Cosecant Exponential Formulation

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Theorem

Let $z$ be a complex number.

Let $\csc z$ denote the cosecant function and $i$ denote the imaginary unit: $i^2 = -1$.

Then:

$\csc z = \dfrac {2 i} {e^{i z} - e^{-i z} }$


Proof

\(\displaystyle \csc z\) \(=\) \(\displaystyle \frac 1 {\sin z}\) $\quad$ Definition of Complex Cosecant Function $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 1 / \frac {e^{i z} - e^{-i z} } {2 i}\) $\quad$ Sine Exponential Formulation $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {2 i} {e^{i z} - e^{-i z} }\) $\quad$ multiplying top and bottom by $2 i$ $\quad$

$\blacksquare$


Also see


Sources