Hyperbolic Cotangent Function is Odd
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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
- Hyperbolic Sine Function is Odd
- Hyperbolic Cosine Function is Even
- Hyperbolic Tangent Function is Odd
- Hyperbolic Secant Function is Even
- Hyperbolic Cosecant Function is Odd
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $8.19$: Functions of Negative Arguments