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Hyperbolic Secant

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Sech
SechReIm
SechContours

The hyperbolic secant is defined as

sechz=1/(coshz)
(1)
=2/(e^z+e^(-z)),
(2)

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

On the real line, it has a maximum at x=0 and inflection points at x=+/-cosh^(-1)(sqrt(2))=+/-0.8813735... (OEIS A091648). It has a fixed point at x=0.76500995... (OEIS A069814).

The derivative is given by

d/(dz)sechz=-sechztanhz,
(3)

where tanhz is the hyperbolic tangent, and the indefinite integral by

intsechzdz=2tan^(-1)[tanh(1/2z)]+C,
(4)

where C is a constant of integration.

sechz has the Taylor series

sechz=sum_(n=0)^(infty)(E_(2n))/((2n)!)z^(2n)
(5)
=1-1/2z^2+5/(24)z^4-(61)/(720)z^6+(277)/(8064)z^8-...
(6)

(OEIS A046976 and A046977) for |z|<1/2pi, where E_n is an Euler number and n! is a factorial.

Equating coefficients of theta^0, theta^4, and theta^8 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
(7)

for real theta with |theta|<pi gives the identities

sum_(n=1)^(infty)sech(pin)=1/2{(sqrt(pi))/([Gamma(3/4)]^2)-1}
(8)
sum_(n=1)^(infty)n^4sech(pin)=(18[Gamma(3/4)]^2)/(sqrt(pi))[sum_(n=1)^(infty)n^2sech(pin)]^2
(9)
sum_(n=1)^(infty)n^8sech(pin)=(168[Gamma(3/4)]^2)/(sqrt(pi))[sum_(n=1)^(infty)n^2sech(pin)]sum_(n=1)^(infty)n^6sech(pin)-(63000[Gamma(3/4)]^6)/(pi^(3/2))[sum_(n=1)^(infty)n^2sech(pin)]^4.
(10)

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 sech(x) and Cosecant csch(x) 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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Hyperbolic Secant

Cite this as:

Weisstein, Eric W. "Hyperbolic Secant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HyperbolicSecant.html

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