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Hyperbolic Cosine

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The hyperbolic cosine is defined as

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

The notation chx is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix). This function describes the shape of a hanging cable, known as the catenary. It is implemented in the Wolfram Language as Cosh[z].

Special values include

cosh0=1
(2)
cosh(lnphi)=1/2sqrt(5),
(3)

where phi is the golden ratio.

The derivative is given by

d/(dz)coshz=sinhz,
(4)

where sinhz is the hyperbolic sine, and the indefinite integral by

intcoshzdz=sinhz+C,
(5)

where C is a constant of integration.

The hyperbolic cosine has Taylor series

coshz=sum_(n=0)^(infty)(z^(2n))/((2n)!)
(6)
=1+1/2z^2+1/(24)z^4+1/(720)z^6+1/(40320)z^8+...
(7)

(OEIS A010050).

See also

Bipolar Coordinates, Bipolar Cylindrical Coordinates, Bispherical Coordinates, Catenary, Catenoid, Chi, Conical Function, Correlation Coefficient--Bivariate Normal Distribution, Cosine, Cubic Equation, de Moivre's Identity, Elliptic Cylindrical Coordinates, Elsasser Function, Hyperbolic Functions, Hyperbolic Geometry, Hyperbolic Lemniscate Function, Hyperbolic Sine, Hyperbolic Secant, Hyperbolic Tangent, Inversive Distance, Inverse Hyperbolic Cosine, Laplace's Equation--Bipolar Coordinates, Laplace's Equation--Bispherical Coordinates, Laplace's Equation--Toroidal Coordinates, Lemniscate Function, Lorentz Group, Mathieu Differential Equation, Mehler's Bessel Function Formula, Mercator Projection, Modified Bessel Function of the First Kind, Oblate Spheroidal Coordinates, Prolate Spheroidal Coordinates, Pseudosphere, Ramanujan Cos/Cosh Identity, Sine-Gordon Equation, Surface of Revolution, Toroidal Coordinates

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Hyperbolic Functions." §4.5 inHandbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 83-86, 1972.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.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. Sequence A010050 in "The On-Line Encyclopedia of Integer Sequences."Spanier, J. and Oldham, K. B. "The Hyperbolic Sine sinh(x) and Cosine cosh(x) Functions." Ch. 28 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 263-271, 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 Cosine

Cite this as:

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

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