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