Real Area Hyperbolic Cosecant of x over a in Logarithm Form

Theorem

For $a > 0$:

$\arcsch \dfrac x a = \map \ln {\dfrac {a + \sqrt {a^2 + x^2} } {\size x} }$

We have that $\arcsch \dfrac x a$ is defined whenever $x \ne 0$.

$\ds \arcsch \frac x a$

$=$

$\ds \map \ln {\frac 1 {x/a} + \frac {\sqrt {1 + \paren {\frac x a}^2} } {\size {\frac x a} } }$

$\ds $

$=$

$\ds \map \ln {\frac a x + \frac {a \paren {\sqrt {\dfrac {a^2 + x^2} {a^2} } } } {\size x} }$

$\ds $

$=$

$\ds \map \ln {\frac a x + \frac {a \paren {\dfrac {\sqrt {a^2 + x^2} } a} } {\size x} }$

$\ds $

$=$

$\ds \map \ln {\frac a x + \frac {\sqrt {a^2 + x^2} } {\size x} }$

$\blacksquare$