# Modulus of Exponential of i z where z is on Circle

## Theorem

Let $C$ be the circle embedded in the complex plane given by the equation:

$z = R e^{i \theta}$

Then:

$\cmod {e^{i z} } = e^{-R \sin \theta}$

## Proof

 $\ds \cmod {e^{i z} }$ $=$ $\ds \cmod {\map \exp {i R \, \map \exp {i \theta} } }$ $\ds$ $=$ $\ds \cmod {\map \exp {i R \paren {\cos \theta + i \sin \theta} } }$ $\ds$ $=$ $\ds \cmod {\map \exp {R \paren {-\sin \theta + i \cos \theta} } }$ $\ds$ $=$ $\ds \cmod {\map \exp {- R \sin \theta} \, \map \exp {i \cos \theta} }$ $\ds$ $=$ $\ds \map \exp {- R \sin \theta}$ Modulus and Argument of Complex Exponential

$\blacksquare$

## Examples

### $6 e^{\pi i / 3}$ on the circle $\cmod z = 6$

Let $z = 6 e^{\pi i / 3}$

Then:

$\cmod {e^{i z} } = e^{-3 \sqrt 3}$