Hyperbolic Cosecant

The hyperbolic cosecant is defined as

(1)

It is implemented in the Wolfram Language as Csch[z].

It is related to the hyperbolic cotangent though

(2)

The derivative is given by

(3)

where  is the hyperbolic cotangent, and the indefinite integral by

(4)

where  is a constant of integration.

It has Taylor series

			(5)
			(6)
			(7)

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

Sums include

			(8)
			(9)

(OEIS A110191; Berndt 1977).

The plot above shows a bifurcation diagram for .

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). "Hyperbolic Functions." §4.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 83-86, 1972.

Berndt, B. C. "Modular Transformations and Generalizations of Several Formulae of Ramanujan." Rocky Mtn. J. Math. 7, 147-189, 1977.

Jeffrey, A. "Hyperbolic Identities." §2.5 in Handbook of Mathematical Formulas and Integrals, 2nd ed. Orlando, FL: Academic Press, pp. 117-122, 2000.

Sloane, N. J. A. Sequences A036280, A036281, and A110191 in "The On-Line Encyclopedia of Integer Sequences."

Spanier, J. and Oldham, K. B. "The Hyperbolic Secant  and Cosecant  Functions." Ch. 29 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 273-278, 1987.

Zwillinger, D. (Ed.). "Hyperbolic Functions." §6.7 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 476-481 1995.