View source for Fresnel integral

{{Use American English|date = January 2019}}
{{Short description|Special function defined by an integral}}
[[image:Fresnel Integrals (Unnormalised).svg|250px|thumb|
Plots of {{math|{{color|#b30000|''S''(''x'')}}}} and {{math|{{color|#00b300|''C''(''x'')}}}}. The maximum of {{math|''C''(''x'')}} is about {{math|0.977451424}}. If the integrands of {{mvar|S}} and {{mvar|C}} were defined using {{math|&pi;''t''<sup>2</sup>/2}} instead of {{math|''t''<sup>2</sup>}}, then the image would be scaled vertically and horizontally (see below).]]

The '''Fresnel integrals''' ''S''(''x'') and ''C''(''x'') are two [[transcendental function]]s 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 and far field|near-field]] [[Fresnel diffraction]] phenomena and are defined through the following [[integral]] representations:

: <math>S(x) = \int_0^x \sin(t^2)\,dt, \quad C(x) = \int_0^x \cos(t^2)\,dt.</math>

The simultaneous [[parametric equation|parametric plot]] of ''S''(''x'') and ''C''(''x'') is the [[Euler spiral]] (also known as the Cornu spiral or clothoid). Recently, they have been used in the design of highways and other engineering projects.<ref name=Stewart>{{cite book|author=James Stewart|title=Calculus Early Transcendentals|url=https://books.google.com/?id=v4-1Rk7uaEcC&pg=PA383&lpg=PA383&vq=%22design+of+highways%22#v=onepage&q=%22design%20of%20highways%22&f=false|year=2008|publisher=Cengage Learning EMEA|isbn=978-0-495-38273-7|page=383}}</ref>

== Definition ==
[[image:Fresnel Integrals (Normalised).svg|250px|thumb|
Fresnel integrals with arguments {{pi}}''t''<sup>2</sup>/2 instead of ''t''<sup>2</sup> converge to 0.5.]]
The Fresnel integrals admit the following [[power series expansion]]s that converge for all ''x'':

:<math>
\begin{align}
S(x) & =\int_0^x \sin(t^2)\,dt=\sum_{n=0}^{\infin}(-1)^n\frac{x^{4n+3}}{(2n+1)!(4n+3)}. \\
C(x) & =\int_0^x \cos(t^2)\,dt=\sum_{n=0}^{\infin}(-1)^n\frac{x^{4n+1}}{(2n)!(4n+1)}.
\end{align}
</math>

Some authors, including [[Abramowitz and Stegun]], (eqs 7.3.1 &ndash; 7.3.2) use <math>\pi t^2 / 2</math> for the argument of the integrals defining ''S''(''x'') and ''C''(''x'').  This changes their [[Limit of a function#Limits at infinity|limits at infinity]] from <math>\textstyle (1/2) \cdot \sqrt{\pi/2}</math> to <math>1/2</math> and the arc length for the first spiral turn from <math>(2\pi)</math> to 2 (at <math>t=2</math>), all smaller by a factor <math>\textstyle \sqrt{2/\pi}</math>. The alternative functions are also called ''Normalized Fresnel integrals''.

== Euler spiral ==
{{Main|Euler spiral}}
[[Image:Cornu Spiral.svg|250px|thumb|
Euler spiral <math>(x,y) = (C(t), S(t))</math>. The spiral converges to the centre of the holes in the image as <math>t</math> tends to positive or negative infinity.]]
The '''Euler [[spiral]]''', also known as '''Cornu spiral''' or '''clothoid''', is the curve generated by a [[parametric plot]] of <math>S(t)</math> against <math>C(t)</math>. The Cornu spiral was created by [[Marie Alfred Cornu]] as a [[nomogram]] for diffraction computations in science and engineering.

From the definitions of Fresnel integrals, the [[infinitesimal]]s <math>dx</math> and <math>dy</math> are thus:
:<math>\begin{align}
dx &= C'(t)\,dt = \cos(t^2)\,dt, \\
dy &= S'(t)\,dt = \sin(t^2)\,dt.
\end{align}</math>

Thus the length of the spiral measured from the [[Origin (mathematics)|origin]] can be expressed as

: <math>L = \int_0^{t_0} \sqrt {dx^2 + dy^2} = \int_0^{t_0} dt = t_0. </math>

That is, the parameter <math>t</math> is the curve length measured from the origin <math>(0,0)</math>, and the Euler spiral has [[Infinity|infinite]] length. The vector <math>(\cos(t^2), \sin(t^2))</math> also expresses the [[unit vector|unit]] [[tangent vector]] along the spiral, giving <math>\theta = t^2</math>. Since ''t'' is the curve length, the curvature <math>\kappa</math> can be expressed as
: <math> \kappa = \frac{1}{R} = \frac{d\theta}{dt} = 2t. </math>

And the rate of change of curvature with respect to the curve length is
: <math>\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 [[track transition curve|transition curve]] in highway and railway engineering: If a vehicle follows the spiral at unit speed, the parameter <math>t</math> in the above derivatives also represents the time. That is, 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 roller-coaster loops to make what are known as [[clothoid loop]]s.

