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=0
n!
(n-k)! k!
(-i)k xn-k yk

We can derive this from the binomial theorm:

(a + b)n=
n
k=0
n!
(n-k)! k!
(-1)k a n-k bk

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

Square

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=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 + i y and w = u + i v

lets take the example of the square function w = z²

so:

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=0
n!
(n-k)! k!
(-i)k xn-k yk

where:

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

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