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Date: 30-3-2019
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There are a number of slightly different definitions of the Fresnel integrals. In physics, the Fresnel integrals denoted and are most often defined by
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so
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These Fresnel integrals are implemented in the Wolfram Language as FresnelC[z] and FresnelS[z].
and are entire functions.
The and integrals are illustrated above in the complex plane.
They have the special values
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and
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An asymptotic expansion for gives
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Therefore, as , and . The Fresnel integrals are sometimes alternatively defined as
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Letting so , and
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In this form, they have a particularly simple expansion in terms of spherical Bessel functions of the first kind. Using
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where is a spherical Bessel function of the second kind
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Related functions , , , and are defined by
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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 and ." Ch. 39 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 373-383, 1987.
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