In physics, the Fresnel integrals are most often defined by
![\begin{displaymath}
C(u)+iS(u) \equiv \int^u_0 e^{i\pi x^2/2}\,dx = \int^u_0 \co...
... x^2)\,dx + i\int^u_0 \sin({\textstyle{1\over 2}}\pi x^2)\,dx,
\end{displaymath}](f_2007.gif) |
(1) |
so
![\begin{displaymath}
C(u) \equiv \int^u_0 \cos({\textstyle{1\over 2}}\pi x^2)\,dx
\end{displaymath}](f_2008.gif) |
(2) |
![\begin{displaymath}
S(u) \equiv \int^u_0 \sin({\textstyle{1\over 2}}\pi x^2)\,dx.
\end{displaymath}](f_2009.gif) |
(3) |
They satisfy
Related functions are defined as
An asymptotic expansion for
gives
![\begin{displaymath}
C(u) \approx {1\over 2} + {1\over \pi u} \sin({\textstyle{1\over 2}}\pi u^2)
\end{displaymath}](f_2023.gif) |
(10) |
![\begin{displaymath}
S(u) \approx {1\over 2} - {1\over \pi u} \cos({\textstyle{1\over 2}}\pi u^2).
\end{displaymath}](f_2024.gif) |
(11) |
Therefore, as
,
and
. The Fresnel integrals are sometimes alternatively
defined as
![\begin{displaymath}
x(t) = \int^t_0 \cos(v^2)\,dv
\end{displaymath}](f_2028.gif) |
(12) |
![\begin{displaymath}
y(t) = \int^t_0 \sin(v^2)\,dv.
\end{displaymath}](f_2029.gif) |
(13) |
Letting
so
, and
![\begin{displaymath}
x(t) = {\textstyle{1\over 2}}\int_0^{\sqrt{t}} x^{-1/2}\cos x\,dx
\end{displaymath}](f_2033.gif) |
(14) |
![\begin{displaymath}
y(t) = {\textstyle{1\over 2}}\int_0^{\sqrt{t}} x^{-1/2}\sin x\,dx.
\end{displaymath}](f_2034.gif) |
(15) |
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
See also Cornu Spiral
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.
© 1996-9 Eric W. Weisstein
1999-05-26