined using their corresponding infinite series.

Examples p.62.

!4!3!21

!

!7!5!3)!12(

)1(sin

:Examples

43

0

2

75

0

312

zzzz

n

ze

zzzz

n

zz

n

nz

n

nn

2.9 Euler’s formula

i

i

reiriyxz

ii

iiiiie

)sin(cos

sincos!5!3!4!2

1

!5

)(

!4

)(

!3

)(

!2

)(1

5342

5432

14

Multiplication and division of complex numbers using Euler’s formula:

.

.

)(

2

1

2

1

2

1

)(212121

21

2

1

2121

ii

i

iii

er

r

er

er

z

z

errererzz

Example p.63; Problems 9.13,22,38.

2.10 Powers and roots of complex numbers

Power of a complex number: ).sin(cos ninrrez nnin

Example p.64.1.

Roots of a complex number:

Fundamental theorem of algebra:

roots. has )0,0( )( polynomialAny 0

nanzazP n

n

m

mm

.1,,2,1,0 ,2

sin2

cos

.1,,2,1,0 ,

/12

/1/1)2(/1

)2(

nkn

k

ni

n

k

nrerrez

nkrerez

nn

k

ni

nnkin

kii

15

Notes about roots of a complex number: 1)There are altogether n values for

2)The first root is

3)All roots are evenly distributed on the circle with a radius of . Each root has a incremental phase change of

.n z

./1 ni

ner

nr /1

./2 n

Example p.65.2-4; Problem 10.18.

16

Read: Chapter 2: 8-10Homework:2.9.7,13,23,38;2.10.2,18,21,27.Due: February 28

17

February 19 Exponential and trigonometric functions

2.11 The exponential and trigonometric functions

i

eez

i

ee

eez

ee

ie

ie

yiyeeeee

irreiyxz

izizii

izizii

i

i

xiyxiyxz

i

2sin

2sin

2cos

2cos

sincos

sincos

)sin(cos

)sin(cos

Examples p.68.1-4; Problems 11.6,9.

Notes on trigonometric functions:1)sinz and cosz are generally complex numbers. They can be more than 1 even if they are real.2)The trigonometric identities (such as ) and the derivative rules (such as ) still hold.

1cossin 22 zzzz cos)'(sin

18

2.12 Hyperbolic functions

yiy

yiiyz

zz

eez

eez

zz

zz

coshcos

sinhsin

etc. ,cosh

sinhtanh

2cosh

2sinh

Example p.70; Problems 12.1,9,15.

19

Read: Chapter 2: 11-12Homework:2.11.6,8,10,11;2.12.1,3,11,32,36.Due: February 28

20

February 21 Complex roots and powers

2.13 Logarithms

,2,1,0 ),2(LnlnLn]ln[)ln(ln

)sin(cos)2()2(

nnirerrerez

irreiyxzninii

i

Notes on logarithms:1)We use Lnr to represent the ordinary real logarithm of r.2)Because a complex number can have an infinite number of phases, it will have an infinite number of logarithms, differing by multiples of 2i.3)The logarithm with the imaginary part 0<2 is called the principle value.

Examples p.72. 1-2; Problems 14.3,6,7.

21

2.14 Complex roots and powers

,2,1,0 ,)2(Lnexp)2(Lnexp

)2(Ln)(exp

ln)(exp )2(ln

nnbrbinbrb

niribb

eribbea

ibbb

era

axayayax

aayx

niayx

abb

yx

ia

a

a

Examples p.73. 1-3; Problems 14.8,12,14.

Notes on roots and powers:1)There are many amplitudes as well as many phases for ab.2)For the amplitude in most cases only the principle value (0a<2 and n=0) is needed.3)by=0, bx=m or 1/m gives us the real powers and real roots of a complex number.

22

2.15 Inverse trigonometric and hyperbolic functions

.2

cos if ,arccosiziz ee

zwwz

Example p.75. 1; Problems 15.3.

23

Read: Chapter 2: 13-15Homework:2.14.3,4,8,14,17;2.15.1,3,15.Due: February 28