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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.

Chi

,

Cosine Integral

,

Exponential Integral

,

Nielsen’s
Spiral

,

Shi

,

Sinc Function

## Related Wolfram sites

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

## 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.

Sine Integral

## Cite this as:

Weisstein, Eric W. “Sine Integral.” From
MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/SineIntegral.html

## Subject classifications

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