Euler's Cosecant Identity

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Theorem

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

where:

$z \in \C$ is a complex number
$\csc z$ denotes the cosecant function
$i$ denotes the imaginary unit: $i^2 = -1$


Proof

\(\ds \csc z\) \(=\) \(\ds \frac 1 {\sin z}\) Definition of Complex Cosecant Function
\(\ds \) \(=\) \(\ds 1 / \frac {e^{i z} - e^{-i z} } {2 i}\) Euler's Sine Identity
\(\ds \) \(=\) \(\ds \frac {2 i} {e^{i z} - e^{-i z} }\) multiplying top and bottom by $2 i$

$\blacksquare$


Also see


Source of Name

This entry was named for Leonhard Paul Euler.


Sources

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