The inverse hyperbolic functions, sometimes also called the area hyperbolic functions (Spanier and Oldham 1987, p. 263) are the multivalued
function that are the inverse functions of
the hyperbolic functions. They are denoted
,
,
,
,
,
and .
Variants of these notations beginning with a capital letter are commonly used to
denote their principal values (e.g., Harris and
Stocker 1998, p. 263).

These functions are multivalued, and hence require branch cuts in the complex
plane. Differing branch cut conventions are possible, but those adopted in this
work follow those used by the Wolfram
Language, summarized below.

The inverse hyperbolic functions as defined in this work have the following ranges for domains on the real line ,
again following the convention of the Wolfram
Language.