Fresnel integral
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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:
<math display="block">S(x) = \int_0^x \sin\left(t^2\right)\,dt, \quad C(x) = \int_0^x \cos\left(t^2\right)\,dt.</math>
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
<math display="block">\int_{-\infty}^\infty e^{\pm iax^2} dx = \sqrt{\frac{\pi}{a}}e^{\pm i\pi/4} </math>
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.
DefinitionEdit
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The Fresnel integrals admit the following Maclaurin series that converge for all Template:Mvar: <math display="block">\begin{align}
S(x) &= \int_0^x \sin\left(t^2\right)\,dt = \sum_{n=0}^{\infin}(-1)^n \frac{x^{4n+3}}{(2n+1)!(4n+3)}, \\
C(x) &= \int_0^x \cos\left(t^2\right)\,dt = \sum_{n=0}^{\infin}(-1)^n \frac{x^{4n+1}}{(2n)!(4n+1)}.
\end{align}</math>
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 spiralEdit
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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: <math display="block">\begin{align} dx &= C'(t)\,dt = \cos\left(t^2\right)\,dt, \\ dy &= S'(t)\,dt = \sin\left(t^2\right)\,dt. \end{align}</math>
Thus the length of the spiral measured from the origin can be expressed as <math display="block">L = \int_0^{t_0} \sqrt {dx^2 + dy^2} = \int_0^{t_0} dt = t_0. </math>
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 <math display="block"> \kappa = \frac{1}{R} = \frac{d\theta}{dt} = 2t. </math>
Thus the rate of change of curvature with respect to the curve length is <math display="block">\frac{d\kappa}{dt} = \frac {d^2\theta}{dt^2} = 2. </math>
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.
PropertiesEdit
Template:Math and Template:Math are odd functions of Template:Mvar,
<math display=block>C(-x) = -C(x), \quad S(-x) = -S(x).</math>
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:
<math display="block">\begin{align} S(x) & =\sqrt{\tfrac18\pi} \sgn x - \left[ 1 + O\left(x^{-4}\right) \right] \left( \frac{\cos\left(x^2\right)}{2x} + \frac{\sin\left(x^2\right)}{ 4x^3 } \right), \\[6px] C(x) & =\sqrt{\tfrac18\pi} \sgn x + \left[ 1 + O\left(x^{-4}\right) \right] \left( \frac{\sin\left(x^2\right)}{2x} - \frac{\cos\left(x^2\right)}{ 4x^3 } \right) . \end{align}</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:<ref>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 Template:Math.</ref>
<math display="block">\begin{align} S(z) & =\sqrt{\frac{\pi}{2}} \cdot\frac{1+i}{4} \left[ \operatorname{erf}\left(\frac{1+i}{\sqrt{2}}z\right) -i \operatorname{erf}\left(\frac{1-i}{\sqrt{2}}z\right) \right], \\[6px] C(z) & =\sqrt{\frac{\pi}{2}} \cdot\frac{1-i}{4} \left[ \operatorname{erf}\left(\frac{1+i}{\sqrt{2}}z\right) + i \operatorname{erf}\left(\frac{1-i}{\sqrt{2}}z\right) \right]. \end{align}</math>
or
<math display="block">\begin{align} C(z) + i S(z) & = \sqrt{\frac{\pi}{2}}\cdot\frac{1+i}{2} \operatorname{erf}\left(\frac{1-i}{\sqrt{2}}z\right), \\[6px] S(z) + i C(z) & = \sqrt{\frac{\pi}{2}}\cdot\frac{1+i}{2} \operatorname{erf}\left(\frac{1+i}{\sqrt{2}}z\right). \end{align}</math>
Limits as Template:Math approaches infinityEdit
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: <math display="block">\int_0^\infty \cos \left(t^2\right)\,dt = \int_0^\infty \sin \left(t^2\right) \, dt = \frac{\sqrt{2\pi}}{4} = \sqrt{\frac{\pi}{8}} \approx 0.6267.</math>
This can be derived with any one of several methods. One of them<ref>Another method based on parametric integration is described for example in Template:Harvnb.</ref> uses a contour integral of the function <math display="block"> e^{-z^2}</math> 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 <math display="block">\left|\int_{\gamma_2}e^{-z^2}\,dz\right| = \left|\int_0^\frac{\pi}{4}e^{-R^2(\cos t + i \sin t)^2}\,Re^{it}dt\right| \leq R\int_0^\frac{\pi}{4}e^{-R^2\cos2t}\,dt \leq R\int_0^\frac{\pi}{4}e^{-R^2\left(1-\frac{4}{\pi}t\right)}\,dt = \frac{\pi}{4R}\left(1-e^{-R^2}\right),</math> 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 <math display="block">\int_{\gamma_1} e^{-z^2} \, dz = \int_0^\infty e^{-t^2} \, dt = \frac{\sqrt{\pi}}{2}.</math>
