Inverse Hyperbolic Cosecant

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The inverse hyperbolic cosecant csch^(-1)z (Zwillinger 1995, p. 481), sometimes called the area hyperbolic cosecant (Harris and Stocker 1998, p. 271) and sometimes denoted cosech^(-1)z (Beyer 1987, p. 181) or arccschz (Abramowitz and Stegun 1972, p. 87; Jeffrey 2000, p. 124), is the multivalued function that is the inverse function of the hyperbolic cosecant. The variants Arccschz (Abramowitz and Stegun 1972, p. 87) and Arcschz (Harris and Stocker 1998, p. 263) are sometimes used to refer to explicit principal values. Worse yet, the notation arccschz is sometimes used for the principal value, with Arccschz being used for the multivalued function (Abramowitz and Stegun 1972, p. 87). Note that in the notation csch^(-1)z, cschz is the hyperbolic cosecant and the superscript -1 denotes an inverse function, not the multiplicative inverse.

The inverse hyperbolic cosecant is a multivalued function and hence requires a branch cut in the complex plane, which the Wolfram Language's convention places at (-i,i).

The principal value of csch^(-1)z is implemented in the Wolfram Language as ArcCsch[z].

It has special value


where phi is the golden ratio.


The inverse hyperbolic cosecant 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 (-i,i). This follows from the definition of csch^(-1)z as


The derivative of the inverse hyperbolic cosecant is


and the indefinite integral is


For real x, it satisfies

 csch^(-1)x={ln((1-sqrt(1+x^2))/x)   for x<0; ln((1+sqrt(1+x^2))/x)   for x>0.

The inverse hyperbolic cosecant has Puiseux series


(OEIS A052468 and A052469) about 0, and Taylor series about infty of


(OEIS A055786 and A002595), where P_k(x) is a Legendre polynomial and (1/2)_n is a Pochhammer symbol.

See also

Hyperbolic Cosecant, Inverse Hyperbolic Functions

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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.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. Sequences A052468, A052469, A055786 and A002595/M4233 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 Cosecant

Cite this as:

Weisstein, Eric W. "Inverse Hyperbolic Cosecant." From MathWorld--A Wolfram Web Resource.

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