Definition:Real Hyperbolic Cosecant/Definition 1

Definition

The real hyperbolic cosecant function is defined on the real numbers as:

$\csch: \R_{\ne 0} \to \R$:

$\forall x \in \R_{\ne 0}: \csch x := \dfrac 2 {e^x - e^{-x} }$

where it is noted that at $x = 0$:

$e^x - e^{-x} = 0$

and so $\csch x$ is not defined at that point.

Also see

• Equivalence of Definitions of Real Hyperbolic Cosecant

• Definition:Real Hyperbolic Sine
• Definition:Real Hyperbolic Cosine
• Definition:Real Hyperbolic Tangent
• Definition:Real Hyperbolic Cotangent
• Definition:Real Hyperbolic Secant

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $8.6$: 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 Cosecant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicCosecant.html