Arcsecant Logarithmic Formulation

Theorem

Let $x$ be a real number.

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

Then:

$\ds \arcsec x = -i \map \Ln {i \sqrt {1 - \frac 1 {x^2} } + \frac 1 x}$

where:

$\arcsec$ is the arcsecant function

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

Proof

\(\ds \arcsec x\)

\(=\)

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

Arcsecant of Reciprocal equals Arccosine

\(\ds \)

\(=\)

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

Arccosine Logarithmic Formulation

\(\ds \)

\(=\)

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

$\blacksquare$

Also see

• Arcsine Logarithmic Formulation
• Arccosine Logarithmic Formulation
• Arctangent Logarithmic Formulation
• Arccotangent Logarithmic Formulation
• Arccosecant Logarithmic Formulation