Cotangent of Complex Number

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Theorem

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

Let $i$ be the imaginary unit.


Then:

Formulation 1

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


Formulation 2

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


Formulation 3

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


where:

$\cot$ denotes the cotangent function (real and complex)
$\sin$ 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