Hyperbolic Cotangent of Complex Number/Formulation 2

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

Let $i$ be the imaginary unit.

Then:
 * $\map \coth {a + b i} = \dfrac {1 - i \coth a \cot b} {\coth a - i \cot b}$

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