Cotangent of Complex Number

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

• Sine of Complex Number
• Cosine of Complex Number
• Tangent of Complex Number
• Secant of Complex Number
• Cosecant of Complex Number