The hyperbolic cosine is defined as
(1)
The notation
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
where
is the golden ratio .
The derivative is given by
(4)
where
is the hyperbolic sine , and the indefinite
integral by
(5)
where
is a constant of integration .
The hyperbolic cosine has Taylor series
(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
Explore with Wolfram|Alpha
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 and Cosine 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. Referenced on Wolfram|Alpha Hyperbolic Cosine
Cite this as:
Weisstein, Eric W. "Hyperbolic Cosine."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicCosine.html
Subject classifications