Inverse Cosecant

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The inverse cosecant is the multivalued function csc^(-1)z (Zwillinger 1995, p. 465), also denoted arccscz (Abramowitz and Stegun 1972, p. 79; Spanier and Oldham 1987, p. 332; Harris and Stocker 1998, p. 315; Jeffrey 2000, p. 125), that is the inverse function of the cosecant. The variants Arccscz (e.g., Beyer 1987, p. 141; Bronshtein and Semendyayev, 1997, p. 70) and Csc^(-1)z are sometimes used to refer to explicit principal values of the inverse cosecant, although this distinction is not always made (e.g,. Zwillinger 1995, p. 466). Worse yet, the notation arccscz is sometimes used for the principal value, with Arccscz being used for the multivalued function (Abramowitz and Stegun 1972, p. 80). Note that in the notation csc^(-1)z (commonly used in North America and in pocket calculators worldwide), cscz is the cosecant and the superscript -1 denotes an inverse function, not the multiplicative inverse.

The principal value of the inverse cosecant is implemented as ArcCsc[x] in the Wolfram Language.


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


The derivative of csc^(-1)z is given by


which simplifies to


for x>0. Its indefinite integral is


which simplifies to


for x>0.

The inverse cosecant has Taylor series about infinity of


(OEIS A055786 and A002595), where P_n(x) is a Legendre polynomial and (x)_n is a Pochhammer symbol.

The inverse cosecant satisfies


for z!=0,


for all complex z, and

csc^(-1)x={sec^(-1)(x/(sqrt(x^2-1)))-pi for x<-1; sec^(-1)(x/(sqrt(x^2-1))) for x>1
={-cos^(-1)((sqrt(x^2-1))/x) for x<-1; cos^(-1)((sqrt(x^2-1))/x) for x>1
={-cot^(-1)(sqrt(x^2-1)) for x<-1; cot^(-1)(sqrt(x^2-1)) for x>1.

See also

Cosecant, Inverse Cosine, Inverse Cotangent, Inverse Secant, Inverse Sine, Inverse Tangent, Inverse Trigonometric Functions, Sine

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Abramowitz, M. and Stegun, I. A. (Eds.). "Inverse Circular Functions." §4.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 79-83, 1972.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 142-143, 1987.Bronshtein, I. N. and Semendyayev, K. A. Handbook of Mathematics, 3rd ed. New York: Springer-Verlag, 1997.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 315, 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 A002595/M4233 and A055786 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 Circular Functions." §6.3 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 465-467, 1995.

Referenced on Wolfram|Alpha

Inverse Cosecant

Cite this as:

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

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