Fresnel integral

The Fresnel integrals $S (x)$ and $C (x)$ are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function ( $erf$ ). They arise in the description of near-field Fresnel diffraction phenomena and are defined through the following integral representations:

S(x)=\int _{0}^{x}\sin \left(t^{2}\right)\,dt,\quad C(x)=\int _{0}^{x}\cos \left(t^{2}\right)\,dt.

The simultaneous parametric plot of $S (x)$ and $C (x)$ is the Euler spiral (also known as the Cornu spiral or clothoid).

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