Lecture 2 Complex Analysis

complex number:-
A complex number Z can be defined as an ordered pair of real number x,y.
Z=(x,y)

Let Z_1=(x_1,y_1 ) and Z_2=(x_2,y_2 ) be any two complex numbers, then their sum is
Z_1+Z_2=(x_1,y_1 )+(x_2,y_2 )=(x_1+x_2 ,y_1+y_2 )
and their product is
Z_1 Z_2=(x_1,y_1 )(x_2,y_2 )=(x_1 x_2-y_1 y_2 ,x_1 y_2+y_1 x_2 )

Remarks:-
The order pair (0,y) is called a pure imaginary numbers.
The order pair (x,0) is called real numbers and we write it as x, so the set of real numbers R is subset of the set of complex numbers C.
x is called real part of complex number Z=(x,y) and we denote it as x=R(Z) .
y is called imaginary part of complex number Z=(x,y) and we write it as y=I(Z) , so y=I(Z) is real numbers.
We can write a complex number Z=(x,y) as algebraic (ordinary) form Z=(x,y)=(x,0)+(0,y)
=(x,0)(0,1)(y,0)
Z=x+iy
where i=(0,1) is called imaginary unit.
The algebraic form of complex numbers is a best the Cartesian form of complex numbers, especially in expression of complex number and satisfied its properties.
By using product of complex numbers, we get
? i?^2=i?i=(0,1)?(0,1)=(0-1,0?1+1?0)
=(-1,0)=-1
i^3=i^2?i=(i)i=-i
i^4=i^2?i^2=(-1)(-1)=1
In general case, If n is positive integer number, then
i^n={?( 1 if remainder of quotient of n and 4 is 0@ i if remainder of quotient of n and 4 is 1 @-1 if remainder of quotient of n and 4 is 2 @–i if remainder of quotient of n and 4 is 3)?
Property (1):- the complex number Z=0 iff (if and only if) real and imaginary parts are zero.
Property (2):- if Z_1=x_1+iy_1 and Z_2=x_2+iy_2 , then
Z_1=Z_2?x_1=x_2 & y_1=y_2
EX (1):- Find value of x and y which satisfy the following equation (x-y-6)+i(y^2-x)=0
Sol:-
By property (2)
x-y-6=0 ………… (1)
y^2-x=0 …………(2)
From eq. (1), we get x=y+6
Put in eq. (2) y^2-y-6=0?(y-3)(y+2)=0
?ei y=3 or y=-2
So that x=9 or x=4