Definition:Hyperbolic Cosecant/Real

Definition

Definition 1

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.

Definition 2

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

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

$\forall x \in \R_{\ne 0}: \csch x := \dfrac 1 {\sinh x}$

where $\sinh$ is the real hyperbolic sine.

It is noted that at $x = 0$ we have that $\sinh x = 0$, 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

• Results about the hyperbolic cosecant function can be found here.

Sources

• Weisstein, Eric W. "Hyperbolic Cosecant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicCosecant.html