Hyperbolic Tangent of Complex Number/Formulation 1

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

Let $i$ be the imaginary unit.

Then:
 * $\tanh \paren {a + b i} = \dfrac {\sinh a \cos b + i \cosh a \sin b} {\cosh a \cos b + i \sinh a \sin b}$

where:
 * $\tanh$ denotes the hyperbolic tangent 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.

Also see

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