Definition:Real Hyperbolic Cotangent/Definition 1
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Definition
The real hyperbolic cotangent function is defined on the real numbers as:
- $\coth: \R_{\ne 0} \to \R$:
- $\forall x \in \R_{\ne 0}: \coth x := \dfrac {e^x + e^{-x} } {e^x - e^{-x} }$
where it is noted that at $x = 0$:
- $e^x - e^{-x} = 0$
and so $\coth x$ is not defined at that point.
Also see
- Definition:Real Hyperbolic Sine
- Definition:Real Hyperbolic Cosine
- Definition:Real Hyperbolic Tangent
- Definition:Real Hyperbolic Secant
- Definition:Real Hyperbolic Cosecant
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $8.4$: Definition of Hyperbolic Functions
- 1972: Frank Ayres, Jr. and J.C. Ault: Theory and Problems of Differential and Integral Calculus (SI ed.) ... (previous) ... (next): Chapter $15$: Differentiation of Hyperbolic Functions: Definition of Hyperbolic Functions
- Weisstein, Eric W. "Hyperbolic Cotangent." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicCotangent.html