Cosine Integral

There are (at least) three types of cosine integrals,'' denoted , , and :

 (1) (2) (3) (4) (5) (6)

Here, is the Exponential Integral, E is the En-Function, and is the Euler-Mascheroni Constant. is the function returned by the Mathematica (Wolfram Research, Champaign, IL) command CosIntegral[x] and displayed above.

To compute the integral of an Even power times a cosine,

 (7)

use Integration by Parts. Let
 (8)

 (9)

so
 (10)

Using Integration by Parts again,
 (11)

 (12)

and
 (13)
Letting ,

 (14)

To find a closed form for an integral power of a cosine function,

 (15)

perform an Integration by Parts so that
 (16)

 (17)

Therefore

 (18)
so

 (19)

 (20)

Now, if is Even so , then

 (21)
Now let , so ,

 (22)

Now if is Odd so , then

 (23)

Now let ,

 (24)

The general result is then

 (25)

The infinite integral of a cosine times a Gaussian can also be done in closed form,

 (26)

See also Chi, Damped Exponential Cosine Integral, Nielsen's Spiral, Shi, Sici Spiral, Sine Integral

References

Abramowitz, M. and Stegun, C. 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.

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.