# Equivalence of Definitions of Hyperbolic Cotangent

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## Theorem

The following definitions of the concept of Hyperbolic Cotangent are equivalent:

### Definition 1

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

$\coth: X \to \C$:
$\forall z \in X: \coth z := \dfrac {e^z + e^{-z} } {e^z - e^{-z}}$

where:

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

### Definition 2

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

$\coth: X \to \C$:
$\forall z \in X: \coth z := \dfrac {\cosh z} {\sinh z}$

where:

$\sinh$ is the hyperbolic sine
$\cosh$ is the hyperbolic cosine
$X = \set {z : z \in \C, \ \sinh z \ne 0}$

### Definition 3

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

$\coth: X \to \C$:
$\forall z \in X: \coth z := \dfrac {e^{2 z} + 1} {e^{2 z} - 1}$

where:

$X = \set {z : z \in \C, \ e^{2 z} - 1 \ne 0}$

### Definition 4

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

$\coth: X \to \C$:
$\forall z \in X: \coth z := \dfrac 1 {\tanh z}$

where:

$\tanh$ is the hyperbolic tangent
$X = \set {z : z \in \C, \ \sinh z \ne 0}$
where $\sinh$ is the hyperbolic sine.

## Proof

 $\ds \forall z \in \set {z \in \C: \ e^{2 z} - 1 \ne 0}: \,$ $\ds$  $\ds \frac {e^{2 z} + 1} {e^{2 z} - 1}$ Definition 3 of Hyperbolic Cotangent $\ds \leadstoandfrom \ \$ $\ds \forall z \in \set {z \in \C: \ e^z - e^{-z} \ne 0}: \,$ $\ds$ $=$ $\ds \frac {e^z \paren {e^z + e^{-z} } } {e^z \paren {e^z - e^{-z} } }$ $\ds \leadstoandfrom \ \$ $\ds \forall z \in \set {z \in \C: \ e^z - e^{-z} \ne 0}: \,$ $\ds$ $=$ $\ds \frac {e^z + e^{-z} } {e^z - e^{-z} }$ Definition 1 of Hyperbolic Cotangent $\ds \leadstoandfrom \ \$ $\ds \forall z \in \set {z \in \C: \ \frac {e^z - e^{-z} } 2 \ne 0}: \,$ $\ds$ $=$ $\ds \frac {\paren {\dfrac {e^z + e^{-z} } 2} } {\paren {\dfrac {e^z - e^{-z} } 2} }$ $\ds \leadstoandfrom \ \$ $\ds \forall z \in \set {z \in \C: \ \sinh z \ne 0}: \,$ $\ds$ $=$ $\ds \frac {\cosh z} {\sinh z}$ Definition of Hyperbolic Sine, Definition of Hyperbolic Cosine, Definition 2 of Hyperbolic Cotangent $\ds \leadstoandfrom \ \$ $\ds \forall z \in \set {z \in \C: \ \sinh z \ne 0}: \,$ $\ds$ $=$ $\ds \dfrac 1 {\frac {\sinh z} {\cosh z} }$ $\ds \leadstoandfrom \ \$ $\ds \forall z \in \set {z \in \C: \ \sinh z \ne 0}: \,$ $\ds$ $=$ $\ds \dfrac 1 {\tanh z}$ Definition 2 of Hyperbolic Tangent, Definition 4 of Hyperbolic Cotangent

$\blacksquare$