Fresnel Integrals
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
"Fresnel Integrals." §7.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
22-11-2018
1716
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
so
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
and
An asymptotic expansion for 
gives
Therefore, as 
, 
and 
. The Fresnel integrals are sometimes alternatively defined as
Letting 
so 
, and 
In this form, they have a particularly simple expansion in terms of spherical Bessel functions of the first kind. Using
where 
is a spherical Bessel function of the second kind
Related functions 
, 
, 
, and 
are defined by
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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