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Shi

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The Shi function, also called the hyperbolic sine integral, is defined for a complex number z by

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

where the integrand is assigned its removable value 1 at t=0.

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

It has Maclaurin series

Shi(z)=sum_(n=0)^(infty)(z^(2n+1))/((2n+1)^2(2n)!)
(2)
=z+1/(18)z^3+1/(600)z^5+1/(35280)z^7+1/(3265920)z^9+...
(3)

(OEIS A061079).

It has derivative

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

for z!=0, with derivative 1 at z=0, and indefinite integral

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

See also

Chi, Cosine Integral, Sine Integral, Sinhc Function

Related Wolfram sites

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

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Sine and Cosine Integrals." §5.2 in Handbook 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 Resource. https://mathworld.wolfram.com/Shi.html

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