Min   Max       Re       Im

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 multiplicative inverse.

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

(1)

The derivative of  is

(2)

which simplifies to

(3)

for . The indefinite integral is

(4)

which simplifies to

(5)

for .

The inverse secant has a Taylor series about infinity of

			(6)
			(7)

(OEIS A055786 and A002595).

The inverse secant satisfies

(8)

for , and

			(9)
			(10)
			(11)

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

		{pi+csc^(-1)(x/(sqrt(x^2-1))) for x<-1; csc^(-1)(x/(sqrt(x^2-1))) for x>1" src="http://mathworld.wolfram.com/images/equations/InverseSecant/Inline34.gif" style="height:120px; width:203px" />	(12)
		{pi-cot^(-1)(1/(sqrt(x^2-1))) for x<-1; cot^(-1)(1/(sqrt(x^2-1))) for x>1" src="http://mathworld.wolfram.com/images/equations/InverseSecant/Inline37.gif" style="height:120px; width:202px" />	(13)
		{pi+sin^(-1)((sqrt(x^2-1))/x) for x<-1; sin^(-1)((sqrt(x^2-1))/x) for x>1" src="http://mathworld.wolfram.com/images/equations/InverseSecant/Inline40.gif" style="height:122px; width:201px" />	(14)
		{pi-tan^(-1)(sqrt(x^2-1)) for x<-1; tan^(-1)(sqrt(x^2-1)) for x>1." src="http://mathworld.wolfram.com/images/equations/InverseSecant/Inline43.gif" style="height:76px; width:197px" />	(15)

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.