Modulus of Exponential of i z where z is on Circle/Examples/6 e^pi i over 3
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Examples of Use of Modulus of Exponential of i z where z is on Circle
Let $z = 6 e^{\pi i / 3}$
Then:
- $\cmod {e^{i z} } = e^{-3 \sqrt 3}$
Proof
From Modulus of Exponential of i z where z is on Circle:
- $(1): \quad \cmod {e^{i z} } = e^{-R \sin \theta}$
where:
- $z = R e^{i \theta}$
Here we have:
- $R = 6$
- $\theta = \dfrac \pi 3$
Hence:
- $\sin \theta = \dfrac {\sqrt 3} 2$
and the result follows by substituting for $R$ and $\sin \theta$ in $(1)$.
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Miscellaneous Problems: $131$