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Inverse Hyperbolic Cotangent


ArcCoth
ArcCothReImAbs
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The inverse hyperbolic cotangent coth^(-1)z (Beyer 1987, p. 181; Zwillinger 1995, p. 481), sometimes called the area hyperbolic cotangent (Harris and Stocker 1998, p. 267), is the multivalued function that is the inverse function of the hyperbolic cotangent.

The variants arccothz and Arcothz (Harris and Stocker 1998, p. 263) are sometimes used to refer to explicit principal values of the inverse hyperbolic cotangent, although this distinction is not always made. Worse yet, the notation arccothz is sometimes used for the principal value, with arccothz being used for the multivalued function (Abramowitz and Stegun 1972, p. 87). The function is sometimes denoted arccothz (Jeffrey 2000, p. 124) or Arcthz (Gradshteyn and Ryzhik 2000, p. xxx). Note that in the notation coth^(-1)z, cothz is the hyperbolic tangent and the superscript -1 denotes an inverse function, not the multiplicative inverse.

The principal value of coth^(-1)z is implemented in the Wolfram Language as ArcCoth[z]

InverseHyperbolicCotangentBranchCut

The inverse hyperbolic cotangent is a multivalued function and hence requires a branch cut in the complex plane, which the Wolfram Language's convention places at the line segment [-1,1]. This follows from the definition of coth^(-1)z as

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

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

 coth^(-1)z=1/icot^(-1)(-iz)
(2)

(Gradshteyn and Ryzhik 2000, p. xxx). For x<0 or x>1, this simplifies to

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

The derivative is

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

and its indefinite integral is

 intcoth^(-1)zdz=zcoth^(-1)z+1/2ln(z^2-1).
(5)

It has the special values

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

It has series expansions

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

(OEIS A005408).


See also

Hyperbolic Cotangent, Inverse Hyperbolic Functions, Inverse Hyperbolic Tangent

Related Wolfram sites

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

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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.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 Cotangent

Cite this as:

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

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