1

Chapter 2 Complex numbers ( 복소수 )

Mathematical methods in the physical sciences 3rd edition Mary L. Boas

Lecture 4 Introduction of complex numbers

고등수학 10- 가 2 장 실수와 복소수에 나옴

2

1. Introduction

.4168282) ,1

1number imaginary consider sLet'

22

iiiexi

i

ex.

.12

42

2

842

0222

iz

zz

negative. becan '4' sometimes, ,2

4

0

22

2

acba

acbbz

cbzaz

3

(READING)

Once the new kind of number is admitted into our number system, fascinating possibilities open up. Can we attach any meaning to marks like sin i, e^i, ln (1+i)? We’ll see later that we can and that, in fact, such expressions may turn up in problems in physics, chemistry, and engineering, as well as, mathematics.

When people first considered taking square roots of negative numbers, they felt very uneasy about the problem. They thought that such numbers could not have any meaning or any connection with reality (hence the term “imaginary”). They certainly would not have believed that the new numbers could be of any practical use. Yet complex numbers are of good importance in a variety of applied fields; for example, the electrical engineer would, to say the least, be severely handicapped without them. The complex notation often simplifies setting up and solving vibration problems in either dynamical or electrical systems, and is useful in solving many differential equations which arise from problems in various branches of physics.

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2. Real and imaginary parts of a complex number ( 복소수의 실수와 허수 부분 )

iyxz

x: real party: imaginary part (not imaginary!!)

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3. Complex plane ( 복소수 평면 )

- Complex plane: similar to the xy plane

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form).(polar sincossincos

.tan,sin,cos 22

ireirirriyxz

x

yyxrryrx

-Rectangular form (x,y) vs. Polar form (r,) ( 직교형태 VS 극좌표 형태 )

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Example)

cf. : radian

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4. Terminology and notation

: of angle

, valuemodulus)(or Absolute

Im :part imaginary

Re :part real

z

rz

yz

xz

ex) iz 1

principal angle

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.sincossincos

. conjugatecomplex *

ireirirz

iyxiyxzz

- Complex conjugate ( 켤레 복소수 )

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5. Complex algebra ( 복소수 연산 )

A. Simplifying to x+iy form

.212121111 ex1. 22 iiiiiii

squaring

i

i

i

iii

ii

ii

i

i

2

1

2

1

10

55

9

236

33

32

3

2 ex2.

2

2

20sin20cos

2

1

2

1

2

1

20sin20cos2

1 ex4. 20

20ie

eii

i

.2221 form)(polar ex3. 2/24/2 ieei ii

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B. Complex conjugate

)not ( ,2121 igfgifzigfzzzzz

C. Absolute value

.,,222 zzzzrerzzrezrez iiii

note) We can get the conjugate of an expression containing i’s by just

changing the signs of all the i terms.

.4

32

4

32

i

iz

i

iz

12

D. Complex equations

.122

022

,2

22

22

2

yxxy

yxiyixyx

iiyx

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E. Graphs

ex.1

9,3,

,3

22

yxiyxiyxz

z

ex2. 222 91,91 yxz

ex3.

xyx

y

z

1tan

4/: of Angle

ex.4 2

1Re xz

xy

2

1

Chapter 2 Complex numbers

Mathematical methods in the physical sciences 3rd edition Mary L. Boas

Lecture 5 Euler formula & roots and powers

6. Complex infinite series ( 복소수 무한 급수 )

.lim,lim where,lim YYXXiYXSS

iYXS

nn

nn

nn

nnn

In this case, we call the complex series convergent.

7. Complex power series; Disk of convergence ( 복소수 멱급수 ; 수렴 원판 )

numberscomplex :, nn

n aza

.11

lim e,convergenc absoluteFor

4321

432

z

n

nz

zzzz

n

ex.

cf. real vs. complex

2.8 Elementary function of complex numbers ( 복소수 기본함수 )

2121)

!3!21)

111211,12)32

22

zzzz

z

eeeiii

zzzeii

iiifzzzfi

- elementary functions: powers, roots, trigonometric, inverse trigonometric, logarithmic, exponential, and combinations of these.

