This document provides information about the Engineering Mathematics 1 course taught by Dr. Ir. Arman Djohan Diponegoro. It includes details about the class code, credits, schedule, topics to be covered including complex variables and vector analysis. It also outlines the grading breakdown and test schedule. Several concepts from complex numbers are then defined and explained, such as the complex plane, polar coordinates, trigonometric form, and operations like addition, subtraction and multiplication of complex numbers. Examples are provided to demonstrate converting between rectangular and trigonometric forms.

1. ENGINEERING MATHAMATICS 1
Dr. Ir. Arman Djohan Diponegoro
5th August 2013

2. 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

3. GENERAL TOPICS ON
ENGINEERING MATHEMATICS 1
 Complex Variables
 Vector Analysis

4. GRADING
 Exercises and Quizzes : 80 %
 Mid Test : 10 %
 Final Test : 10 %

5. 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

6. COMPLEX NUMBERS

7. 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

8. COMPLEX NUMBERS

9. 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 by
z* = x – iy.
Two complex numbers are equal if and only if their real parts are equal and their
imaginary parts are equal.
1i
iyx 

10. 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 by
z* = x – iy.
Two complex numbers are equal if and only if their real parts are equal and their
imaginary parts are equal.
1iiyx 

11. 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.

12. 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 

13. 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).

14. RELATIONSHIPS AMONG X, Y,
R, AND 

x  rcos
y  rsin
r  x2
 y2
tan 
y
x
, if x  0

15. 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  

16. COMPLEX PLANE
Complex plane, polar form of a complex number






 
x
y1
tan
P
z = 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,

17. 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
y
z
,...2,1,0,2tan 1






 
nn
x
y


18. EULER FORMULA – AN
ALTERNATE POLAR FORM
The polar form of a complex number can be rewritten as :
This leads to the complex exponential function :


j
re
jyxjrz

 )sin(cos
 
 



jj
jj
ee
j
ee




2
1
sin
2
1
cos
Further leads to :
 yjye
eeee
x
jyxjyxz
sincos 
 

19. EULER FORMULA
 Remember the well-known Taylor Expansions :

20. EULER FORMULA
 So, we can conlude that :

21. GRAPHIC REPRESENTATION

22. A complex number, z = 1 + j , has a magnitude
EXAMPLE
4
2
4
sin
4
cos2

 j
ejz 






2)11(|| 22
z
rad2
4
2
1
1
tan 1












 


 nnzand argument :
Hence its principal argument is : Arg / 4z  rad
Hence in polar form :

23. A complex number, z = 1 - j , has a magnitude
2)11(|| 22
z
rad2
4
2
1
1
tan 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








4
sin
4
cos22 4

jez
j
EXAMPLE

24. What about z1=0+j, z2=0-j, z3=2+j0, z4=-2?


5.01
1
10
5.0
1



j
e
jz


5.01
1
10
5.0
2



 j
e
jz
02
2
02
0
3



j
e
jz






2
2
024
j
e
jz
EXAMPLE

25. ●
●
●
Im
Re
z1 = + j
z2 = - j
z3 = 2z4 = -2
●
5.0
EXAMPLE (CONTINUED)

26. 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.
1
cos120
2
 
3
sin120
2

1 3
2(cos120 sin120 ) 2 ,
2 2
1 3
i i
i
 
   
 
  

27. 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
x
x  

28. 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

29. 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.

30. 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
  
  
   
 

31. 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.

32. 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!

33. For a complex number z2 ≠ 0,
)(
2
1))((
2
1
2
1
2
1 2121
2
1




 jj
j
j
e
r
r
e
r
r
er
er
z
z
magnitudes divide!
phases subtract!
2
1
2
1
r
r
z
z
 2121 )(  z

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

35. 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:

36. 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:

37. AXIOMS SATISFIED BY THE
COMPLEX NUMBERS SYSTEM

38. DE MOIVRE’S THEOREM

39. 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. 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 n
r i r n i n       
   cis cis .
n n
r r n 

41. 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 sin225i i   
   
 
 
55
5
2 5 225 5 2
1 2 cos225 sin225
cos( ) sin( )
4 2 cos1125 sin1125
2 2
4 2
2 2
25
4 4
i i
i
i
i
i
     
     
 
 
  
 
 

42. 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  

43. 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 n
r i r  
360 360
or = , 0,1,2,..., 1.
k k
k n
n n n
 
 
  
   

44. 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
 2
1
2
cos
60
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. 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.
120 360
30 90
4 4
k
k


    

46. 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

47. 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

 
 


48. 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.

49. NEXT AGENDA
 Limit, Functions, and Continuity of Complex Variables.