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