The inverse hyperbolic secant (Beyer 1987, p. 181; Zwillinger 1995, p. 481),
sometimes called the area hyperbolic secant (Harris and Stocker 1998, p. 271)
and sometimes also denoted (Jeffrey 2000, p. 124), is the multivalued
function that is the inverse function of
the hyperbolic secant. The variants or (Harris and Stocker 1998, p. 263) are sometimes
used to refer to explicit principal values of
the inverse hyperbolic secant, although this distinction is not always made. Worse
yet, the notation is sometimes used for the principal value, with
being used for the multivalued function (Abramowitz and Stegun 1972, p. 87).
Note that in the notation , is the hyperbolic secant
and the superscript denotes an inverse function,
not the multiplicative inverse.