Periodicity of Hyperbolic Tangent

Theorem

Let $k \in \Z$.

Then:

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

Proof

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

\(=\)

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

Definition of Hyperbolic Tangent

\(\ds \)

\(=\)

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

Periodicity of Hyperbolic Sine

\(\ds \)

\(=\)

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

Periodicity of Hyperbolic Cosine

\(\ds \)

\(=\)

\(\ds \tanh x\)

Definition of Hyperbolic Tangent

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.88$: Periodicity of Hyperbolic Functions