Real Area Hyperbolic Cotangent of x over a in Logarithm Form

Theorem

$\arcoth \dfrac x a = \dfrac 1 2 \map \ln {\dfrac {x + a} {x - a} }$

Proof

\(\ds \arcoth \frac x a\)

\(=\)

\(\ds \frac 1 2 \map \ln {\frac {\frac x a + 1} {\frac x a - 1} }\)

Definition of Real Area Hyperbolic Cotangent

\(\ds \)

\(=\)

\(\ds \frac 1 2 \map \ln {\frac {x + a} {x - a} }\)

multiplying top and bottom by $a$

$\blacksquare$

Also see

• $\arsinh \dfrac x a$ in Logarithm Form

• $\arcosh \dfrac x a$ in Logarithm Form

• $\artanh \dfrac x a$ in Logarithm Form

• $\arsech \dfrac x a$ in Logarithm Form

• $\arcsch \dfrac x a$ in Logarithm Form