Sine Integral
المؤلف:
Abramowitz, M. and Stegun, I. A
المصدر:
"Sine and Cosine Integrals." §5.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
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25-11-2018
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Sine Integral
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
![Si(2x)=2xsum_(n=0)^infty[j_n(x)]^2](http://mathworld.wolfram.com/images/equations/SineIntegral/NumberedEquation6.gif) |
(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.
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