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: The simultaneous parametric plot of S(x) and C(x) is the Euler spiral (also known as the Cornu spiral or clothoid). The Fresnel integrals admit the following power series expansions that converge for all x: Some widely used tables use π/2t2 instead of t2 for the argument of the integrals defining S(x) and C(x). This changes their limits at infinity from 1/2· to 1/2 and the arc length for the first spiral turn from to 2 (at t = 2). These alternative functions are usually known as normalized Fresnel integrals. Euler spiral The Euler spiral, also known as Cornu spiral or clothoid, is the curve generated by a parametric plot of S(t) against C(t). The Cornu spiral was created by Marie Alfred Cornu as a nomogram for diffraction computations in science and engineering. From the definitions of Fresnel integrals, the infinitesimals dx and dy are thus: Thus the length of the spiral measured from the origin can be expressed as That is, the parameter t is the curve length measured from the origin (0, 0), and the Euler spiral has infinite length. The vector (cos(t2), sin(t2)) also expresses the unit tangent vector along the spiral, giving θ = t2. Since t is the curve length, the curvature κ can be expressed as Thus the rate of change of curvature with respect to the curve length is An Euler spiral has the property that its curvature at any point is proportional to the distance along the spiral, measured from the origin. This property makes it useful as a transition curve in highway and railway engineering: if a vehicle follows the spiral at unit speed, the parameter t in the above derivatives also represents the time. Consequently, a vehicle following the spiral at constant speed will have a constant rate of angular acceleration.

About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related courses (2)
MATH-101(e): Analysis I
Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.
PHYS-101(g): General physics : mechanics
Le but du cours de physique générale est de donner à l'étudiant les notions de base nécessaires à la compréhension des phénomènes physiques. L'objectif est atteint lorsque l'étudiant est capable de pr
Related lectures (92)
Fresnel Integrals: Curvilinear Integrals in R^2
Explores Fresnel integrals through curvilinear integrals in R^2, focusing on cos(x²-y) and sin(x) functions.
Two-loop Computations: External Legs and Internal Propagators
Covers the computation of two-loop diagrams with external legs and internal propagators.
Integration Techniques: Change of Variable and Integration by Parts
Explores advanced integration techniques such as change of variable and integration by parts to simplify complex integrals and solve challenging integration problems.
Show more
Related people (1)
Related concepts (5)
Incomplete gamma function
In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, which are defined similarly to the gamma function but with different or "incomplete" integral limits. The gamma function is defined as an integral from zero to infinity. This contrasts with the lower incomplete gamma function, which is defined as an integral from zero to a variable upper limit.
Error function
In mathematics, the error function (also called the Gauss error function), often denoted by erf, is a complex function of a complex variable defined as: Some authors define without the factor of . This nonelementary integral is a sigmoid function that occurs often in probability, statistics, and partial differential equations. In many of these applications, the function argument is a real number. If the function argument is real, then the function value is also real.
Trigonometric integral
In mathematics, trigonometric integrals are a family of integrals involving trigonometric functions. The different sine integral definitions are Note that the integrand is the sinc function, and also the zeroth spherical Bessel function. Since sinc is an even entire function (holomorphic over the entire complex plane), Si is entire, odd, and the integral in its definition can be taken along any path connecting the endpoints. By definition, Si(x) is the antiderivative of sin x / x whose value is zero at x = 0, and si(x) is the antiderivative whose value is zero at x = ∞.
Show more