Fresnel integral

File:Fresnel Integrals (Unnormalised).svg

Plots of Template:Math and Template:Math. The maximum of Template:Math is about Template:Val. If the integrands of Template:Mvar and Template:Mvar were defined using Template:Math instead of Template:Math, then the image would be scaled vertically and horizontally (see below).

The Fresnel integrals Template:Math and Template:Math are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function (Template:Math). They arise in the description of near-field Fresnel diffraction phenomena and are defined through the following integral representations:

$$S(x)=\underset{0}{\overset{x}{\int }}\sin \left(t^2\right)dt,C(x)=\underset{0}{\overset{x}{\int }}\cos \left(t^2\right)dt.$$

The parametric curve Template:Tmath is the Euler spiral or clothoid, a curve whose curvature varies linearly with arclength.

The term Fresnel integral may also refer to the complex definite integral

$$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{\pm ia{x}^2}dx=\sqrt{\frac{\pi }{a}}{e}^{\pm i\pi /4}$$

where Template:Math is real and positive; this can be evaluated by closing a contour in the complex plane and applying Cauchy's integral theorem.

Definition

File:Fresnel Integrals (Normalised).svg

Fresnel integrals with arguments Template:Math instead of Template:Math converge to Template:Sfrac instead of Template:Math.

The Fresnel integrals admit the following Maclaurin series that converge for all Template:Mvar: $$\begin{array}{rl}S(x)& =\underset{0}{\overset{x}{\int }}\sin \left(t^2\right)dt=\sum _{n=0}^{\mathrm{\infty }}(-1{)}^n\frac{x^{4n+3}}{(2n+1)!(4n+3)},\\ C(x)& =\underset{0}{\overset{x}{\int }}\cos \left(t^2\right)dt=\sum _{n=0}^{\mathrm{\infty }}(-1{)}^n\frac{x^{4n+1}}{(2n)!(4n+1)}.\end{array}$$

Some widely used tablesTemplate:SfnTemplate:Sfn use Template:Math instead of Template:Math for the argument of the integrals defining Template:Math and Template:Math. This changes their limits at infinity from Template:Math to Template:SfracTemplate:Sfn and the arc length for the first spiral turn from Template:Math to 2 (at Template:Math). These alternative functions are usually known as normalized Fresnel integrals.

Euler spiral

File:Cornu Spiral.svg

Euler spiral Template:Math. The spiral converges to the centre of the holes in the image as Template:Mvar tends to positive or negative infinity.

File:CornuSpiralAnimation.gif

Animation depicting evolution of a Cornu spiral with the tangential circle with the same radius of curvature as at its tip, also known as an osculating circle.

The Euler spiral, also known as a Cornu spiral or clothoid, is the curve generated by a parametric plot of Template:Math against Template:Math. The Euler spiral was first studied in the mid 18th century by Leonhard Euler in the context of Euler–Bernoulli beam theory. A century later, Marie Alfred Cornu constructed the same spiral as a nomogram for diffraction computations.

From the definitions of Fresnel integrals, the infinitesimals Template:Mvar and Template:Mvar are thus: $$\begin{array}{rl}dx& =C^{\prime }(t)dt=\cos \left(t^2\right)dt,\\ dy& =S^{\prime }(t)dt=\sin \left(t^2\right)dt.\end{array}$$

Thus the length of the spiral measured from the origin can be expressed as $$L=\underset{0}{\overset{t_0}{\int }}\sqrt{d{x}^2+d{y}^2}=\underset{0}{\overset{t_0}{\int }}dt=t_0.$$

That is, the parameter Template:Mvar is the curve length measured from the origin Template:Math, and the Euler spiral has infinite length. The vector Template:Math, where Template:Math, also expresses the unit tangent vector along the spiral. Since Template:Mvar is the curve length, the curvature Template:Mvar can be expressed as $$\kappa =\frac{1}{R}=\frac{d\theta }{dt}=2t.$$

Thus the rate of change of curvature with respect to the curve length is $$\frac{d\kappa }{dt}=\frac{d^2\theta }{d{t}^2}=2.$$

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 Template:Mvar 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.

Sections from Euler spirals are commonly incorporated into the shape of rollercoaster loops to make what are known as clothoid loops.

Properties

Template:Math and Template:Math are odd functions of Template:Mvar,

$$C(-x)=-C(x),S(-x)=-S(x).$$

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.

Asymptotics of the Fresnel integrals as Template:Math are given by the formulas:

$$\begin{array}{rl}S(x)& =\sqrt{\frac{1}{8}\pi }\mathrm{sgn}x-[1+O\left(x^{-4}\right)](\frac{\cos \left(x^2\right)}{2x}+\frac{\sin \left(x^2\right)}{4x^3}),\\ [6px]C(x)& =\sqrt{\frac{1}{8}\pi }\mathrm{sgn}x+[1+O\left(x^{-4}\right)](\frac{\sin \left(x^2\right)}{2x}-\frac{\cos \left(x^2\right)}{4x^3}).\end{array}$$

File:Fresnel S with domain coloring.svg

Complex Fresnel integral Template:Math

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 Template:Mvar.

