Definition:Inverse Hyperbolic Cotangent/Complex/Definition 2

From ProofWiki
Jump to navigation Jump to search


Let $S$ be the subset of the complex plane:

$S = \C \setminus \set {-1 + 0 i, 1 + 0 i}$

The inverse hyperbolic cotangent is a multifunction defined on $S$ as:

$\forall z \in S: \map {\coth^{-1} } z := \set {\dfrac 1 2 \map \ln {\dfrac {z + 1} {z - 1} } + k \pi i: k \in \Z}$

where $\ln$ denotes the complex natural logarithm considered as a multifunction.

Also known as

The principal branch of the inverse hyperbolic cotangent is also known as the area hyperbolic cotangent, as it can be used, among other things, for evaluating areas of regions bounded by hyperbolas.

Some sources refer to it as hyperbolic arccotangent, but this is strictly a misnomer, as there is nothing arc related about an inverse hyperbolic cotangent.

Also see