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Hyperbolic Sine


SinhReal
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SinhReImAbs
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The hyperbolic sine is defined as

 sinhz=1/2(e^z-e^(-z)).
(1)

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

Special values include

sinh0=0
(2)
sinh(lnphi)=1/2,
(3)

where phi is the golden ratio.

The value

 sinh1=1.17520119...
(4)

(OEIS A073742) has Engel expansion 1, 6, 20, 42, 72, 110, ... (OEIS A068377), which has closed form 2n(2n+1) for n>1.

The derivative is given by

 d/(dz)sinhz=coshz,
(5)

where coshz is the hyperbolic cosine, and the indefinite integral by

 intsinhzdz=coshz+C,
(6)

where C is a constant of integration.

sinhz has the Taylor series

sinhz=sum_(n=0)^(infty)(z^(2n+1))/((2n+1)!)
(7)
=z+1/6z^3+1/(120)z^5+1/(5040)z^7+1/(362880)z^9+...
(8)

(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 sinh(x) and Cosine cosh(x) 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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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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