The Fresnel integrals can be expressed using the error function as follows:

File:Fresnel C with domain coloring.svg

Complex Fresnel integral Template:Math

$$\begin{array}{rl}S(z)& =\sqrt{\frac{\pi }{2}}\cdot \frac{1+i}{4}[\mathrm{erf}\left(\frac{1+i}{\sqrt{2}}z\right)-i\mathrm{erf}\left(\frac{1-i}{\sqrt{2}}z\right)],\\ [6px]C(z)& =\sqrt{\frac{\pi }{2}}\cdot \frac{1-i}{4}[\mathrm{erf}\left(\frac{1+i}{\sqrt{2}}z\right)+i\mathrm{erf}\left(\frac{1-i}{\sqrt{2}}z\right)].\end{array}$$

or

$$\begin{array}{rl}C(z)+iS(z)& =\sqrt{\frac{\pi }{2}}\cdot \frac{1+i}{2}\mathrm{erf}\left(\frac{1-i}{\sqrt{2}}z\right),\\ [6px]S(z)+iC(z)& =\sqrt{\frac{\pi }{2}}\cdot \frac{1+i}{2}\mathrm{erf}\left(\frac{1+i}{\sqrt{2}}z\right).\end{array}$$

Limits as Template:Math approaches infinity

The integrals defining Template:Math and Template:Math cannot be evaluated in the closed form in terms of elementary functions, except in special cases. The limits of these functions as Template:Mvar goes to infinity are known: $$\underset{0}{\overset{\mathrm{\infty }}{\int }}\cos \left(t^2\right)dt=\underset{0}{\overset{\mathrm{\infty }}{\int }}\sin \left(t^2\right)dt=\frac{\sqrt{2\pi }}{4}=\sqrt{\frac{\pi }{8}}\approx 0.6267.$$

File:Fresnel Integral Contour.svg

The sector contour used to calculate the limits of the Fresnel integrals

This can be derived with any one of several methods. One of them uses a contour integral of the function $$e^{-z^2}$$ around the boundary of the sector-shaped region in the complex plane formed by the positive Template:Math-axis, the bisector of the first quadrant Template:Math with Template:Math, and a circular arc of radius Template:Math centered at the origin.

As Template:Math goes to infinity, the integral along the circular arc Template:Math tends to Template:Math $$\left|\underset{{\gamma }_2}{\int }{e}^{-z^2}dz\right|=\left|\underset{0}{\overset{\frac{\pi }{4}}{\int }}{e}^{-R^2(\cos t+i\sin t{)}^2}R{e}^{it}dt\right|\le R\underset{0}{\overset{\frac{\pi }{4}}{\int }}{e}^{-R^2\cos 2t}dt\le R\underset{0}{\overset{\frac{\pi }{4}}{\int }}{e}^{-R^2(1-\frac{4}{\pi }t)}dt=\frac{\pi }{4R}(1-e^{-R^2}),$$ where polar coordinates Template:Math were used and Jordan's inequality was utilised for the second inequality. The integral along the real axis Template:Math tends to the half Gaussian integral $$\underset{{\gamma }_1}{\int }{e}^{-z^2}dz=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-t^2}dt=\frac{\sqrt{\pi }}{2}.$$

Note too that because the integrand is an entire function on the complex plane, its integral along the whole contour is zero. Overall, we must have $$\underset{{\gamma }_3}{\int }{e}^{-z^2}dz=\underset{{\gamma }_1}{\int }{e}^{-z^2}dz=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-t^2}dt,$$ where Template:Math denotes the bisector of the first quadrant, as in the diagram. To evaluate the left hand side, parametrize the bisector as $$z=t{e}^{i\frac{\pi }{4}}=\frac{\sqrt{2}}{2}(1+i)t$$ where Template:Mvar ranges from 0 to Template:Math. Note that the square of this expression is just Template:Math. Therefore, substitution gives the left hand side as $$\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-i{t}^2}\frac{\sqrt{2}}{2}(1+i)dt.$$

Using Euler's formula to take real and imaginary parts of Template:Math gives this as $$\begin{array}{rl}& \underset{0}{\overset{\mathrm{\infty }}{\int }}(\cos \left(t^2\right)-i\sin \left(t^2\right))\frac{\sqrt{2}}{2}(1+i)dt\\ [6px]& =\frac{\sqrt{2}}{2}\underset{0}{\overset{\mathrm{\infty }}{\int }}[\cos \left(t^2\right)+\sin \left(t^2\right)+i(\cos \left(t^2\right)-\sin \left(t^2\right))]dt\\ [6px]& =\frac{\sqrt{\pi }}{2}+0i,\end{array}$$ where we have written Template:Math to emphasize that the original Gaussian integral's value is completely real with zero imaginary part. Letting $$I_C=\underset{0}{\overset{\mathrm{\infty }}{\int }}\cos \left(t^2\right)dt,I_S=\underset{0}{\overset{\mathrm{\infty }}{\int }}\sin \left(t^2\right)dt$$ and then equating real and imaginary parts produces the following system of two equations in the two unknowns Template:Math and Template:Math: $$\begin{array}{rl}{I}_C+I_S& =\sqrt{\frac{\pi }{2}},\\ I_C-I_S& =0.\end{array}$$

