Hyperbolic Sine
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
"Hyperbolic Functions." §4.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover,
الجزء والصفحة:
...
3-6-2019
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Hyperbolic Sine
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
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
(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.
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