Hyperbolic Cosine
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The hyperbolic cosine is defined as
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(1)
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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
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(2)
|
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(3)
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where 
is the golden
ratio.
The derivative is given by
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(4)
|
where 
is the hyperbolic
sine, and the indefinite integral by
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(5)
|
where 
is a constant
of integration.
The hyperbolic cosine has Taylor series
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(6)
|
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(7)
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(OEIS A010050).
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.
Weisstein, Eric W. "Hyperbolic Cosine." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HyperbolicCosine.html
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