The most common "sine integral" is defined as
|
(1)
|
is the function implemented in the
Wolfram Language as the function SinIntegral[z].
is an entire
function.
A closed related function is defined by
where
is the exponential integral, (3)
holds for ,
and
|
(6)
|
The derivative of is
|
(7)
|
where
is the sinc function and the integral
is
|
(8)
|
A series for
is given by
|
(9)
|
(Havil 2003, p. 106).
It has an expansion in terms of spherical
Bessel functions of the first kind as
|
(10)
|
(Harris 2000).
The half-infinite integral of the sinc function
is given by
|
(11)
|
To compute the integral of a sine function times a power
|
(12)
|
use integration by parts. Let
|
(13)
|
|
(14)
|
so
|
(15)
|
Using integration by parts again,
|
(16)
|
|
(17)
|
|
(18)
|
Letting ,
so
|
(19)
|
General integrals of the form
|
(20)
|
are related to the sinc function and can be computed
analytically.
See also
Chi,
Cosine Integral,
Exponential Integral,
Nielsen's
Spiral,
Shi,
Sinc Function
Related Wolfram sites
http://functions.wolfram.com/GammaBetaErf/SinIntegral/
Explore with Wolfram|Alpha
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.Arfken, G. Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 342-343,
1985.Harris, F. E. "Spherical Bessel Expansions of Sine, Cosine,
and Exponential Integrals." Appl. Numer. Math. 34, 95-98, 2000.Havil,
J. Gamma:
Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 105-106,
2003.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.;
and Vetterling, W. T. "Fresnel Integrals, Cosine and Sine Integrals."
§6.79 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 248-252, 1992.Spanier, J. and Oldham,
K. B. "The Cosine and Sine Integrals." Ch. 38 in An
Atlas of Functions. Washington, DC: Hemisphere, pp. 361-372, 1987.Referenced
on Wolfram|Alpha
Sine Integral
Cite this as:
Weisstein, Eric W. "Sine Integral." From
MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SineIntegral.html
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