Hyperbolic Cosine

Min Max

Min   Max       Re       Im

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

			(2)
			(3)

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

			(6)
			(7)

(OEIS A010050).

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.