== Properties ==
* ''C''(''x'') and  ''S''(''x'') are [[odd function]]s of ''x''.
* Asymptotics of the Fresnel integrals as <math>x \to \infty</math> are given by the formulas:

:: <math>
\begin{align}
S(x) & =\sqrt{\frac{\pi}{2}} \left( \frac{\mbox{sign}{(x)}}{2} - \left[1+O(x^{-4}) \right] \left( \frac{\cos{(x^{2})}}{x \sqrt{2 \pi}} + \frac{\sin{(x^2)}}{ x^3 \sqrt{8 \pi}} \right) \right), \\
C(x) & =\sqrt{\frac{\pi}{2}} \left( \frac{\mbox{sign}{(x)}}{2} + \left[1+O(x^{-4}) \right] \left( \frac{\sin{(x^2)}}{x \sqrt{2 \pi}} - \frac{\cos{(x^{2})}}{ x^3 \sqrt{8 \pi}} \right) \right).
\end{align}
</math>

[[File:Fresnel-sin.png|right|thumb|300px|Complex Fresnel integral S(z)]]
* Using the power series expansions above, the Fresnel integrals can be extended to the domain of [[complex number]]s, and they become [[analytic function]]s of a complex variable.
* The Fresnel integrals can be expressed using the [[error function]] as follows:<ref>functions.wolfram.com, [http://functions.wolfram.com/GammaBetaErf/FresnelS/27/01/ Fresnel integral S: Representations through equivalent functions] and [http://functions.wolfram.com/GammaBetaErf/FresnelC/27/01/ 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 <math>\sqrt{\pi/2}</math></ref>

[[File:Fresnel-cos.png|right|thumb|300px|Complex Fresnel integral ''C''(''z'')]]

:: <math>
\begin{align}
S(z) & =\sqrt{\frac{\pi}{2}} \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], \\
C(z) & =\sqrt{\frac{\pi}{2}}\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>
\begin{align}
C(z) + i S(z) & = \sqrt{\frac{\pi}{2}}\frac{1+i}{2} \operatorname{erf}\left(\frac{1-i}{\sqrt{2}}z\right), \\
S(z) + i C(z) & = \sqrt{\frac{\pi}{2}}\frac{1+i}{2} \operatorname{erf}\left(\frac{1+i}{\sqrt{2}}z\right).
\end{align}
</math>

* ''C'' and  ''S'' are [[entire function]]s.

===Limits as ''x'' approaches infinity===
The integrals defining ''C''(''x'') and ''S''(''x'') cannot be evaluated in the [[closed-form expression|closed form]] in terms of [[elementary function]]s, except in special cases. The [[limit of a function|limits]] of these functions as ''x'' goes to infinity are known:

::<math>\int_0^\infty \cos t^2\,dt = \int_0^\infty \sin t^2 \, dt = \frac{\sqrt{2\pi}}{4} = \sqrt{\frac{\pi}{8}} \approx 0.6267.</math>

[[Image:Fresnel Integral Contour.svg|right|250px|thumb|The sector contour used to calculate the limits of the Fresnel integrals]]
The limits of {{math|''C''}} and {{math|''S''}} as the argument tends to infinity can be found by the methods of [[complex analysis]]. This uses the [[contour integral]] of the function

:<math>e^{-t^2}</math>

around the boundary of the [[Circular sector|sector]]-shaped region in the [[complex plane]] formed by the positive {{math|''x''}}-axis, the bisector of the first quadrant {{math|''y'' {{=}} ''x''}} with {{math|''x'' ≥ 0}}, and a circular arc of radius {{math|''R''}} centered at the origin.

