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 ) . {\displaystyle {\begin{aligned}S(x)&=\int _{0}^{x}\sin \left(t^{2}\right)\,dt,\\C(x)&=\int _{0}^{x}\cos \left(t^{2}\right)\,dt,\\F(x)&=\left({\frac {1}{2}}{\sqrt {\frac {\pi }{2}}}-S\left(x\right)\right)\cos \left(x^{2}\right)-\left({\frac {1}{2}}{\sqrt {\frac {\pi }{2}}}-C\left(x\right)\right)\sin \left(x^{2}\right),\\G(x)&=\left({\frac {1}{2}}{\sqrt {\frac {\pi }{2}}}-S\left(x\right)\right)\sin \left(x^{2}\right)+\left({\frac {1}{2}}{\sqrt {\frac {\pi }{2}}}-C\left(x\right)\right)\cos \left(x^{2}\right).\end{aligned}}} The parametric curve ⁠ ( S ( t ) , C ( t ) ) {\displaystyle {\bigl (}S(t),C(t){\bigr )}} ⁠ is the Euler spiral or clothoid, a curve whose curvature varies linearly with arclength. The term Fresnel integral may also refer to the complex definite integral ∫ − ∞ ∞ e ± i a x 2 d x = π a e ± i π / 4 {\displaystyle \int _{-\infty }^{\infty }e^{\pm iax^{2}}dx={\sqrt {\frac {\pi }{a}}}e^{\pm i\pi /4}} where a is real and positive; this can be evaluated by closing a contour in the complex plane and applying Cauchy's integral theorem.