Periodicity of Hyperbolic Cotangent

Theorem

Let $k \in \Z$.

Then:

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

Proof

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

\(=\)

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

Definition 2 of Hyperbolic Cotangent

\(\ds \)

\(=\)

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

Periodicity of Hyperbolic Sine

\(\ds \)

\(=\)

\(\ds \frac {\cosh x} {\sinh x}\)

Periodicity of Hyperbolic Cosine

\(\ds \)

\(=\)

\(\ds \coth x\)

Definition 2 of Hyperbolic Cotangent

$\blacksquare$

Sources

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