Periodicity of Hyperbolic Cotangent

Theorem

Let $k \in \Z$.

Then:

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

$\ds \map \coth {x + 2 k \pi i}$

$=$

$\ds \frac {\map \cosh {x + 2 k \pi i} } {\map \sinh {x + 2 k \pi i} }$

Definition 2 of Hyperbolic Cotangent

$\ds $

$=$

$\ds \frac {\map \cosh {x + 2 k \pi i} } {\sinh x}$

$\ds $

$=$

$\ds \frac {\cosh x} {\sinh x}$

$\ds $

$=$

$\ds \coth x$

Definition 2 of Hyperbolic Cotangent

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.91$: Periodicity of Hyperbolic Functions