Fresnel integral

Plots of S(x) and C(x). The maximum of C(x) is about 0.977451424. If the integrands of S and C were defined using ⁠π/2⁠t² instead of t², then the image would be scaled vertically and horizontally (see below).

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:

$${\begin{aligned}S(x)&=\int _{0}^{x}\sin \left(t^{2}\right)\,dt,\\C(x)&=\int _{0}^{x}\cos \left(t^{2}\right)\,dt,\\F(x)&=\left({\frac {1}{2}}{\sqrt {\frac {\pi }{2}}}-S\left(x\right)\right)\cos \left(x^{2}\right)-\left({\frac {1}{2}}{\sqrt {\frac {\pi }{2}}}-C\left(x\right)\right)\sin \left(x^{2}\right),\\G(x)&=\left({\frac {1}{2}}{\sqrt {\frac {\pi }{2}}}-S\left(x\right)\right)\sin \left(x^{2}\right)+\left({\frac {1}{2}}{\sqrt {\frac {\pi }{2}}}-C\left(x\right)\right)\cos \left(x^{2}\right).\end{aligned}}$$

The parametric curve ⁠${\bigl (}S(t),C(t){\bigr )}$⁠ 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

$$\int _{-\infty }^{\infty }e^{\pm iax^{2}}dx={\sqrt {\frac {\pi }{a}}}e^{\pm i\pi /4}$$

where a is real and positive; this can be evaluated by closing a contour in the complex plane and applying Cauchy's integral theorem.