Hyperbolic Cotangent Function is Odd

Theorem

Let $\coth: \C \to \C$ be the hyperbolic cotangent function on the set of complex numbers.


Then $\coth$ is odd:

$\map \coth {-x} = -\coth x$


Proof

\(\ds \map \coth {-x}\) \(=\) \(\ds \frac {\map \cosh {-x} } {\map \sinh {-x} }\) Definition 2 of Hyperbolic Cotangent
\(\ds \) \(=\) \(\ds \frac {\map \cosh {-x} } {-\sinh x}\) Hyperbolic Sine Function is Odd
\(\ds \) \(=\) \(\ds \frac {\cosh x} {-\sinh x}\) Hyperbolic Cosine Function is Even
\(\ds \) \(=\) \(\ds -\coth x\)

$\blacksquare$


Also see


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $8.19$: Functions of Negative Arguments
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.