Solving this for Template:Math and Template:Math gives the desired result. Template:Collapse bottom

Generalization

The integral $$\int x^m{e}^{i{x}^n}dx=\int \sum _{l=0}^{\mathrm{\infty }}\frac{i^l{x}^{m+nl}}{l!}dx=\sum _{l=0}^{\mathrm{\infty }}\frac{i^l}{(m+nl+1)}\frac{x^{m+nl+1}}{l!}$$ is a confluent hypergeometric function and also an incomplete gamma functionTemplate:Sfn $$\begin{array}{rl}\int x^m{e}^{i{x}^n}dx& =\frac{x^{m+1}}{m+1}{}_1{F}_1(\begin{array}{c}\frac{m+1}{n}\\ 1+\frac{m+1}{n}\end{array}\mid i{x}^n)\\ [6px]& =\frac{1}{n}{i}^{\frac{m+1}{n}}\gamma (\frac{m+1}{n},-i{x}^n),\end{array}$$ which reduces to Fresnel integrals if real or imaginary parts are taken: $$\int x^m\sin (x^n)dx=\frac{x^{m+n+1}}{m+n+1}{}_1{F}_2(\begin{array}{c}\frac{1}{2}+\frac{m+1}{2n}\\ \frac{3}{2}+\frac{m+1}{2n},\frac{3}{2}\end{array}\mid -\frac{x^{2n}}{4}).$$ The leading term in the asymptotic expansion is $${}_1{F}_1(\begin{array}{c}\frac{m+1}{n}\\ 1+\frac{m+1}{n}\end{array}\mid i{x}^n)\sim \frac{m+1}{n}\mathrm{\Gamma }\left(\frac{m+1}{n}\right)e^{i\pi \frac{m+1}{2n}}{x}^{-m-1},$$ and therefore $$\underset{0}{\overset{\mathrm{\infty }}{\int }}{x}^m{e}^{i{x}^n}dx=\frac{1}{n}\mathrm{\Gamma }\left(\frac{m+1}{n}\right)e^{i\pi \frac{m+1}{2n}}.$$

For Template:Math, the imaginary part of this equation in particular is $$\underset{0}{\overset{\mathrm{\infty }}{\int }}\sin \left(x^a\right)dx=\mathrm{\Gamma }(1+\frac{1}{a})\sin \left(\frac{\pi }{2a}\right),$$ with the left-hand side converging for Template:Math and the right-hand side being its analytical extension to the whole plane less where lie the poles of Template:Math.

The Kummer transformation of the confluent hypergeometric function is $$\int x^m{e}^{i{x}^n}dx=V_{n,m}(x)e^{i{x}^n},$$ with $$V_{n,m}:=\frac{x^{m+1}}{m+1}{}_1{F}_1(\begin{array}{c}1\\ 1+\frac{m+1}{n}\end{array}\mid -i{x}^n).$$

Numerical approximation

For computation to arbitrary precision, the power series is suitable for small argument. For large argument, asymptotic expansions converge faster.Template:Sfn Continued fraction methods may also be used.Template:Sfn

For computation to particular target precision, other approximations have been developed. CodyTemplate:Sfn developed a set of efficient approximations based on rational functions that give relative errors down to Template:Val. 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.Template:Sfn Boersma developed an approximation with error less than Template:Val.Template:Sfn

Applications

The Fresnel integrals were originally used in the calculation of the electromagnetic field intensity in an environment where light bends around opaque objects.Template:Sfn More recently, they have been used in the design of highways and railways, specifically their curvature transition zones, see track transition curve.Template:Sfn Other applications are rollercoastersTemplate:Sfn or calculating the transitions on a velodrome track to allow rapid entry to the bends and gradual exit.

Gallery

• Plot of the Fresnel integral function S(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
  Plot of the Fresnel integral function S(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
• Plot of the Fresnel integral function C(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
  Plot of the Fresnel integral function C(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
• Plot of the Fresnel auxiliary function G(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
  Plot of the Fresnel auxiliary function G(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
• Plot of the Fresnel auxiliary function F(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
  Plot of the Fresnel auxiliary function F(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D

See also

• Böhmer integral
• Fresnel zone
• Track transition curve
• Euler spiral
• Zone plate
• Dirichlet integral

References

• Template:AS ref
• (Uses Template:Math instead of Template:Math.)
• Template:Dlmf

External links

• Cephes, free/open-source C++/C code to compute Fresnel integrals among other special functions. Used in SciPy and ALGLIB.
• 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.
• Template:Springer