Fresnel integrals, S(x) and C(x), are two transcendental functions named after Augustin-Jean Fresnel that are used in optics, which are closely related to the error function (erf). They arise in the description of far 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). Recently, they have been used in the design of highways and other engineering projects.
The Fresnel integrals admit the following power series expansions that converge for all x:
Some authors, including Abramowitz and Stegun, (eqs 7.3.1 – 7.3.2) use for the argument of the integrals defining S(x) and C(x). To get these functions, multiply the above integrals by and multiply the argument x by .
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, can be expressed as:
And 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. That is, a vehicle following the spiral at constant speed will have a constant rate of angular acceleration.
Sections from Euler spirals are commonly incorporated into the shape of roller-coaster loops to make what are known as "clothoid loops".
- C(x) and S(x) are odd functions of x.
- Using the power series expansions above, the Fresnel integrals can be extended to the domain of complex numbers, and they become analytic functions of a complex variable.
- The Fresnel integrals can be expressed using the error function as follows:
- C and S are entire functions.
- The integrals defining C(x) and S(x) cannot be evaluated in the closed form in terms of elementary functions, except in special cases. The limits of these functions as x goes to infinity are known:
As R goes to infinity, the integral along the circular arc tends to 0, the integral along the real axis tends to the Gaussian integral
and after routine transformations, the integral along the bisector of the first quadrant can be related to the limit of the Fresnel integrals.
which reduces to Fresnel integrals if real or imaginary parts are taken:
The leading term in the asymptotic expansion is
and therefore .
For m=0, the imaginary part of this equation in particular is
with the left-hand side converging for a>1 and the right-hand side being its analytical extension to the whole plane less where lie the poles of .
The Kummer transformation of the confluent hypergeometric function is
The Fresnel integrals were originally used in the calculation of the field intensity in an environment related to the bending of light around opaque objects. More recently, they have been used in the design of highways and railways, specifically their curvature transition zones and roller coasters.
- Stewart, James (2007). Essential Calculus. Belmont, Calif.: Thomson Brooks/Cole. p. 230. ISBN 0-495-01442-7.
- functions.wolfram.com, Fresnel integral S: Representations through equivalent functions and Fresnel integral C: Representations through equivalent functions. Note: Wolfram uses the Abramowitz & Stegun convention, which differs from the one in this article by factors of
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- Faddeeva Package, free/open-source C++/C code to compute complex error functions (from which the Fresnel integrals can be obtained), with wrappers for Matlab, Python, and other languages.
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