Complex Analysis Notes

  Polar form of a complex number. If z= x + iy is a complex number. Then z = r ( cosθ + sinθ ) is called polar form or trigonometric form of a complex number.   By comparing real and imaginary parts of a complex number,  we get x = r cosθ           (1) y = r sinθ  …

Locus of complex number play a very important role in the complex analysis. Today we learn how to represent and find the locus of a complex number. We can also prepare Multiple choice questions about this topic and complex analysis.          Let  P(Z) be the property that satisfied by a complex number  z = x + iy. …

  Modulus:    It is defined as                        Argument: Arg(Z):  It is defined as                        it is also known as amplitude of Z    symbolically amp(Z) Note:  If |Z|=0 then x=0 and y=0 The Argument θ of a complex number is …

The polar coordinates of a P are (r,θ) where r is known as modulus and  θ is the argument of a complex number Z. From above figure   a   =   r cosθ  b   =   r sinθ or  The plane is known as Argand plane or Argand Diagram or complex plane or Gaussian plane.

Let      Z1=( x1, y1 ) =             Addition:  The sum of two complex numbers is                                                                                   Subtraction:  The …

Need Of Complex Numbers Some quadratic equations which have no solutions in real numbers.  For Example 1.     2.    3.    In order to find the solutions of above given or similar quadratic equations, the symbol ” i  ”  was used.  Euler was the first person to introduce the symbol ” i “. where                 …

Distributive laws  Left Distributive Law  Right Distributive Law

Properties of complex Numbers If     be the three complex numbers then                                             Associative law of multiplication      Multiplicative identity.. Commutative law of multiplication For each non-zero 

Definition( Complex Number) A complex number is an element (x, y) of the set  = {  } obeying the following rules of addition and multiplication If   then 1.      2.      as   Z= 3+4i

  There are following properties of complex numbers. Associative, additive identity, additive inverse and commutative properties. If     be the three complex numbers then    .   Associative property of addition 2.   Z + 0 = 0 + Z = Z  Additive identity 3.   Z + ( -Z ) = ( 0,0) = 0 Additive inverse property 4.    .  .  .  .  …