The inverse hyperbolic sine (Beyer 1987, p. 181; Zwillinger 1995, p. 481),
sometimes called the area hyperbolic sine (Harris and Stocker 1998, p. 264)
is the multivalued function that is the inverse function of the hyperbolic
sine.
The variants
or
(Harris and Stocker 1998, p. 263) are sometimes used to refer to explicit principal values of the inverse hyperbolic sine,
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). The notations (Jeffrey 2000, p. 124) and (Gradshteyn and Ryzhik 2000, p. xxx) are sometimes
also used. Note that in the notation , is the hyperbolic sine
and the superscript denotes an inverse function,
not the multiplicative inverse.
(Beyer 1987, p. 181; Zwillinger 1995, p. 481),
sometimes called the area hyperbolic sine (Harris and Stocker 1998, p. 264)
is the multivalued function that is the inverse function of the hyperbolic
sine.
or 
(Harris and Stocker 1998, p. 263) are sometimes used to refer to explicit principal values of the inverse hyperbolic sine,
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). The notations 
(Jeffrey 2000, p. 124) and 
(Gradshteyn and Ryzhik 2000, p. xxx) are sometimes
also used. Note that in the notation 
, 
is the hyperbolic sine
and the superscript 
denotes an inverse function,
not the multiplicative inverse.
is implemented in the Wolfram
Language as ArcSinh[z]
and in the GNU C library as asinh(double x).
and 
. This follows from the definition of 
as
is a Legendre polynomial. It has a Taylor
series about infinity of
