1. introduction to complex numbers

ENGINEERING MATHAMATICS 1Dr. Ir. Arman Djohan Diponegoro

5th August 2013

ENGINEERING MATHEMATICS 1

Class : Engineering Mathematics 1 – 02

Code : ENEE 600006

Credits : 3 SKS

Lecturer : Dr. Ir. Arman Djohan Diponegoro, M.Sc

Assistant : Adhitya Satria Pratama

Schedule : Thursday, 13.00-15.30

Venue : GK. 301

References :

1. Kreyszig, Erwin. 2011. Advanced Engineering Mathematics, 10th edition. John

Wiley and Sons Inc.Other relevant and related references are welcomed

GENERAL TOPICS ON ENGINEERING MATHEMATICS 1

Complex Variables

Vector Analysis

GRADING

Exercises and Quizzes : 80 %

Mid Test : 10 %

Final Test : 10 %

SCHEDULE

SUN MON TUE WED THU FRI SAT

1 2 3 4 5 6 7

8 9 10 11 12 13 14

15 16 17 18 19 20 21

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17 18 19 20 21 22 23

24 25 26 27 28 29 30


1 2 3 4 5 6 7

8 9 10 11 12 13 14

15 16 17 18 19 20 21

22 23 24 25 26 27 28

29 30 31

September 2013 October 2013

November 2013 December 2013

MIDTERM

TEST

FINAL TEST

COMPLEX NUMBERS

COMPLEX NUMBERS

In the early days of modern mathematics, people were puzzled by equations like

this one:

The equation looks simple enough, but in the sixteenth century people had no idea

how to solve it. This is because to the common-sense mind the solution seems to

be without meaning:

For this reason, mathematicians dubbed an imaginary number. We

abbreviate this by writing “i” in its place, that is:

1

COMPLEX NUMBERS

DEFINITIONA complex number z is a number of the form

where

x is the real part and y the imaginary part, written as x = Re z, y = Im z.

i is called the imaginary unit

If x = 0, then z = iy is a pure imaginary number.

The complex conjugate of a complex number, z = x + iy, denoted by z* , is given byz* = x – iy.

Two complex numbers are equal if and only if their real parts are equal and theirimaginary parts are equal.

1i

iyx

DEFINITION

A complex number z is a number of the form

where

x is the real part and y the imaginary part, written as x = Re z, y = Im z.

i is called the imaginary unit

If x = 0, then z = iy is a pure imaginary number.

The complex conjugate of a complex number, z = x + iy, denoted by z* , is given byz* = x – iy.

Two complex numbers are equal if and only if their real parts are equal and theirimaginary parts are equal.

1iiyx

COMPLEX PLANE

A complex number can be plotted on a plane with two perpendicular coordinate

axes

The horizontal x-axis, called the real axis

The vertical y-axis, called the imaginary axis

P

z = x + iy

x

y

O

Represent z = x + jy geometrically

as the point P(x,y) in the x-y plane,

or as the vector from the

origin to P(x,y).

OP

The complex plane

x-y plane is also known as

the complex plane.

POLAR COORDINATES

siny r With cos ,x r

z takes the polar form:

r is called the absolute value or modulus or

magnitude of z and is denoted by |z|.

*22 zzyxrz

22

* ))((

yx

jyxjyxzz

Note that :

)sin(cos jrz

TRIGONOMETRIC FORM FOR COMPLEX NUMBERS

We modify the familiar coordinate system by calling the horizontal axis the real

axis and the vertical axis the imaginary axis.

Each complex number a + bi determines a unique position vector with initial point

(0, 0) and terminal point (a, b).

RELATIONSHIPS AMONG X, Y, R, AND

x r cos

y r sin

r x2 y2

tan y

x, if x 0

TRIGONOMETRIC (POLAR) FORM OF A COMPLEX NUMBER

The expression

is called the trigonometric form or (polar form) of the complex number x + yi.

The expression cos + i sin is sometimes abbreviated cis .

Using this notation

(cos sin )r i

(cos sin ) is written cis .r i r

COMPLEX PLANE

Complex plane, polar form of a complex number

x

y1tan

Pz = x + iy

x

y

O

Im

Re

|z| =

r

θ

Geometrically, |z| is the distance of the

point z from the origin while θ is the

directed angle from the positive x-axis to

OP in the above figure.

