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The hyperbolic secant is defined as

			(1)
			(2)

where  is the hyperbolic cosine. It is implemented in the Wolfram Language as Sech[z].

On the real line, it has a maximum at  and inflection points at  (OEIS A091648). It has a fixed point at  (OEIS A069814).

The derivative is given by

(3)

where  is the hyperbolic tangent, and the indefinite integral by

(4)

where  is a constant of integration.

 has the Taylor series

			(5)
			(6)

(OEIS A046976 and A046977), where  is an Euler number and  is a factorial.

Equating coefficients of , , and  in the Ramanujan cos/cosh identity

(7)

gives the amazing identities

{(sqrt(pi))/([Gamma(3/4)]^2)-1} sum_(n=1)^inftyn^4sech(pin)=(18[Gamma(3/4)]^2)/(sqrt(pi))[sum_(n=1)^inftyn^2sech(pin)]^2 sum_(n=1)^inftyn^8sech(pin)=(168[Gamma(3/4)]^2)/(sqrt(pi))[sum_(n=1)^inftyn^2sech(pin)]×sum_(n=1)^inftyn^6sech(pin)-(63000[Gamma(3/4)]^6)/(pi^(3/2))[sum_(n=1)^inftyn^2sech(pin)]^4. " src="http://mathworld.wolfram.com/images/equations/HyperbolicSecant/NumberedEquation4.gif" style="height:230px; width:506px" />

(8)

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.

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 A046976, A046977, A069814, and A091648 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.