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 <math display="block">\int_{\gamma_3} e^{-z^2} \, dz = \int_{\gamma_1} e^{-z^2} \, dz = \int_0^\infty e^{-t^2} \, dt,</math> where Template:Math denotes the bisector of the first quadrant, as in the diagram. To evaluate the left hand side, parametrize the bisector as <math display="block">z = te^{i\frac{\pi}{4}} = \frac{\sqrt{2}}{2}(1 + i)t</math> 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 <math display="block">\int_0^\infty e^{-it^2}\frac{\sqrt{2}}{2}(1 + i) \, dt.</math>
Using Euler's formula to take real and imaginary parts of Template:Math gives this as <math display="block">\begin{align} & \int_0^\infty \left(\cos\left(t^2\right) - i\sin\left(t^2\right)\right)\frac{\sqrt{2}}{2}(1 + i) \, dt \\[6px] &\quad = \frac{\sqrt{2}}{2} \int_0^\infty \left[\cos\left(t^2\right) + \sin\left(t^2\right) + i \left(\cos\left(t^2\right) - \sin\left(t^2\right)\right) \right] \, dt \\[6px] &\quad = \frac{\sqrt{\pi}}{2} + 0i, \end{align}</math> where we have written Template:Math to emphasize that the original Gaussian integral's value is completely real with zero imaginary part. Letting <math display="block">I_C = \int_0^\infty \cos\left(t^2\right) \, dt, \quad I_S = \int_0^\infty \sin\left(t^2\right) \, dt</math> and then equating real and imaginary parts produces the following system of two equations in the two unknowns Template:Math and Template:Math: <math display="block">\begin{align} I_C + I_S & = \sqrt{\frac{\pi}{2}}, \\ I_C - I_S & = 0. \end{align}</math>
Solving this for Template:Math and Template:Math gives the desired result. Template:Collapse bottom
GeneralizationEdit
The integral <math display="block">\int x^m e^{ix^n}\,dx = \int\sum_{l=0}^\infty\frac{i^lx^{m+nl}}{l!}\,dx
= \sum_{l=0}^\infty \frac{i^l}{(m+nl+1)}\frac{x^{m+nl+1}}{l!}</math>
is a confluent hypergeometric function and also an incomplete gamma functionTemplate:Sfn <math display="block">\begin{align} \int x^m e^{ix^n}\,dx & =\frac{x^{m+1}}{m+1}\,_1F_1\left(\begin{array}{c} \frac{m+1}{n}\\1+\frac{m+1}{n}\end{array}\mid ix^n\right) \\[6px] & =\frac{1}{n} i^\frac{m+1}{n}\gamma\left(\frac{m+1}{n},-ix^n\right), \end{align}</math> which reduces to Fresnel integrals if real or imaginary parts are taken: <math display="block">\int x^m\sin(x^n)\,dx = \frac{x^{m+n+1}}{m+n+1} \,_1F_2\left(\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}\right).</math> The leading term in the asymptotic expansion is <math display="block"> _1F_1 \left(\begin{array}{c}\frac{m+1}{n}\\1+\frac{m+1}{n} \end{array}\mid ix^n\right)\sim \frac{m+1}{n}\,\Gamma\left(\frac{m+1}{n}\right) e^{i\pi\frac{m+1}{2n}} x^{-m-1},</math> and therefore <math display="block">\int_0^\infty x^m e^{ix^n}\,dx = \frac{1}{n} \,\Gamma\left(\frac{m+1}{n}\right)e^{i\pi\frac{m+1}{2n}}.</math>
For Template:Math, the imaginary part of this equation in particular is <math display="block">\int_0^\infty\sin\left(x^a\right)\,dx = \Gamma\left(1+\frac{1}{a} \right) \sin\left(\frac{\pi}{2a}\right),</math> 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 <math display="block"> \int x^m e^{ix^n}\,dx = V_{n,m}(x)e^{ix^n},</math> with <math display="block">V_{n,m} := \frac{x^{m+1}}{m+1}\,_1F_1\left(\begin{array}{c} 1 \\ 1 + \frac{m+1}{n} \end{array}\mid -ix^n\right).</math>
Numerical approximationEdit
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
ApplicationsEdit
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.Template:Citation needed
GalleryEdit
- 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.svg
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.svg
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 auxillary function G(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg
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 auxillary function F(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg
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 alsoEdit
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NotesEdit
ReferencesEdit
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External linksEdit
- 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.
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