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Complex Functions Review Guide

The document reviews complex functions and their properties. It defines imaginary numbers and shows how to represent complex numbers in rectangular and polar forms. It also covers complex number operations like addition, multiplication, and division. Exponential and trigonometric relations to complex functions are presented along with their use in representing periodic functions. Complex variables and functions are briefly introduced.

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Ragib Dihan
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44 views2 pages

Complex Functions Review Guide

The document reviews complex functions and their properties. It defines imaginary numbers and shows how to represent complex numbers in rectangular and polar forms. It also covers complex number operations like addition, multiplication, and division. Exponential and trigonometric relations to complex functions are presented along with their use in representing periodic functions. Complex variables and functions are briefly introduced.

Uploaded by

Ragib Dihan
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Review of complex functions:

Imaginary number: j  1 j 2  1

Complex number Relations:

complex plane
z  x  jy

Magnitude of z: z  x2  y 2
y
Angle of z:   tan 1
x

Rectangular form: z  x  jy ; z  z (cos   j sin  )


Polar form: z  z  ; z  z e j

Complex conjugate: z  x  jy  z   

Complex algebra:
z1  x1  jy1  z1  , z2  x2  jy2  z2 
Addition and substraction: z1  z2  ( x1  x2 )  j ( y1  y2 )
Multiplication: z1  z2  ( x1  jy1 )( x2  jy2 )  x1 x2  y1 y2   j x1 y2  x2 y1   z1 z2   


z 2  x  jy   x 2  y 2  j 2 xy 
2

x  jy x  jy   x 2
 y 2   j  xy  xy   x2  y 2

1 1 1 x  jy x  jy
   2
z x  jy x  jy x  jy x  y 2
z1 x1  jy1 x1  jy1 x2  jy 2  x1 x2  y1 y2   j x2 y1  x1 y2  z1
Division:       
z2 x2  jy 2 x2  y2
2 2
x2  y2
2 2
z2

Power and roots:


z n  ( z  )n  z n
n

1 1 1
z n  ( z  ) n  z n  / n
Relation between exponential and trigonometric functions:
x 0 x 2 x 4 x 6 x8 x10
cos( x)       
0! 2! 4! 6! 8! 10!
x1 x3 x5 x 7 x9 x11
sin( x)       
1! 3! 5! 7! 9! 11!
cos( x)  j sin( x) 
x x2 x3 x 4 x5 x6 x 7 x8 x9 x10 x11
e 1 j   j   j   j   j 
jx
j 
1! 2! 3! 4! 5! 6! 7! 8! 9! 10! 11!
 eix  cos( x)  j sin( x) Euler formula
eix  cos( x)  j sin( x)
e jx  e jx  2 cos( x)  cos( x)  e jx  e jx 
1
2
e jx  e jx  2 j sin( x)  sin( x)   j e jx  e jx 
1
2
Complex variable:

s    j

Complex function:

F (s)  Fx  jFy
Complex function for a linear system has a typical form:
K ( s  z1 )( s  z2 )....(s  zm )
F ( s) 
( s  p1 )(s  p2 )....(s  pn )

Zeros: -z1, -z2…-zm F(s) = 0


Poles: -p1, -p2….-pn F(s) = infinity

Single pole (k=1) and multiple pole: ( s  p)


k

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