Loterre: Mathematics: Fresnel integral

mathematical analysis > calculus > integral calculus > Fresnel integral

mathematical physics > special function > hypergeometric function > Fresnel integral

mathematical analysis > functional analysis > special function > hypergeometric function > Fresnel integral

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:
${\\displaystyle S(x)=\\int _{0}^{x}\\sin \\left(t^{2}\ ight)\\,dt,\\quad C(x)=\\int _{0}^{x}\\cos \\left(t^{2}\ ight)\\,dt.}$
The simultaneous parametric plot of $S (x)$ and $C (x)$ is the Euler spiral (also known as the Cornu spiral or clothoid).
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Fresnel_integral)

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