Maths - Powers Of Double Numbers

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 zn

where:

Square

z plane   w plane
z plane

-->

w=z²

w squared

how this plot was produced.

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:

We can use the binomial theorem:

(a + D b)n=
n
k=0
n!
(n-k)! k!
(-D)k a n-k bk

 

 


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