Inverse Secant -- From Wolfram MathWorld

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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 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

		${pi+csc^(-1)(x/(sqrt(x^2-1))) for x<-1; csc^(-1)(x/(sqrt(x^2-1))) for x>1$	(12)
		${pi-cot^(-1)(1/(sqrt(x^2-1))) for x<-1; cot^(-1)(1/(sqrt(x^2-1))) for x>1$	(13)
		${pi+sin^(-1)((sqrt(x^2-1))/x) for x<-1; sin^(-1)((sqrt(x^2-1))/x) for x>1$	(14)
		${pi-tan^(-1)(sqrt(x^2-1)) for x<-1; tan^(-1)(sqrt(x^2-1)) for x>1.$	(15)