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,](/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](/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.](/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.Referenced on Wolfram|Alpha
Hyperbolic Secant
Cite this as:
Weisstein, Eric W. "Hyperbolic Secant."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicSecant.html
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