Hyperbolic Cotangent in terms of Cotangent

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Theorem

Let $z \in \C$ be a complex number.

Then:

$\coth z = -\cot \paren {i z}$

where:

$\cot$ denotes the cotangent function
$\coth$ denotes the hyperbolic cotangent
$i$ is the imaginary unit: $i^2 = -1$.


Proof

\(\ds i \coth z\) \(=\) \(\ds \frac {i \cosh z} {\sinh z}\) Definition 2 of Hyperbolic Cotangent
\(\ds \) \(=\) \(\ds \frac {-\cosh z} {i \sinh z}\) $i^2 = -1$
\(\ds \) \(=\) \(\ds \frac {-\cos \paren {i z} } {i \sinh z}\) Hyperbolic Cosine in terms of Cosine
\(\ds \) \(=\) \(\ds \frac {-\cos \paren {i z} } {\sin \paren {i z} }\) Hyperbolic Sine in terms of Sine
\(\ds \) \(=\) \(\ds -\cot \paren {i z}\) Definition of Complex Cotangent Function

$\blacksquare$


Also see


Sources

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