Periodicity of Hyperbolic Cotangent
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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