Imaginary Part of Integer Power of Complex Number is Harmonic

Theorem
Let $z \in \C$ be a complex number.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $z^n$ denote $z$ raised to the $n$th power.

Then the imaginary part $\map \Im {z^n}$ of $z^n$ is a harmonic polynomial.

Proof
Let $z = x + i y$.

Then:

Then we have:

and:

It remains to be shown that for $j \in \set {0, 1, \ldots, n - 2}$ that:


 * $\paren {-1}^{\paren {j - 1} / 2} \dbinom n j \paren {n - j} \paren {n - j - 1} = -\paren {-1}^{\paren {j + 1} / 2} \dbinom n {j + 2} \paren {j + 2} \paren {j + 1}$

First we have that:

Hence it remains to be shown that:


 * $\dbinom n j \paren {n - j} \paren {n - j - 1} = \dbinom n {j + 2} \paren {j + 2} \paren {j + 1}$

We have:

and:

Hence the result.