Fresnel integral

The Fresnel integral is an oscillatory Gaussian integral. In the angular-frequency convention, its distributional form isfor nonzero real $\omega$.

For the optical convention of Fresnel integrals, $C(w)={\int }_0^w\cos (\pi u^2/2)du$ and $S(w)={\int }_0^w\sin (\pi u^2/2)du$. Both functions are odd and tend to $1/2$ as $w\to +\infty$. If the unscaled convention uses phase $t^2$, multiply its argument by $\sqrt{\pi /2}$ and its value by $\sqrt{2/\pi }$ to obtain this convention. An aperture factor with phase $\pi x^2/(\lambda R)$ uses $u=x\sqrt{2/(\lambda R)}$. Stating the convention prevents missing factors of $\pi$ or $\sqrt{2}$ in diffraction calculations.

In the unscaled convention, the finite Fresnel integrals are $C(\lambda )={\int }_0^{\lambda }\cos (t^2)dt$ and $S(\lambda )={\int }_0^{\lambda }\sin (t^2)dt$. Their Taylor series follow by termwise integration. At infinity both tend to $\sqrt{\pi /8}$, and their oscillatory tails have asymptotic expansions obtained by integration by parts. This convention differs from the convention with phase $\pi t^2/2$.

For $A=\sqrt{\pi /8}$, define real tail functions by $Q+iP=e^{-i{\lambda }^2}{\int }_{\lambda }^{\infty }{e}^{i{t}^2}dt$. Then $C=A+P\sin {\lambda }^2-Q\cos {\lambda }^2$ and $S=A-P\cos {\lambda }^2-Q\sin {\lambda }^2$. Repeated integration by parts giveswhere $(-1)!!=1$. These are asymptotic expansions, rather than convergent infinite sums at a fixed argument.

For $\alpha >0$, polar coordinates reduce the integral to $2\pi {\int }_0^{\infty }r{e}^{-(\alpha -i\lambda )r^2}dr$. The result is $\pi /(\alpha -i\lambda )$. Its damped limit for real $\lambda >0$ is $i\pi /\lambda$, which fixes the sign of the oscillatory complex Gaussian integral. The Fresnel integral convention can be compared with NIST definitions.

For $t>0$, contour rotation of the Gaussian integral gives