Mathematics – II
(MATH F112)
BITS Pilani
K K Birla Goa Campus
Dr. Amit Setia (Assistant Professor)
Department of Mathematics
BITS Pilani
K K Birla Goa Campus
Chapter 2
Analytic Functions
Function of a complex variable
Let S be a set of complex numbers.
A function f defined on S is a rule that assigns to each z
in S a complex number w.
The number w is called the value of f at z and is denoted
by f (z) i.e. w = f (z).
The set S is called the domain of definition of f.
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BITS Pilani, K K Birla Goa Campus
Note:
When the domain of definition is not mentioned,
we agree that the largest possible set is to be taken.
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Example
1
Let w  ,
z A
z
where A  {z  : z  1 & z  0}
then find the domain of definition of f .
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Solution
The domain of definition of f is A
where A  {z 
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: z  1 & z  0}
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BITS Pilani, K K Birla Goa Campus
Example
1
Let w 
z
then find the domain of definition of f .
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Solution
The domain of definition of f is
z 
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: z  0
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Example
z  3i
Let w  2
z 9
then find the domain of definition of f .
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Solution
The domain of definition of f is
z 
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: z  3i
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Let w  f  z 
where z  x  iy, w  u  iv
 u  iv  f ( x  iy )  f  z 
i.e. f  z   u  x, y   i v  x, y 
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Example
i
Let f  z  
z
then express it in the form
f  z   u  x, y   i v  x , y 
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Solution
i  x  iy 
i
i
y
x
Let f  z   
 2
 2
i 2
2
2
z x  iy x  y
x y
x  y2
 f  z   u  x, y   i v  x , y 
y
x
where u  x, y   2
, v  x, y   2
2
2
x y
x y
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Let w  f  z 
where z  rei , w  u  iv
 u  iv  f (rei )  f  z 
i.e. f  z   u  r ,    i v  r ,  
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Example
i
Let f  z  
z
then express it in the form
f  z   u  r ,   i v  r , 
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Solution
i
i
i  i
Let f  z    i  e
z re
r
i  cos   i sin   sin 
cos 


i
r
r
r
 f  z   u  r ,   i v  r , 
sin 
cos 
where u  r ,   
, v  r ,  
r
r
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Complex Polynomial
If n = 0 or positive integer and if a0, a1, a2,…, an are
complex constants with an ≠ 0, the function
P(z) = a0 + a1 z + a2 z2 + … + an zn
is a polynomial of degree n.
For example,
f  z   3  4 z  (5  3i ) z
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Rational function
It is defined as ratio of two polynomials
P( z )
f z 
, Q( z )  0  z
Q( z )
where P( z ), Q( z ) are polynomials
z2
e.g . f  z   3
is a rational function,
z 5
2
z
what about g  z   z ?
e
BITS Pilani, K K Birla Goa Campus
Multiple-valued functions
It is is a rule that assigns more than one value
to a point z in the domain of definition.
For example,
f  z  z
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1/2
is a double valued function.
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BITS Pilani, K K Birla Goa Campus
Example
f z  z
1/2
where z  re
i
 f  z    r ei  /2 ,
z0
where r  z ,
     is called the principal value of arg z
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Example
Can we make it single valued f  z   z ?
1/2
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Solution
Yes, we can make it single valued
f  z  r e
i  /2
,
where r  0 and      
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Limit of a function
lim f  z   w0
z  z0
i.e., for each   0,
there exists a positive number  such that
0  | z  z0 | 
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
f  z   w0  
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Theorem
When a limit of a function f ( z ) exists at a point z 0 ,
it is unique.
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