Fresnel integral
The Fresnel integrals S(x) and C(x), and their auxiliary functions F(x) and G(x) are 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 ) = ∫ 0 x sin ( t 2 ) d t , C ( x ) = ∫ 0 x cos ( t 2 ) d t , F ( x ) = ( 1 2 π 2 − S ( x ) ) cos ( x 2 ) − ( 1 2 π 2 − C ( x ) ) sin ( x 2 ) , G ( x ) = ( 1 2 π 2 − S ( x ) ) sin ( x 2 ) + ( 1 2 π 2 − C ( x ) ) cos ( x 2 ) .