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).
See also
Beta Exponential Function,
Bipolar Coordinates,
Bipolar
Cylindrical Coordinates,
Bispherical Coordinates,
Catenary,
Catenoid,
Conical Function,
Cubic
Equation,
de Moivre's Identity,
Dixon-Ferrar
Formula,
Elliptic Cylindrical
Coordinates,
Elsasser Function,
Gudermannian,
Helicoid,
Helmholtz
Differential Equation--Elliptic Cylindrical Coordinates,
Hyperbolic
Cosecant,
Hyperbolic Functions,
Inverse
Hyperbolic Sine,
Laplace's
Equation--Bispherical Coordinates,
Laplace's
Equation--Toroidal Coordinates,
Lebesgue Constants,
Lorentz Group,
Mercator
Projection,
Miller Cylindrical Projection,
Modified Bessel Function of
the Second Kind,
Modified
Spherical Bessel Function of the First Kind,
Modified
Struve Function,
Nicholson's Formula,
Oblate Spheroidal Coordinates,
Parabola
Involute,
Partition Function P,
Poinsot's
Spirals,
Prolate Spheroidal Coordinates,
Schläfli's Formula,
Shi,
Sine,
Sine-Gordon Equation,
Surface of Revolution,
Tau
Function,
Toroidal Coordinates,
Toroidal
Function,
Tractrix,
Watson's
Formula
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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.Referenced on Wolfram|Alpha
Hyperbolic Sine
Cite this as:
Weisstein, Eric W. "Hyperbolic Sine."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicSine.html
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