Hyperbolic Cotangent of Complex Number/Formulation 1

Theorem

Let $a$ and $b$ be real numbers.

Let $i$ be the imaginary unit.

Then:

$\coth \paren {a + b i} = \dfrac {\cosh a \cos b + i \sinh a \sin b} {\sinh a \cos b + i \cosh a \sin b}$

where:

$\coth$ denotes the hyperbolic cotangent function

$\sin$ denotes the real sine function

$\cos$ denotes the real cosine function

$\sinh$ denotes the hyperbolic sine function

$\cosh$ denotes the hyperbolic cosine function.

$\ds \coth \paren {a + b i}$

$=$

$\ds \frac {\cosh \paren {a + b i} } {\sinh \paren {a + b i} }$

Definition of Hyperbolic Cotangent

$\ds $

$=$

$\ds \dfrac {\cosh a \cos b + i \sinh a \sin b} {\sinh a \cos b + i \cosh a \sin b}$

$\blacksquare$