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Shi

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The hyperbolic sine integral, often called the "Shi function" for short, is defined by

Shi(z)=int_0^z(sinht)/tdt.
(1)

The function is implemented in the Wolfram Language as the function SinhIntegral[z].

It has Maclaurin series

Shi(z)=sum_(n=0)^(infty)(x^(2n+1))/((2n+1)^2(2n)!)
(2)
=z+1/(18)z^3+1/(600)z^5+1/(35280)z^7+1/(326592)z^9+...
(3)

(OEIS A061079).

It has derivative

(dShi(z))/(dz)=(sinhz)/z
(4)

and indefinite integral

intShi(z),dz=zShi(z)-coshz.
(5)

See also

Chi, Cosine Integral, Sine Integral, Sinhc Function

Related Wolfram sites

http://functions.wolfram.com/GammaBetaErf/SinhIntegral/

Explore with Wolfram|Alpha

References

Abramowitz, M. and Stegun, I. A. (Eds.). "Sine and Cosine Integrals." §5.2 inHandbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 231-233, 1972.Sloane, N. J. A. Sequence A061079 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Shi

Cite this as:

Weisstein, Eric W. "Shi." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Shi.html

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