Fresnel Integrals

In physics, the Fresnel integrals are most often defined by

 (1)

so
 (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 alternatively defined as
 (12)

 (13)

Letting so , and
 (14)

 (15)

In this form, they have a particularly simple expansion in terms of Spherical Bessel Functions of the First Kind. Using
 (16) (17)

where is a Spherical Bessel Function of the Second Kind
 (18) (19)

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.