Fresnel Integrals

In physics, the Fresnel integrals are most often defined by

$\begin{displaymath} C(u)+iS(u) \equiv \int^u_0 e^{i\pi x^2/2}\,dx = \int^u_0 \co... ... x^2)\,dx + i\int^u_0 \sin({\textstyle{1\over 2}}\pi x^2)\,dx, \end{displaymath}$

(1)

$\begin{displaymath} C(u) \equiv \int^u_0 \cos({\textstyle{1\over 2}}\pi x^2)\,dx \end{displaymath}$

(2)

$\begin{displaymath} S(u) \equiv \int^u_0 \sin({\textstyle{1\over 2}}\pi x^2)\,dx. \end{displaymath}$

(3)

They satisfy

$\displaystyle C(\pm\infty)$	$\textstyle =$	$\displaystyle -{\textstyle{1\over 2}}$	(4)
$\displaystyle S(\pm\infty)$	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}.$	(5)

Related functions are defined as

$\displaystyle C_1(z)$	$\textstyle \equiv$	$\displaystyle \sqrt{2\over\pi} \int_0^x \cos t^2\,dt$	(6)
$\displaystyle S_1(z)$	$\textstyle \equiv$	$\displaystyle \sqrt{2\over\pi} \int_0^x \sin t^2\,dt$	(7)
$\displaystyle C_2(z)$	$\textstyle \equiv$	$\displaystyle {1\over\sqrt{2\pi}} \int {\cos t\over\sqrt{t}}\,dt$	(8)
$\displaystyle S_2(z)$	$\textstyle \equiv$	$\displaystyle {1\over\sqrt{2\pi}} \int {\sin t\over\sqrt{t}}\,dt.$	(9)

An asymptotic expansion for $x \gg 1$ gives

$\begin{displaymath} C(u) \approx {1\over 2} + {1\over \pi u} \sin({\textstyle{1\over 2}}\pi u^2) \end{displaymath}$

(10)

$\begin{displaymath} S(u) \approx {1\over 2} - {1\over \pi u} \cos({\textstyle{1\over 2}}\pi u^2). \end{displaymath}$

(11)

Therefore, as $u\to\infty$ ,

and

. The Fresnel integrals are sometimes alternatively defined as

$\begin{displaymath} x(t) = \int^t_0 \cos(v^2)\,dv \end{displaymath}$

(12)

$\begin{displaymath} y(t) = \int^t_0 \sin(v^2)\,dv. \end{displaymath}$

(13)

Letting $x\equiv v^2$ so $dx=2v\,dv=2\sqrt{x}\,dv$ , and $dv=x^{-1/2}\,dx/2$

$\begin{displaymath} x(t) = {\textstyle{1\over 2}}\int_0^{\sqrt{t}} x^{-1/2}\cos x\,dx \end{displaymath}$

(14)

$\begin{displaymath} y(t) = {\textstyle{1\over 2}}\int_0^{\sqrt{t}} x^{-1/2}\sin x\,dx. \end{displaymath}$

(15)

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

$\displaystyle j_0(x)$	$\textstyle =$	$\displaystyle {\sin x\over x}$	(16)
$\displaystyle n_1(x)$	$\textstyle =$	$\displaystyle -j_{-1}(x) = -{\cos x\over x},$	(17)

where

is a Spherical Bessel Function of the Second Kind

$\displaystyle x(t^2)$	$\textstyle =$	$\displaystyle -{\textstyle{1\over 2}}\int^t_0 n_1(x)x^{1/2}\,dx$
	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}\int^t_0 j_{-1}(x)x^{1/2}\,dx = x^{1/2} \sum_{n=0}^\infty j_{2n}(x)$	(18)
$\displaystyle y(t^2)$	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}\int^t_0 j_0(x)x^{1/2}\,dx$
	$\textstyle =$	$\displaystyle x^{1/2} \sum_{n=0}^\infty j_{2n+1}(x).$	(19)

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.