Fresnel Integrals

There are a number of slightly different definitions of the Fresnel integrals. In physics, the Fresnel integrals denoted and are most often defined by

			(1)
			(2)

			(3)
			(4)

These Fresnel integrals are implemented in the Wolfram Language as FresnelC[z] and FresnelS[z].

and are entire functions.

	Min	Max
Re
Im

	Min	Max
Re
Im

The and integrals are illustrated above in the complex plane.

They have the special values

			(5)
			(6)
			(7)

and

			(8)
			(9)
			(10)

An asymptotic expansion for gives

			(11)
			(12)

Therefore, as , and . The Fresnel integrals are sometimes alternatively defined as

			(13)
			(14)

Letting so , and

			(15)
			(16)

In this form, they have a particularly simple expansion in terms of spherical Bessel functions of the first kind. Using

			(17)
			(18)
			(19)

where is a spherical Bessel function of the second kind

			(20)
			(21)
			(22)
			(23)
			(24)

Related functions , , , and are defined by

			(25)
			(26)
			(27)
			(28)

Related Wolfram sites

http://functions.wolfram.com/GammaBetaErf/FresnelC/, http://functions.wolfram.com/GammaBetaErf/FresnelS/

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References

Abramowitz, M. and Stegun, I. 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.Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Generalized Fresnel Integrals

and

." §1.3 in Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, p. 24, 1990.Spanier, J. and Oldham, K. B. "The Fresnel Integrals