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Inverse Hyperbolic Tangent

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The inverse hyperbolic tangent tanh^(-1)z (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 arctanhz (Jeffrey 2000, p. 124) or Arthz (Gradshteyn and Ryzhik 2000, p. xxx). The variants Arctanhz or Artanhz (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 arctanhz is sometimes used for the principal value, with Arctanhz being used for the multivalued function (Abramowitz and Stegun 1972, p. 87). Note that in the notation tanh^(-1)z, tanhz is the hyperbolic tangent and the superscript -1 denotes an inverse function, not the multiplicative inverse.

The principal value of tanh^(-1)z is implemented in the Wolfram Language as ArcTanh[z] and in the GNU C library as atanh(double x).

InverseHyperbolicTangentBranchCut

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 (-infty,-1] and [1,infty). This follows from the definition of tanh^(-1)z as

tanh^(-1)z=1/2[ln(1+z)-ln(1-z)].
(1)

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

tanh^(-1)z=1/itan^(-1)(iz)
(2)

(Gradshteyn and Ryzhik 2000, p. xxx). For real x<1, this simplifies to

tanh^(-1)x=1/2ln((1+x)/(1-x)).
(3)

The derivative of the inverse hyperbolic tangent is

d/(dz)tanh^(-1)z=1/(1-z^2),
(4)

and the indefinite integral is

inttanh^(-1)zdz=ztanh^(-1)z+1/2ln(z^2-1)+C.
(5)

It has special values

tanh^(-1)0=0
(6)
tanh^(-1)1=infty
(7)
tanh^(-1)infty=-1/2pii
(8)
tanh^(-1)i=1/4pii.
(9)

It has Maclaurin series

tanh^(-1)z=sum_(n=1)^(infty)(z^(2n-1))/(2n-1)
(10)
=z+1/3z^3+1/5z^5+1/7z^7+1/9z^9+...
(11)
tanh^(-1)z=-1/2pii+sum_(n=1)^(infty)(z^(-2n+1))/(2n-1)
(12)
=-1/2pii+z+1/3z^3+1/5z^5+1/7z^7+...
(13)

(OEIS A005408).

An indefinite integral involving tanh^(-1)z is given by

int(dx)/(xsqrt(a+bx))=ln[(sqrt(a+bx)-sqrt(a))/(sqrt(a+bx)+sqrt(a))]
(14)
=ln[((sqrt(a+bx)-sqrt(a))^2)/((a+bx)-a)]
(15)
=ln[((2a+bx)-2sqrt(a(a+bx)))/(bx)]
(16)
=2tanh^(-1)(-sqrt(a/(a+bx)))
(17)

when a>0.

See also

Hyperbolic Tangent, Inverse Hyperbolic Cotangent, Inverse Hyperbolic Functions

Related Wolfram sites

http://functions.wolfram.com/ElementaryFunctions/ArcTanh/

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Inverse Hyperbolic Functions." §4.6 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 86-89, 1972.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 142-143, 1987.GNU C Library. "Mathematics: Inverse Trigonometric Functions." http://www.gnu.org/manual/glibc-2.2.3/html_chapter/libc_19.html#SEC391.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. xxx, 2000.Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, 1998.Jeffrey, A. "Inverse Trigonometric and Hyperbolic Functions." §2.7 in Handbook of Mathematical Formulas and Integrals, 2nd ed. Orlando, FL: Academic Press, pp. 124-128, 2000.Sloane, N. J. A. Sequence A005408/M2400 in "The On-Line Encyclopedia of Integer Sequences."Spanier, J. and Oldham, K. B. "Inverse Trigonometric Functions." Ch. 35 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 331-341, 1987.Zwillinger, D. (Ed.). "Inverse Hyperbolic Functions." §6.8 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 481-483, 1995.

Referenced on Wolfram|Alpha

Inverse Hyperbolic Tangent

Cite this as:

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

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