Cosecant

Min Max Re Im

The cosecant is the function defined by

			(1)
			(2)

where is the sine. The cosecant is implemented in the Wolfram Language as Csc[z].

The notation is sometimes also used (Gellert et al. 1989, p. 222; Gradshteyn and Ryzhik 2000, p. xxix). Note that the cosecant does not appear to be in consistent widespread use in Europe, although it does appear explicitly in various German and Russian handbooks (e.g., Gellert et al. 1989, p. 222; Gradshteyn and Ryzhik 2000, pp. xxix and p. 43). Interestingly, while is treated on par with the other trigonometric functions in some tabulations (Gellert et al. 1989, p. 222), it is not in others (Gradshteyn and Ryzhik 2000, who do not list it in their table of "basic functional relations" on p. 28, but do give identities involving it on p. 43).

Harris and Stocker (1998, p. 300) call secant and cosecant "rarely used functions," but then devote an entire section to them. Because these functions do seem to be in widespread use in the United States (e.g., Abramowitz and Stegun 1972, p. 72), reports of their demise seem to be a bit premature.

The derivative is

(3)

and the indefinite integral is

(4)

where is a constant of integration. For on the real axis, this simplifies to

			(5)
			(6)

The Laurent series of the cosecant function is

			(7)
			(8)

(OEIS A036280 and A036281), where is a Bernoulli number.

The positive integer values of giving incrementally largest values of are given by 1, 3, 22, 333, 355, 103993, ... (OEIS A046947), which are precisely the numerators of the convergents of and correspond to the values 1.1884, 7.08617, 112.978, 113.364, 33173.7, ....