Short Notes: Complex Functions | Short Notes for Electrical Engineering - Electrical Engineering (EE) PDF Download

 Page 1


 
 
 
 
 
COMPLEX FUNCTIONS 
 
?  Exponential function of complex variable  
          ? ?
? ?
x iy
z
f z e e
?
??   
   
? ? ? ?
? ? ?
iyxx
f z e e e cosy i siny  = u + iv 
 
?  Logarithmic function of complex variable  
   If 
w
ez ? ; then w is logarithmic function of z  
   log z = w + 2inp  
   This logarithm of complex number has infinite numbers of values.  
   The general value of logarithm is denoted by Log z & the principal value is log z & is     
   found from general value by taking n = 0.   
 
? Analytic function  
         A function f(z) which is single valued and possesses a unique derivative with respect to z at     
         all points of region R is called as an analytic function.  
         If u & v are real, single valued functions of x & y s. t. 
u u v v
,,,
x y x y
????
? ? ? ?
 are continuous       
         throughout a region R, then Cauchy – Riemann equations 
u v v u
;  
x y x y
? ? ? ? ?
??
? ? ? ?
   
         are necessary & sufficient condition for f(z) = u + iv to be analytic in R. 
 
? Line integral of a complex function  
   
? ? ? ? ? ? ? ?
?
??
b
Ca
f z dz f z t z' t dt  
     where C is a smooth curve represented by z = z(t), where a = t = b.  
 
? Cauchy’s Theorem  
If f(z) is an analytic function and f’(z) is continuous at each point within and on a 
closed curve C. then   
      ? ?
C
f z dz 0 ?
?
   
 
 
 
 
Page 2


 
 
 
 
 
COMPLEX FUNCTIONS 
 
?  Exponential function of complex variable  
          ? ?
? ?
x iy
z
f z e e
?
??   
   
? ? ? ?
? ? ?
iyxx
f z e e e cosy i siny  = u + iv 
 
?  Logarithmic function of complex variable  
   If 
w
ez ? ; then w is logarithmic function of z  
   log z = w + 2inp  
   This logarithm of complex number has infinite numbers of values.  
   The general value of logarithm is denoted by Log z & the principal value is log z & is     
   found from general value by taking n = 0.   
 
? Analytic function  
         A function f(z) which is single valued and possesses a unique derivative with respect to z at     
         all points of region R is called as an analytic function.  
         If u & v are real, single valued functions of x & y s. t. 
u u v v
,,,
x y x y
????
? ? ? ?
 are continuous       
         throughout a region R, then Cauchy – Riemann equations 
u v v u
;  
x y x y
? ? ? ? ?
??
? ? ? ?
   
         are necessary & sufficient condition for f(z) = u + iv to be analytic in R. 
 
? Line integral of a complex function  
   
? ? ? ? ? ? ? ?
?
??
b
Ca
f z dz f z t z' t dt  
     where C is a smooth curve represented by z = z(t), where a = t = b.  
 
? Cauchy’s Theorem  
If f(z) is an analytic function and f’(z) is continuous at each point within and on a 
closed curve C. then   
      ? ?
C
f z dz 0 ?
?
   
 
 
 
 
 
 
 
 
 
? Cauchy’s Integral formula 
 If f(z) is analytic within & on a closed curve C, & a is any point within C.  
   ? ?
? ?
C
fz
1
f a dz
2 i z a
?
??
?
   
    
? ?
? ?
? ?
? ?
?
?
?
?
?
C
n
n1
fz
n!
f a  dz
2i
za
   
 
Singularities of an Analytic Function  
? Isolated singularity  
  ? ? ? ?
n
n
n
f z a z a
?
? ??
??
?
  ;     
? ?
? ?
n
n1
ft
1
a dt
2i
ta
?
?
?
?
?
    
       z = 
0
z is an isolated singularity if there is no singularity of f(z) in the  neighborhood of z 
= 
0
z .   
? Removable singularity  
If all the negative power of (z – a) are zero in the expansion of f(z), 
   f(z) = ? ?
n
n
n0
a z a
?
?
?
?
  
The singularity at z = a can be removed by defined f(z) at z = a such that f(z) is 
analytic at  z = a.  
 
? Poles  
If all negative powers of (z – a) after n
th
 are missing, then z = a is a pole of order ‘n’.  
 
? Essential singularity  
If the number of negative power of (z – a) is infinite, the z = a is essential 
singularity & cannot be removed.   
 
