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

单词

Fresnel Integrals

释义

Fresnel Integrals

In physics, the Fresnel integrals are most often defined by

(1)

(2)

(3)

They satisfy

			(4)
			(5)

Related functions are defined as

			(6)
			(7)
			(8)
			(9)

An asymptotic expansion for

gives

(10)

(11)

Therefore, as

and

. The Fresnel integrals are sometimes alternativelydefined as

(12)

(13)

Letting

, and

(14)

(15)

In this form, they have a particularly simple expansion in terms of Spherical Bessel Functions of the FirstKind. Using

			(16)
			(17)

where

is a Spherical Bessel Function of the Second Kind


			(18)

			(19)

See also Cornu Spiral

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Fresnel Integrals.'' §7.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 300-302, 1972.

Leonard, I. E. ``More on Fresnel Integrals.'' Amer. Math. Monthly 95, 431-433, 1988.

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 Fresnel Integrals and .'' Ch. 39 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 373-383, 1987.

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