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
\(\ds \cmod {e^{i z} }\) | \(=\) | \(\ds \cmod {\map \exp {i R \, \map \exp {i \theta} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {\map \exp {i R \paren {\cos \theta + i \sin \theta} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {\map \exp {R \paren {-\sin \theta + i \cos \theta} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {\map \exp {- R \sin \theta} \, \map \exp {i \cos \theta} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \exp {- R \sin \theta}\) | 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:
- $\cmod {e^{i z} } = e^{-3 \sqrt 3}$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Polar Form of Complex Numbers: $87$