Inverse Sine

Min Max Re Im

The inverse sine is the multivalued function (Zwillinger 1995, p. 465), also denoted (Abramowitz and Stegun 1972, p. 79; Harris and Stocker 1998, p. 307; Jeffrey 2000, p. 124), that is the inverse function of the sine. The variants (e.g., Bronshtein and Semendyayev, 1997, p. 69) and are sometimes used to refer to explicit principal values of the inverse sine, 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). Note that in the notation (commonly used in North America and in pocket calculators worldwide), is the sine and the superscript denotes the inverse function, not the multiplicative inverse.

The principal value of the inverse sine is implemented as ArcSin[z] in the Wolfram Language. In the GNU C library, it is implemented as asin(double x).

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

(1)

Special values include

			(2)
			(3)
			(4)

The derivative of is

(5)

and its indefinite integral is

(6)

The inverse sine satisfies

(7)

for ,

			(8)
			(9)
			(10)

for all complex ,

			(11)
			(12)
			(13)
			(14)

and

			(15)
			(16)

for , where equality at points where the denominators are 0 is understood to mean in the limit as or , respectively.

The Maclaurin series for the inverse sine with is given by

			(17)
			(18)

(OEIS A055786 and A002595), where is a Pochhammer symbol.

The inverse sine can be given by the sum

(19)

where is a binomial coefficient (Borwein et al. 2004, p. 51; Borwein and Chamberland 2005; Bailey et al. 2007, pp. 15-16). Similarly,

			(20)
			(21)
			(22)

(Bailey et al. 2007, pp. 16 and 282; Borwein and Chamberland 2007). Ramanujan gave the cases for , 2, 3, and 4 (Berndt 1985, pp. 262-263), and the general cases are given in terms of multiple sums by Bailey et al. (2006, pp. 15-16 and 282) and Borwein and Chamberland (2007).

The inverse sine has continued fraction

(23)

(Wall 1948, p. 345).

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