Hyperbolic Secant (original) (raw)
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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,](http://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](http://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.](http://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
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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.
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Cite this as:
Weisstein, Eric W. "Hyperbolic Secant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HyperbolicSecant.html