Hyperbolic Cotangent of Complex Number/Formulation 3

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

Let $i$ be the imaginary unit.

Then:
 * $\map \coth {a + b i} = \dfrac {\coth a + \coth a \cot^2 b} {\coth^2 a + \cot^2 b} + \dfrac {\cot b - \coth^2 a \cot b} {\coth^2 a + \cot^2 b} i$

where:
 * $\cot$ denotes the real cotangent function
 * $\coth$ denotes the hyperbolic cotangent function.

Also see

 * Hyperbolic Sine of Complex Number
 * Hyperbolic Cosine of Complex Number
 * Hyperbolic Tangent of Complex Number
 * Hyperbolic Cosecant of Complex Number
 * Hyperbolic Secant of Complex Number