From the figure,

COMPLEX NUMBERS

θ is called the argument of z and is denoted by arg z. Thus,

For z = 0, θ is undefined.

A complex number z ≠ 0 has infinitely many possible arguments, each one

differing from the rest by some multiple of 2π. In fact, arg z is actually

The value of θ that lies in the interval (-π, π] is called the principle

argument of z (≠ 0) and is denoted by Arg z.

0tanarg 1

z

x

yz

,...2,1,0,2tan 1

nn

x

y

EULER FORMULA – AN ALTERNATE POLAR FORMThe polar form of a complex number can be rewritten as :

This leads to the complex exponential function :

jre

jyxjrz

)sin(cos

jj

jj

eej

ee

2

1sin

2

1cos

Further leads to :

yjye

eeee

x

jyxjyxz

sincos

EULER FORMULA

Remember the well-known Taylor Expansions :

EULER FORMULA

So, we can conlude that :

GRAPHIC REPRESENTATION

A complex number, z = 1 + j , has a magnitude

EXAMPLE

424

sin4

cos2

j

ejz

2)11(|| 22 z

rad24

21

1tan 1

nnzand argument :

Hence its principal argument is : Arg / 4z rad

Hence in polar form :

A complex number, z = 1 - j , has a magnitude

2)11(|| 22 z

rad24

21

1tan 1

nnzand argument :

Hence its principal argument is : rad

Hence in polar form :

In what way does the polar form help in manipulating complex numbers?

4

zArg

4sin

4cos22 4

jezj

EXAMPLE

What about z1=0+j, z2=0-j, z3=2+j0, z4=-2?

5.01

1

10

5.0

1

je

jz

5.01

1

10

5.0

2

je

jz

02

2

02

0

3

je

jz

2

2

024

je

jz

EXAMPLE

●

●

●

Im

Re

z1 = + j

z2 = - j

z3 = 2z4 = -2●

5.0

EXAMPLE (CONTINUED)

EXAMPLE

Express 2(cos 120 + i sin 120) in rectangular form.

Notice that the real part is negative and the imaginary part is positive,this is consistent with 120 degrees being a quadrant II angle.

1cos120

2

3sin120

2

1 32(cos120 sin120 ) 2 ,

2 2

1 3

i i

i

CONVERTING FROM RECTANGULAR FORM TO TRIGONOMETRIC FORM

Step 1 Sketch a graph of the number x + yi in the complex plane.

Step 2 Find r by using the equation

Step 3 Find by using the equation

choosing the quadrant indicated in Step 1.

2 2 .r x y

tan , 0y

xx

ADDITION AND SUBTRACTION OF COMPLEX NUMBERS

For complex numbers a + bi and c + di,

Examples

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

a bi c di a c b d i

a bi c di a c b d i

(10 4i) (5 2i)

= (10 5) + [4 (2)]i

= 5 2i

(4 6i) + (3 + 7i)

= [4 + (3)] + [6 + 7]i

= 1 + i

MULTIPLICATION OF COMPLEX NUMBERS

For complex numbers a + bi and c + di,

The product of two complex numbers is found by multiplying as if the numbers

were binomials and using the fact that i2 = 1.

(a bi)(c di) (ac bd) (ad bc)i.

EXAMPLES: MULTIPLYING

(2 4i)(3 + 5i) (7 + 3i)2

2

2(3) 2(5 ) 4 (3) 4 (5 )

6 10 12 20

6 2 20( 1)

26 2

i i i i

i i i

i

i

2 2

2

7 2(7)(3 ) (3 )

49 42 9

49 42 9( 1)

40 42

i i

i i

i

i

ARITHMETIC OPERATIONS IN POLAR FORM

The representation of z by its real and imaginary parts is useful

for addition and subtraction.

For multiplication and division, representation by the polar form

has apparent geometric meaning.

Suppose we have 2 complex numbers, z1 and z2 given by :

2

1

2222

1111

j

j

erjyxz

erjyxz

2121

221121

yyjxx

jyxjyxzz

))((

21

2121

21

21

j

jj

err

ererzz

Easier with normal

form than polar form

Easier with polar form

than normal form

magnitudes multiply! phases add!

