Inverse Hyperbolic Sine -- From Wolfram MathWorld

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.

Its principal value of is implemented in the Wolfram Language as ArcSinh[z] and in the GNU C library as asinh(double x).

The inverse hyperbolic sine is a multivalued function and hence requires a branch cut in the complex plane, which the Wolfram Language's convention places at the line segments and . This follows from the definition of as

(1)

The inverse hyperbolic sine is given in terms of the inverse sine by

(2)

(Gradshteyn and Ryzhik 2000, p. xxx).

The derivative of the inverse hyperbolic sine is

(3)

and the indefinite integral is

(4)

It has a Maclaurin series

			(5)
			(6)
			(7)

(OEIS A055786 and A002595), where is a Legendre polynomial. It has a Taylor series about infinity of

			(8)
			(9)

(OEIS A052468 and A052469).