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Fresnel Integrals

In physics, the Fresnel integrals are most often defined by

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)

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

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
C(u) \approx {1\over 2} + {1\over \pi u} \sin({\textstyle{1\over 2}}\pi u^2)
\end{displaymath} (10)

S(u) \approx {1\over 2} - {1\over \pi u} \cos({\textstyle{1\over 2}}\pi u^2).
\end{displaymath} (11)

Therefore, as $u\to\infty$, $C(u) = 1/2$ and $S(u)=1/2$. The Fresnel integrals are sometimes alternatively defined as
x(t) = \int^t_0 \cos(v^2)\,dv
\end{displaymath} (12)

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 $dv=x^{-1/2}\,dx/2$
x(t) = {\textstyle{1\over 2}}\int_0^{\sqrt{t}} x^{-1/2}\cos x\,dx
\end{displaymath} (14)

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 $n_1(x)$ 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


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

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© 1996-9 Eric W. Weisstein