Modulus of Exponential of i z where z is on Circle

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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

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

$\blacksquare$


Examples

Circle $6 e^{\pi i / 3}$

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

Then:

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


Sources