Equivalence of Definitions of Hyperbolic Cotangent
Jump to navigation
Jump to search
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 |
$\blacksquare$