Fresnel integral

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, and their auxiliary functions Template:Math and Template:Math are 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"> \begin{align}

 S(x) &= \int_0^x \sin\left(t^2\right)\,dt, \\
 C(x) &= \int_0^x \cos\left(t^2\right)\,dt, \\
 F(x) &= \left(\frac{1}{2}-S\left(x\right)\right)\cos\left(x^2\right)-\left(\frac{1}{2}-C\left(x\right)\right)\sin\left(x^2\right), \\
 G(x) &= \left(\frac{1}{2}-S\left(x\right)\right)\sin\left(x^2\right)+\left(\frac{1}{2}-C\left(x\right)\right)\cos\left(x^2\right).

\end{align} </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.

Definition

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: <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.

The Auxiliary functions Template:Math and Template:Math provide monotonic bounds for the Fresnel Integrals:<ref name="atlas">Template:Cite book</ref> <math display="block">\begin{align}

\frac{1}{2}-F(x)-G(x) \leq C(x) \leq \frac{1}{2}+F(x)+G(x), \\
\frac{1}{2}-F(x)-G(x) \leq S(x) \leq \frac{1}{2}+F(x)+G(x).

\end{align}</math>

Euler spiral

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.

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: <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.

Properties

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>

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:<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>

Complex Fresnel integral Template:Math

<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 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: <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>

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<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 by the Cauchy integral theorem. 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

Generalization

The integral <math display="block">\int x^m e^{ix^n}\,dx = \int\sum_{k=0}^\infty\frac{i^kx^{m+nk}}{k!}\,dx

= \sum_{k=0}^\infty \frac{i^k}{(m+nk+1)}\frac{x^{m+nk+1}}{k!}</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 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.Template:Citation needed

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 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 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

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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
• Template:Cite web
• Template:Mathworld
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