Hyperbolic Tangent of Complex Number/Formulation 3

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

Let $i$ be the imaginary unit.

Then:
 * $\tanh \paren {a + b i} = \dfrac {\tanh a + \tanh a \tan^2 b} {1 + \tanh^2 a \tan^2 b} + \dfrac {\tan b - \tanh^2 a \tan b} {1 + \tanh^2 a \tan^2 b} i$

where:
 * $\tan$ denotes the real tangent function
 * $\tanh$ denotes the hyperbolic tangent 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