Real and Imaginary Parts of Integer Power of Complex Number are 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 both the real part $\map \Re {z^n}$ and the imaginary part $\map \Im {z^n}$ of $z^n$ are harmonic polynomials.

Proof

Real Part of Integer Power of Complex Number is Harmonic

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 real part $\map \Re {z^n}$ of $z^n$ is a harmonic polynomial.

$\Box$

Imaginary Part of Integer Power of Complex Number is Harmonic

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.

$\blacksquare$

Sources

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.7$ Complex Numbers and Functions: Powers: $3.7.23$