RESIDUES  
  If z = a is an isolated singularity of f(z)   
     
? ? ? ? ? ? ? ? ? ?
2
2 1 2
0 1 2 1
f z a a z a a z a ............. a z a a z a ...........
?
??
?
? ? ? ? ? ? ? ? ? ? ?      
  Then residue of f(z) at z = a is 
?1
a 
 
 
Page 3


 
 
 
 
 
COMPLEX FUNCTIONS 
 
?  Exponential function of complex variable  
          ? ?
? ?
x iy
z
f z e e
?
??   
   
? ? ? ?
? ? ?
iyxx
f z e e e cosy i siny  = u + iv 
 
?  Logarithmic function of complex variable  
   If 
w
ez ? ; then w is logarithmic function of z  
   log z = w + 2inp  
   This logarithm of complex number has infinite numbers of values.  
   The general value of logarithm is denoted by Log z & the principal value is log z & is     
   found from general value by taking n = 0.   
 
? Analytic function  
         A function f(z) which is single valued and possesses a unique derivative with respect to z at     
         all points of region R is called as an analytic function.  
         If u & v are real, single valued functions of x & y s. t. 
u u v v
,,,
x y x y
????
? ? ? ?
 are continuous       
         throughout a region R, then Cauchy – Riemann equations 
u v v u
;  
x y x y
? ? ? ? ?
??
? ? ? ?
   
         are necessary & sufficient condition for f(z) = u + iv to be analytic in R. 
 
? Line integral of a complex function  
   
? ? ? ? ? ? ? ?
?
??
b
Ca
f z dz f z t z' t dt  
     where C is a smooth curve represented by z = z(t), where a = t = b.  
 
? Cauchy’s Theorem  
If f(z) is an analytic function and f’(z) is continuous at each point within and on a 
closed curve C. then   
      ? ?
C
f z dz 0 ?
?
   
 
 
 
 
 
 
 
 
 
? Cauchy’s Integral formula 
 If f(z) is analytic within & on a closed curve C, & a is any point within C.  
   ? ?
? ?
C
fz
1
f a dz
2 i z a
?
??
?
   
    
? ?
? ?
? ?
? ?
?
?
?
?
?
C
n
n1
fz
n!
f a  dz
2i
za
   
 
Singularities of an Analytic Function  
? Isolated singularity  
  ? ? ? ?
n
n
n
f z a z a
?
? ??
??
?
  ;     
? ?
? ?
n
n1
ft
1
a dt
2i
ta
?
?
?
?
?
    
       z = 
0
z is an isolated singularity if there is no singularity of f(z) in the  neighborhood of z 
= 
0
z .   
? Removable singularity  
If all the negative power of (z – a) are zero in the expansion of f(z), 
   f(z) = ? ?
n
n
n0
a z a
?
?
?
?
  
The singularity at z = a can be removed by defined f(z) at z = a such that f(z) is 
analytic at  z = a.  
 
? Poles  
If all negative powers of (z – a) after n
th
 are missing, then z = a is a pole of order ‘n’.  
 
? Essential singularity  
If the number of negative power of (z – a) is infinite, the z = a is essential 
singularity & cannot be removed.   
 
RESIDUES  
  If z = a is an isolated singularity of f(z)   
     
? ? ? ? ? ? ? ? ? ?
2
2 1 2
0 1 2 1
f z a a z a a z a ............. a z a a z a ...........
?
??
?
? ? ? ? ? ? ? ? ? ? ?      
  Then residue of f(z) at z = a is 
?1
a 
 
 
 
 
 
 
 
  Residue Theorem 
   
? ?
? ? ?
?
c
f z dz 2 i  (sum of residues at the singular points within c ) 
    If f(z) has a pole of order ‘n’ at z=a 
   ? ?
? ?
? ? ? ?
n1
n
n1
za
1d
Res f a z a f z
n 1 ! dz
?
?
?
??
??
??
??
??
?
??
 
  
 Evaluation Real Integrals 
   I= 
? ?
?
? ? ?
?
2
0
F cos ,sin d 
   
ii
ee
cos 
2
? ? ?
?
?? ; 
ii
ee
sin 
2i
? ? ?
?
?? 
    Assume z=  
i
e
?
                      
    
??
??
??
??
1
z+
z
cos 
2
; 
??
? ? ?
??
??
11
sin z
2i z
 
    I= ? ? ? ? ? ?
n
k
k=1
c
dz
f z 2 i Res f z
iz
??
??
??
??
?
?
    
   Residue should only be calculated at poles in upper half plane. 
   Residue is calculated for the function: 
? ? fz
iz
??
??
??
   
    
? ? ? ?
?
??
??
?
?
f x dx 2 i Res f z 
Where residue is calculated at poles in upper half plane & poles of f(z) are found 
by substituting z in place of x in f(x).