Hyperbolic Tangent of Complex Number

Theorem

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

Let $i$ be the imaginary unit.

Then:

Formulation 1

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

Formulation 2

$\tanh \paren {a + b i} = \dfrac {\tanh a + i \tan b} {1 + i \tanh a \tan b}$

Formulation 3

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

Formulation 4

$\map \tanh {a + b i} = \dfrac {\sinh 2 a + i \sin 2 b} {\cosh 2 a + \cos 2 b}$

where:

$\tan$ denotes the real 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

$\tanh$ 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