Sine Integral

Min Max

Min   Max       Re       Im

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

			(2)
			(3)
			(4)
			(5)

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.

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.