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The inverse secant (Zwillinger 1995, p. 465), also denoted (Abramowitz and Stegun 1972, p. 79; Harris and Stocker 1998, p. 315; Jeffrey 2000, p. 124), is the inverse function of the secant. The variants (Beyer 1987, p. 141) and are sometimes used to indicate the principal value, although this distinction is not always made (e.g., Zwillinger 1995, p. 466). Worse yet, the notation is sometimes used for the principal value, with being used for the multivalued function (Abramowitz and Stegun 1972, p. 80). In the notation (commonly used in North America and in pocket calculators worldwide), is the secant and the superscript denotes the inverse function, not the reciprocal.

The principal value of the inverse secant is implemented as ArcSec[z] in the Wolfram Language.

The inverse secant is a multivalued function and hence requires a branch cut in the complex plane, which the Wolfram Language's convention places at . This follows from the definition of as

The derivative of is

which simplifies to

for . The indefinite integral is

which simplifies to

for .

The inverse secant has a Laurent series about infinity of

for (OEIS A055786 and A002595).

The inverse secant satisfies

for , and

for all complex number arguments . It is given in terms of other inverse trigonometric functions by

See also

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

Related Wolfram sites

https://functions.wolfram.com/ElementaryFunctions/ArcSec/

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References

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. 141-143, 1987.Bronshtein, I. N. and Semendyayev, K. A. Handbook of Mathematics, 3rd ed. New York: Springer-Verlag, 1997.Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 315, 1998.Jeffrey, A. Handbook of Mathematical Formulas and Integrals, 2nd ed. Orlando, FL: Academic Press, 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.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, 1995.

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Inverse Secant

Cite this as:

Weisstein, Eric W. "Inverse Secant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/InverseSecant.html

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