Hyperbolic Cotangent of Complex Number

Theorem

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

Let $i$ be the imaginary unit.

Then:

Formulation 1

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

Formulation 2

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

Formulation 3

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

where:

$\cot$ denotes the real cotangent function

$\sin$ denotes the real sine function

$\cos$ denotes the real cosine function

$\sinh$ denotes the hyperbolic sine function

$\cosh$ denotes the hyperbolic cosine 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 Secant of Complex Number
• Hyperbolic Cosecant of Complex Number