The hyperbolic secant is defined as
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
![intsechzdz=2tan^(-1)[tanh(1/2z)]+C,](https://mathworld.wolfram.com/images/equations/HyperbolicSecant/NumberedEquation2.svg) | (4) |
where 
is a constant of integration.
has the Taylor series
(OEIS A046976 and A046977), where 
is an Euler number and 
is a factorial.
Equating coefficients of 
, 
, and 
in the Ramanujan cos/cosh identity
![[1+2sum_(n=1)^infty(cos(ntheta))/(cosh(npi))]^(-2)+[1+2sum_(n=1)^infty(cosh(ntheta))/(cosh(npi))]^(-2)=(2Gamma^4(3/4))/pi](https://mathworld.wolfram.com/images/equations/HyperbolicSecant/NumberedEquation3.svg) | (7) |
gives the amazing identities
![sum_(n=1)^inftysech(pin)=1/2{(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.](https://mathworld.wolfram.com/images/equations/HyperbolicSecant/NumberedEquation4.svg) | (8) |
See also
Benson's Formula,
Catenary,
Catenoid,
Euler Number,
Gaussian Function,
Hyperbolic Cosine,
Hyperbolic Functions,
Inverse Hyperbolic Secant,
Lorentzian Function,
Oblate Spheroidal Coordinates,
Pseudosphere,
Secant,
Surface of Revolution,
Tractrix,
Witch of Agnesi Explore with Wolfram|Alpha
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.Referenced on Wolfram|Alpha
Hyperbolic Secant Cite this as:
Weisstein, Eric W. "Hyperbolic Secant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HyperbolicSecant.html
Subject classifications
