Arccosecant Logarithmic Formulation

Theorem

Let $x$ be a real number.

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

Then:

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

where:

$\arccsc$ is the arccosecant function

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

Proof

\(\ds \arccsc x\)

\(=\)

\(\ds \map \arcsin {\frac 1 x}\)

Arccosecant of Reciprocal equals Arcsine

\(\ds \)

\(=\)

\(\ds -i \map \Ln {\sqrt {1 - \paren {\frac 1 x}^2} + i \times \frac 1 x}\)

Arcsine Logarithmic Formulation

\(\ds \)

\(=\)

\(\ds -i \map \Ln {\sqrt {1 - \frac 1 {x^2} } + \frac i x}\)

$\blacksquare$

Also see

• Arcsine Logarithmic Formulation
• Arccosine Logarithmic Formulation
• Arctangent Logarithmic Formulation
• Arccotangent Logarithmic Formulation
• Arcsecant Logarithmic Formulation