Real and Imaginary Parts of Integer Power of Complex Number are Harmonic
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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: $3.7.23$