Fresnel integral

Fresnel integrals, S(x) and C(x), are two transcendental functions named after Augustin-Jean Fresnel that are used in optics. They arise in the description of near field Fresnel diffraction phenomena, and are defined through the following integral representations:

Failed to parse (Missing texvc executable; please see math/README to configure.): S(x)=\int_0^x \sin(t^2)\,dt,\quad C(x)=\int_0^x \cos(t^2)\,dt.

The simultaneous paremetric plot of S(x) and C(x) is the Cornu spiral, or clothoid.

Definition

The Fresnel integrals admit the following power series expansions that converge for all x:

Failed to parse (Missing texvc executable; please see math/README to configure.): S(x)=\int_0^x \sin(t^2)\,dt=\sum_{n=0}^{\infin}(-1)^n\frac{x^{4n+3}}{(4n+3)(2n+1)!},

Failed to parse (Missing texvc executable; please see math/README to configure.): C(x)=\int_0^x \cos(t^2)\,dt=\sum_{n=0}^{\infin}(-1)^n\frac{x^{4n+1}}{(4n+1)(2n)!}.

Some authors, including Abramowitz and Stegun, (eqs 7.3.1 – 7.3.2) use Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\pi}{2}t^2

for the exponent of the integrals defining S(x) and C(x). To get the same functions, multiply the integral by Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{\frac{2}{\pi}}
and divide the argument x by the same factor.

Cornu spiral

The Cornu spiral, also known as 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.

Since

Failed to parse (Missing texvc executable; please see math/README to configure.): C'(t)^2 + S'(t)^2 = \sin^2(t^2) + \cos^2(t^2) = 1,

in this parametrization the tangent vector has unit length and t is the oriented arc length of the curve measured from the origin (0,0). Therefore, both spirals have infinite length.

It has the property that its curvature at any point is proportional to the distance along the curve, measured from the origin. This property makes it useful as a transition curve in highway and railway engineering, because a vehicle following the curve at constant speed will have a constant rate of angular acceleration. Sections from the clothoid spiral are commonly incorporated into the shape of roller-coaster loops to make what are known as "clothoid loops".

Properties

• 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): S(x)=\frac{\sqrt{\pi}}{4} \left( \sqrt{i}\,\operatorname{erf}(\sqrt{i}\,x) + \sqrt{-i}\,\operatorname{erf}(\sqrt{-i}\,x) \right)

Failed to parse (Missing texvc executable; please see math/README to configure.): C(x)=\frac{\sqrt{\pi}}{4} \left( \sqrt{-i}\,\operatorname{erf}(\sqrt{i}\,x) + \sqrt{i}\,\operatorname{erf}(\sqrt{-i}\,x) \right)

.

• The integrals defining C(x) and S(x) cannot be evaluated in the closed form, except in special cases. The limits of these functions as x goes to infinity are known:

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{0}^{\infty} \cos t^2\,dt = \int_{0}^{\infty} \sin t^2\,dt = \frac{\sqrt{2\pi}}{4} = \sqrt{\frac{\pi}{8}}.

Evaluation

The limits of C and S as the argument tends to infinity can be found by the methods of complex analysis. This uses the contour integral of the function

Failed to parse (Missing texvc executable; please see math/README to configure.): e^{-\frac{1}{2}t^2}

around the boundary of the sector-shaped region in the complex plane formed by the positive x-axis, the half-line y = x, x ≥ 0, and the circle of radius R centered at the origin.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{0}^{\infty} e^{-\frac{1}{2}t^2}dt = \frac{\sqrt{2\pi}}{2}= \sqrt{\frac {\pi}{2}},

and after routine transformations, the integral along the bisector of the first quadrant can be related to the limit of the Fresnel integrals.

See also

• Augustin-Jean Fresnel
• Fresnel zone
• Zone plate

References

• Eric W. Weisstein, Fresnel Integrals at MathWorld.
• Eric W. Weisstein, Cornu Spiral at MathWorld.
• R. Nave, The Cornu spiral, Hyperphysics (2002) (Uses πt²/2 instead of t².)
• Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 7)
• Roller Coaster Loop Shapes. Retrieved on 2008-08-13.af:Klotoïde

ca:Clotoide cs:Klotoida de:Klothoide es:Clotoide fr:Clothoïde it:Clotoide nl:Clothoïde ja:クロソイド曲線 pl:Klotoida pt:Espiral de Cornu ru:Клотоида fi:Fresnelin integraalit sv:Klotoid