Arccosecant Logarithmic Formulation

Theorem
Let $x$ be a real number.

Let $x \in \hointl {-\infty} {-1} \cup \hointr 1 {\infty}$.

Then:


 * $\displaystyle \arccsc x = -i \map \Ln {\sqrt {1 - \frac 1 {x^2} } + \frac i x}$

where:


 * $\arccsc$ is the arccosecant function
 * $\Ln$ is the principal branch of the complex logarithm whose imaginary part lies in $\hointl {-\pi} \pi$.

Also see

 * Arcsine Logarithmic Formulation
 * Arccosine Logarithmic Formulation
 * Arctangent Logarithmic Formulation
 * Arccotangent Logarithmic Formulation
 * Arcsecant Logarithmic Formulation