# Reciprocal of Complex Exponential

## Theorem

Let $z \in \C$.

Let $\exp$ denote the complex exponential function.

Then:

$\dfrac 1 {\map \exp z} = \map \exp {-z}$

## Proof

 $\ds \map \exp {-z}$ $=$ $\ds \dfrac {\map \exp {-z} } {\map \exp 0}$ as $\map \exp 0 = 1$ by Exponential of Zero $\ds$ $=$ $\ds \dfrac {\map \exp {-z} } {\map \exp {z - z} }$ $\ds$ $=$ $\ds \dfrac {\map \exp {-z} } {\map \exp z \, \map \exp {-z} }$ Exponential of Sum: Complex Numbers $\ds$ $=$ $\ds \dfrac 1 {\map \exp z}$

$\blacksquare$