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:
is the Euler spiral or clothoid, a curve whose curvature varies linearly with arclength.
where a is real and positive; this can be evaluated by closing a contour in the complex plane and applying Cauchy's integral theorem.
The Fresnel integrals admit the following power series expansions that converge for all x:
Some widely used tables[1][2] 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·√π/2 to 1/2[3] and the arc length for the first spiral turn from √2π to 2 (at t = 2).
A century later, Marie Alfred Cornu constructed the same spiral as a nomogram for diffraction computations.
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
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
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.
which can be readily seen from the fact that their power series expansions have only odd-degree terms, or alternatively because they are antiderivatives of even functions that also are zero at the origin.
Using the power series expansions above, the Fresnel integrals can be extended to the domain of complex numbers, where they become entire functions of the complex variable z.
The integrals defining C(x) and S(x) cannot be evaluated in the closed form in terms of elementary functions, except in special cases.
around the boundary of the sector-shaped region in the complex plane formed by the positive x-axis, the bisector of the first quadrant y = x with x ≥ 0, and a circular arc of radius R centered at the origin.
Note too that because the integrand is an entire function on the complex plane, its integral along the whole contour is zero.
To evaluate the left hand side, parametrize the bisector as
Using Euler's formula to take real and imaginary parts of e−it2 gives this as
where we have written 0i to emphasize that the original Gaussian integral's value is completely real with zero imaginary part.
Solving this for IC and IS gives the desired result.
which reduces to Fresnel integrals if real or imaginary parts are taken:
For computation to arbitrary precision, the power series is suitable for small argument.
For large argument, asymptotic expansions converge faster.
[8] For computation to particular target precision, other approximations have been developed.
Cody[9] developed a set of efficient approximations based on rational functions that give relative errors down to 2×10−19.
A FORTRAN implementation of the Cody approximation that includes the values of the coefficients needed for implementation in other languages was published by van Snyder.
[11] The Fresnel integrals were originally used in the calculation of the electromagnetic field intensity in an environment where light bends around opaque objects.
[13] Other applications are rollercoasters[12] or calculating the transitions on a velodrome track to allow rapid entry to the bends and gradual exit.
