Maths - Powers Of A Complex Variable

The expression for:

(x + i y)n

where n is an integer.

is given by the following sum:

(x + i y)n=
(n-k)! k!
(-i)k xn-k yk

We can derive this from the binomial theorm:

(a + b)n=
(n-k)! k!
(-1)k a n-k bk

with 'a' replaced by x and 'b' replaced by iy


We can look at a simplest we can take the case where n=2 which gives:
(x + i y)2

z plane   w plane
z plane



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 + i y and w = u + i v

lets take the example of the square function w = z²


w = (x + i y)²

multiplying out gives:

w = x² - y² + i 2 x y

so the u and v components are:

u = x² - y²
v = 2 x y

Integer Powers

As shown above:

(x + i y)n=
(n-k)! k!
(-i)k xn-k yk


k (-i)k k!
0 1 0
1 -i 1
2 -1 2
3 i 6
4 1 24

So (-i)k cycles round every 4 entries, each step is a rotation by 90. The even values are real and the odd values are imaginary.

So the powers for n = 1 to 4 are:

n (x + i y)n u v
1 (x + i y)1 x y
2 (x + i y)2 x²-y² 2xy
3 (x + i y)3 x3 - 3y²x 3x²y -y3
4 (x + i y)4 x4 - 6x²y² - y4 -2y3x-2yx3

metadata block
see also:
Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Visual Complex Analysis - If you already know the basics of complex numbers but want to get an in depth understanding using an geometric and intuitive approach then this is a very good book. The book explains how to represent complex transformations such as the Möbius transformations. It also shows how complex functions can be differentiated and integrated.

Terminology and Notation

Specific to this page here:


This site may have errors. Don't use for critical systems.

Copyright (c) 1998-2023 Martin John Baker - All rights reserved - privacy policy.