- Elementary functions of complex numbers behave just like those of real numbers

2.9 Euler’s formula

sincos!5!3!4!2

1

!5!4!3!21

!5!4!3!21

!4!21cos,

!5!3sin

5342

5432

5432

4253

ii

iii

iiiiiei

fomula sEuler' sincos iei

ireiriyxz sincos

Ex. Find the graph expressing a given z.

iniii eeee 22/6/ ,3,,2

.

,

21

2121

2

121

212121

i

iii

er

rzz

errererzz

- Multiplication, division

ex. i

i

1

1 2

.2

2

2

2

2

1

1 4/3

4/

2/

4/

24/2

i

i

i

i

i

ee

e

e

e

i

i

10. Powers and roots of complex numbers ( 복소수의 멱수와 근 )

ninnininnnin

nni

nni

errezerrez

ni

nie

ninie

//1/1/1

/1/1

,

.sincossincos

sincossincos

ex.1 .10/sin10/cos 2/22510/25 ieeei iii

ex.2 Cube roots of 8?

.8888088 3

23/13/1232

ki

ikik eeei

2

3

2

122,2

.2

3

2

122,1

2,0

3

4

3

2

iezk

iezk

zk

i

i

ex.3 Find the plot all values of 4 64

.4

7,

4

5,

4

3,

44

2

4

2264 4/14/1

k

r

ex. 4 6 8i

).5,4,3,2,1,0(346

2/32

6

28 6/16/1

kk

r

Chapter 2 Complex numbers

Mathematical methods in the physical sciences 3rd edition Mary L. Boas

Lecture 6 application

11. The exponential and trigonometric functions ( 지수함수와 삼각함수 )

yiyeeeee xiyxiyxz sincos

.2

cos,2

sin Similarly,

.2

cos,2

sin

sincos,sincos

iziziziz

iiii

ii

eez

i

eez

ee

i

ee

ieie

.1 ex. 2222 eeeee ii

- exponential function

- trigonometric function

12. Hyperbolic functions ( 쌍곡함수 )

2cosh,

2sinh

zzzz eez

eez

- The other hyperbolic functions are named and defined in a similar way to

parallel the trigonometric functions:

.sincos.,sinhcosh

1cossin.,1sinhcosh

.coshcos,sinhsin

.sinh

1h csc,

cosh

1hsec

tanh

1coth,

cosh

sinhtanh

2222

zzdz

dcfzz

dz

d

zzcfzz

yiyyiiy

zz

zz

zz

z

zz

13. Logarithms ( 로그함수 )

.lnln irLnrezw i

- Since has an infinite values (all differing by multiples of 2), a complex

number has infinitely many logarithms. (principal value)

ex. .,3,,211ln iiiniLn

14. Complex roots and powers ( 복소수 근과 멱수 )

abb ea ln

ex. 1 Find all values of i^(-2i)

- For complex a and b,

- Since ln a is multiple values, powers a^b are usually multiple values (cf. principal value).

.,,,

)22/()22/(1lnln9422/2ln22

eeeeeei

niniinniiiii

15. Inverse trigonometric and hyperbolic functions ( 역삼각함수와 역쌍곡함수 )

wzee

zwiziz

arccos2

cos

16. Some applications ( 응용 )

- Electricity

.,,C

I

dt

dV

C

QV

dt

dILVIRV LR

CLR

CLR

VVVV

tIC

VtLIVtRIV

tII

voltageTotal

.cos1

,cos,sin

sin

000

0

(method 1)

‘complicated function’

(method 2)

.11

,

,

solution. theofpart imaginary thecan take we,complex a with describingAfter

.ImImsin

0

0

0

00

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eICi

V

LIieLIiV

RIeRIV

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eIItI

tiC

tiL

tiR

ti

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Impedence

1

CLiRZ

ZIIC

LiRVVVV CLR

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:Resonance cf. C

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