Definition:Hyperbolic Cosecant

Definition

The hyperbolic cosecant is one of the hyperbolic functions:

Definition 1

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

$\csch: X \to \C$:

$\forall z \in X: \csch z := \dfrac 2 {e^z - e^{-z} }$

where:

$X = \set {z: z \in \C, \ e^z - e^{-z} \ne 0}$

Definition 2

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

$\csch: X \to \C$:

$\forall z \in X: \csch z := \dfrac 1 {\sinh z}$

where:

$\sinh$ is the hyperbolic sine

$X = \set {z: z \in \C, \ \sinh z \ne 0}$

Real Hyperbolic Cosecant

On the real numbers it is defined similarly.

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 Hyperbolic Cosecant

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

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

Linguistic Note

The usual symbol csch for hyperbolic cosecant is awkward to pronounce.

Some pedagogues say it as cosetch, and some as cosesh.

Others prefer the mouthful which is hyperbolic cosec.

Sources

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