For a complex number z2 ≠ 0,

)(

2

1))((

2

1

2

1

2

1 2121

2

1

jj

j

j

er

re

r

r

er

er

z

z

magnitudes divide!phases subtract!

2

1

2

1

r

r

z

z

2121 )( z

EXERCISES

Let z = x + iy and w = u + iv be two complex variables. Prove that :

Prove that :

COMPLEX ANALYSIS

In the early days, all of this probably seemed like a neat little trick that could be

used to solve obscure equations, and not much more than that.

It turns out that an entire branch of analysis called complex analysis can be

constructed, which really supersedes real analysis.

For example, we can use complex numbers to describe the behavior of the

electromagnetic field.

Complex numbers are often hidden. For example, as we’ll see later, the

trigonometric functions can be written down in surprising ways like:

AXIOMS SATISFIED BY THE COMPLEX NUMBERS SYSTEM

These axioms should be

familiar since their general

statement is similar to that

used for the reals.

We suppose that u, w, z are

three complex numbers,

that is, u, w, z ∈ C, then

these axioms follow:

AXIOMS SATISFIED BY THE COMPLEX NUMBERS SYSTEM

DE MOIVRE’S THEOREM


De Moivre’s theorem is about the powers of complex numbers and a relationship

that exists to make simplifying a complex number, raised to a power, easier.

The resulting relationship is very useful for proving the trigonometric identities

and finding roots of a complex number.

40


If is a complex number, and if n is any real number,

then

In compact form, this is written

1 1 1cos sinr i

1 1cos sin cos sin .n nr i r n i n

cis cis .n nr r n

41

EXAMPLE: FIND (1 I)5 AND EXPRESS THE RESULT IN RECTANGULAR FORM.

First, find trigonometric notation for 1 i

Theorem

1 2 cos225 sin 225i i

55

5

2 5 225 5 2

1 2 cos225 sin 225

cos( ) sin( )

4 2 cos1125 sin1125

2 24 2

2 2

25

4 4

i i

i

i

i

i

42

NTH ROOTS

For a positive integer n, the complex number a + bi is an nth root of the complex

number x + yi if

.n

a bi x yi

43

NTH ROOT THEOREM

If n is any positive integer, r is a positive real number, and is in degrees, then the

nonzero complex number r(cos + i sin ) has exactly n distinct nth roots, given

by

where

cos sin or cis ,n nr i r

360 360 or = , 0,1,2,..., 1.

k kk n

n n n

44

EXAMPLE: SQUARE ROOTS

Find the square roots of

Trigonometric notation:

For k = 0, root is

For k = 1, root is

1 3 i

1 3 i 2 cos60 isin60

2 cos60 isin60

1

2 21

2 cos60

2 k

360

2

isin

60

2 k

360

2

2 cos 30 k 180 isin 30 k 180

2 cos30 isin30

2 cos210 isin210

45

EXAMPLE: FOURTH ROOT

Find all fourth roots of Write the roots in rectangular form.

Write in trigonometric form.

Here r = 16 and = 120. The fourth roots of this number have absolute value

8 8 3.i

8 8 3 16 cis 120i

4 16 2.

120 36030 90

4 4k

k

46

EXAMPLE: FOURTH ROOT CONTINUED

There are four fourth roots, let k = 0, 1, 2 and 3.

Using these angles, the fourth roots are

0 30 90 30

1 30 90 120

2 30 90 210

3 30 90

0

1

2

3 300

k

k

k

k

2 cis 30 , 2 cis 120 , 2 cis 210 , 2 cis 300

EXAMPLE: FOURTH ROOT CONTINUED

Written in rectangular form

The graphs of the roots are all on a

circle that has center at the origin and

radius 2.

Slide 8-47

3

1 3

3

1 3

i

i

i

i

HOMEWORK

1.

2.

3. Find the fourth roots of 2.

Please do the homework on a paper. This exercise should be submitted on Thursday,

August 12th 2013 before the class begins.

NEXT AGENDA

Limit, Functions, and Continuity of Complex Variables.