Fresnel integrals

Summary

The Fresnel integrals $S(x) = \int_0^x \sin(t^2)\, dt$ and $C(x) = \int_0^x \cos(t^2)\, dt$ are integrals of sine and cosine of a square. They arise in the optics of diffraction near an edge, studied by Augustin-Jean Fresnel. The plane curve $(C(t), S(t))$ is the Euler spiral, whose curvature grows linearly with arc length. Both integrals tend to $\sqrt{\pi/8}$ as $x \to \infty$.