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)

so

			(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)

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