Real Area Hyperbolic Cosecant of x over a in Logarithm Form

Theorem
For $a > 0$:


 * $\csch^{-1} \dfrac x a = \map \ln {\dfrac {a + \sqrt {a^2 + x^2} } {\size x} }$

Proof
We have that $\csch^{-1} \dfrac x a$ is defined whenever $x \ne 0$.

Also see

 * $\sinh^{-1} \dfrac x a$ in Logarithm Form


 * $\cosh^{-1} \dfrac x a$ in Logarithm Form


 * $\tanh^{-1} \dfrac x a$ in Logarithm Form


 * $\coth^{-1} \dfrac x a$ in Logarithm Form


 * $\sech^{-1} \dfrac x a$ in Logarithm Form