Periodicity of Hyperbolic Cosecant

Theorem

Let $k \in \Z$.

Then:

$\map \csch {x + 2 k \pi i} = \csch x$


Proof

\(\ds \map \csch {x + 2 k \pi i}\) \(=\) \(\ds \frac 1 {\map \sinh {x + 2 k \pi i} }\) Definition of Hyperbolic Cosecant
\(\ds \) \(=\) \(\ds \frac 1 {\sinh x}\) Periodicity of Hyperbolic Sine
\(\ds \) \(=\) \(\ds \csch x\) Definition of Hyperbolic Cosecant

$\blacksquare$


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.89$: Periodicity of Hyperbolic Functions
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.