First we will calculate the simple case of a square then we go on to use binomial theorem to calculate the more general case of z^{n}
where:
 n is an integer.
 z is a double number.
Square
z plane  w plane  

> w=z² 
Pure real values always square to a positive value and pure imaginary values always square to a negative value. However real and imaginary parts together cover the whole plane.
Let the components of the input and output planes be:
z = x + D y and w = u + D v
lets take the example of the square function w = z²
so:
w = (x + D y)²
multiplying out gives:
w = x² + y² + D 2 x y
so the u and v components are:
u = x² + y²
v = 2 x y
Using Binomial Theorem
We want to calculate an expression for:
(z)^{n}
where:
 n is an integer.
 z is a double number.
We can use the binomial theorem:
(a + D b)^{n}= 


(D)^{k} a ^{nk} b^{k} 