“Cosine Integral”的概念、定义、习题解题方法及翻译-数学辞典

单词 Cosine Integral

释义

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

(WolframResearch, 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)

(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)

(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.

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