Reciprocal of Power of Complex Number
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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.
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: $3.7.25$