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:

• n is an integer.
• z is a double number.

Square

z plane		w plane
	--> w=z²

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:

• n is an integer.
• z is a double number.

We can use the binomial theorem:

(a + D b)n=

n ∑ k=0

n! (n-k)! k!

(-D)k a n-k bk

 

 

EuclideanSpace

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Prerequisites

On this page we discussed how to represent and evaluate functions where the domain and codomain are complex numbers.