Min Max

Min   Max       Re       Im

The hyperbolic sine is defined as

(1)

The notation  is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix). It is implemented in the Wolfram Language as Sinh[z].

Special values include

			(2)
			(3)

where  is the golden ratio.

The value

(4)

(OEIS A073742) has Engel expansion 1, 6, 20, 42, 72, 110, ... (OEIS A068377), which has closed form for .

The derivative is given by

(5)

where  is the hyperbolic cosine, and the indefinite integral by

(6)

where  is a constant of integration.

 has the Taylor series

			(7)
			(8)

(OEIS A009445).

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.

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. Sequences A009445, A068377, and A073742 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.