Inverse Hyperbolic Tangent

Min Max Re Im

The inverse hyperbolic tangent (Zwillinger 1995, p. 481; Beyer 1987, p. 181), sometimes called the area hyperbolic tangent (Harris and Stocker 1998, p. 267), is the multivalued function that is the inverse function of the hyperbolic tangent.

The function is sometimes denoted (Jeffrey 2000, p. 124) or (Gradshteyn and Ryzhik 2000, p. xxx). The variants or (Harris and Stocker 1998, p. 263) are sometimes used to refer to explicit principal values of the inverse hyperbolic tangent, 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 tangent and the superscript denotes an inverse function, not the multiplicative inverse.

The principal value of is implemented in the Wolfram Language as ArcTanh[z] and in the GNU C library as atanh(double x).

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

(2)

(Gradshteyn and Ryzhik 2000, p. xxx). For real , this simplifies to

(3)

The derivative of the inverse hyperbolic tangent is

(4)

and the indefinite integral is

(5)

It has special values

			(6)
			(7)
			(8)
			(9)

It has Maclaurin series

			(10)
			(11)
			(12)
			(13)

(OEIS A005408).

An indefinite integral involving is given by

			(14)
			(15)
			(16)
			(17)

when .

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