Hyperbolic Cotangent of Complex Number/Formulation 1
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Theorem
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
- $\coth \paren {a + b i} = \dfrac {\cosh a \cos b + i \sinh a \sin b} {\sinh a \cos b + i \cosh a \sin b}$
where:
- $\coth$ denotes the hyperbolic 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.
Proof
\(\ds \coth \paren {a + b i}\) | \(=\) | \(\ds \frac {\cosh \paren {a + b i} } {\sinh \paren {a + b i} }\) | Definition of Hyperbolic Cotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\cosh a \cos b + i \sinh a \sin b} {\sinh a \cos b + i \cosh a \sin b}\) | Hyperbolic Sine of Complex Number and Hyperbolic Cosine of Complex Number |
$\blacksquare$