Periodicity of Hyperbolic Tangent
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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