Reciprocal of Power of Complex Number

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.

The reciprocal of $z^n$ can be expressed as:

$\dfrac 1 {z^n} = \dfrac {\overline z^n} {\cmod z^{2 n} }$

where:

$\overline z$ denotes the complex conjugate of $z$

$\cmod z^2$ denotes the modulus of $z$.

Proof

\(\ds \dfrac 1 {z^n}\)

\(=\)

\(\ds \paren {\dfrac 1 z}^n\)

\(\ds \)

\(=\)

\(\ds \paren {\dfrac {\overline z} {\cmod z^2} }^n\)

\(\ds \)

\(=\)

\(\ds \dfrac {\paren {\overline z}^n} {\paren {\cmod z^2}^n}\)

\(\ds \)

\(=\)

\(\ds \dfrac {\overline z^n} {\cmod z^{2 n} }\)

$\blacksquare$

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: $3.7.25$