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