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 themultivalued function that is theinverse function of thehyperbolic 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 explicitprincipal 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 thehyperbolic tangent and the superscript denotes aninverse function,not the multiplicative inverse.

Theprincipal value of is implemented in theWolfram Language asArcTanh[z] and in the GNU C library asatanh(double x).

The inverse hyperbolic tangent is amultivalued function and hence requires abranch cut in thecomplex plane, which theWolfram 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 theinversetangent by

(2)

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

(3)

Thederivative of the inverse hyperbolic tangent is

(4)

and theindefinite integral is

(5)

It has special values

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

It hasMaclaurin series

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

(OEISA005408).

Anindefinite integral involving is given by

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

when.

See also

Related Wolfram sites

Explore with Wolfram|Alpha

More things to try:

• arctanh (infinity)
• integrate arctanh x
• differentiate arctanh x

References

Referenced on Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Inverse Hyperbolic Tangent."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/InverseHyperbolicTangent.html

Subject classifications