Hyperbolic Cosecant of Complex Number

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Theorem

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

Let $i$ be the imaginary unit.


Then:

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

where:

$\csch$ denotes the hyperbolic cosecant 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

\(\ds \map \csch {a + b i}\) \(=\) \(\ds \frac 1 {\map \sinh {a + b i} }\) Definition of Hyperbolic Cosecant
\(\ds \) \(=\) \(\ds \dfrac 1 {\sinh a \cos b + i \cosh a \sin b}\) Hyperbolic Sine of Complex Number
\(\ds \) \(=\) \(\ds \dfrac {\sinh a \cos b - i \cosh a \sin b} {\paren {\sinh a \cos b + i \cosh a \sin b} \paren {\sinh a \cos b - i \cosh a \sin b} }\) multiplying denominator and numerator by $\sinh a \cos b - i \cosh a \sin b$
\(\ds \) \(=\) \(\ds \dfrac {\sinh a \cos b - i \cosh a \sin b} {\sinh^2 a \cos^2 b - i^2 \cosh^2 a \sin^2 b}\) Difference of Two Squares
\(\ds \) \(=\) \(\ds \dfrac {\sinh a \cos b - i \cosh a \sin b} {\sinh^2 a \cos^2 b + \cosh^2 a \sin^2 b}\) Definition of Imaginary Unit

$\blacksquare$


Also see