Fresnel Integrals

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

so

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

C(u) and S(u) are entire functions.

The C(u) and S(u) integrals are illustrated above in the complex plane.

They have the special values

and

An asymptotic expansion for u>>1 gives

Therefore, as u->infty, C(u)=1/2 and S(u)=1/2. The Fresnel integrals are sometimes alternatively defined as

Letting x=v^2 so dx=2vdv=2sqrt(x)dv, and dv=x^(-1/2)dx/2

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

where n_1(x) is a spherical Bessel function of the second kind

Related functions C_1(z), C_2(z), S_1(z), and S_2(z) are defined by

See also

Cornu Spiral

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 S(x,nu) and C(x,nu)." §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 S(x) and C(x)." Ch. 39 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 373-383, 1987.

Referenced on Wolfram|Alpha

Fresnel Integrals

Cite this as:

Weisstein, Eric W. "Fresnel Integrals." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FresnelIntegrals.html

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