1. introduction to complex numbers
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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
22 23 24 25 26 27 28
29 30
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 22 23 24 25 26
27 28 29 30 31
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 22 23
24 25 26 27 28 29 30
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
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
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.
Slide 8-40
DE MOIVRE’S THEOREM
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
Slide 8-41
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
Slide 8-42
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
Slide 8-43
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
Slide 8-44
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
Slide 8-45
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
Slide 8-46
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.