As {{math|''R''}} goes to infinity, the integral along the circular arc <math>\gamma_2</math> tends to {{math|0}}

<math>\left|\int_{\gamma_2}e^{-t^2}dt\right| \leq \int_{\gamma_2}|e^{-t^2}|dt = 
R\int_0^{\pi/4}e^{-R^2\cos2t}dt \leq R\int_0^{\pi/4}e^{-R^2(1-\frac{4}{\pi}t)}dt =
\frac{\pi}{4R}\left(1-e^{-R^2}\right),</math>

where polar coordinates <math>z=Re^{it}</math> were used and [[Jordan's inequality]] was utilised for the second inequality. The integral along the real axis <math>\gamma_1</math> tends to the half [[Gaussian integral]]

:<math>\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>\int_0^\infty e^{-t^2} \, dt = \int_{\gamma_3} e^{-t^2} \, dt,</math>

where <math>\gamma_3</math> denotes the bisector of the first quadrant, as in the diagram. To evaluate the right hand side, parametrize the bisector as
:<math>t = re^{\pi i /4} = \frac{\sqrt{2}}{2}(1 + i)r</math>
where r ranges from 0 to <math>+\infty</math>. Note that the square of this expression is just <math>+ir^2</math>. Therefore, substitution gives the right hand side as

:<math>\int_0^\infty e^{-ir^2}\frac{\sqrt{2}}{2}(1 + i) \, dr.</math>

Using [[Euler's formula]] to take real and imaginary parts of <math>e^{-ir^2}</math> gives this as

:<math>
\begin{align}
& \int_0^\infty (\cos(r^2) - i\sin(r^2))\frac{\sqrt{2}}{2}(1 + i) \, dr \\
= {} & \frac{\sqrt{2}}{2} \int_0^\infty \left[\cos(r^2) + \sin(r^2) + i \left(\cos(r^2) - \sin(r^2)\right) \right] \, dr = \frac{\sqrt{\pi}}{2} + 0i,
\end{align}
</math>

where we have written <math>0i</math> to emphasize that the original Gaussian integral's value is completely real with zero imaginary part. Letting <math>I_C = \int_0^\infty \cos(r^2) \, dr, I_S = \int_0^\infty \sin(r^2) \, dr</math> and then equating real and imaginary parts produces the following system of two equations in the two unknowns <math>I_C, I_S</math>:

: <math>
\begin{align}
I_C + I_S & = \sqrt{\frac{\pi}{2}}, \\
I_C - I_S & = 0.
\end{align}
</math>

Solving this for <math>I_C</math> and <math>I_S</math> gives the desired result.

== Generalization ==
The integral
: <math>\int x^m \exp(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 function]]<ref>{{cite arxiv|first1 =R. J. | last1=Mathar
|title=Series Expansion of Generalized Fresnel Integrals |year=2012 |eprint=1211.3963
| class=math.CA
}}</ref>

: <math>
\begin{align}
\int x^m \exp(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) \\
& =\frac{1}{n} i^{(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>\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> _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(m+1)/(2n)} x^{-m-1},</math>

and therefore
: <math>\int_0^\infty x^m \exp(ix^n)\,dx = \frac{1}{n} \,\Gamma\left(\frac{m+1}{n}\right)e^{i\pi(m+1)/(2n)}.</math>

For ''m''&nbsp;=&nbsp;0, the imaginary part of this equation in particular is

: <math>\int_0^\infty\sin(x^a)\,dx = \Gamma\left(1+\frac{1}{a} \right) \sin\left(\frac{\pi}{2a}\right),</math>

with the left-hand side converging for ''a''&nbsp;>&nbsp;1 and the right-hand side being its analytical extension to the whole plane less where lie the poles of <math>\Gamma(a^{-1})</math>.

The Kummer transformation of the confluent hypergeometric function is

: <math> \int x^m \exp(ix^n)\,dx = V_{n,m}(x)e^{ix^n},</math>

with
: <math>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<ref>{{dlmf|id=7.12.ii|title=Error Functions, Dawson’s and Fresnel Integrals: Asymptotic expansions|first=N. M. |last=Temme}}</ref> converge faster. Continued fraction methods may also be used.<ref>{{cite book| last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press |  location=New York | isbn=978-0-521-88068-8 | chapter=Section 6.8.1. Fresnel Integrals | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=297}}</ref>

For computation to particular target precision, other approximations have been developed. Cody<ref>{{cite journal |last1=Cody |first1=William J |title=Chebyshev approximations for the Fresnel integrals |journal=Math. Comp. |date=1968 |volume=22 |issue=102 |pages=450–453 |doi=10.1090/S0025-5718-68-99871-2 |url=http://www.ams.org/journals/mcom/1968-22-102/S0025-5718-68-99871-2/S0025-5718-68-99871-2.pdf}}</ref> developed a set of efficient approximations based on rational functions that give relative errors down to {{val|2|e=-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.<ref>{{cite journal |last1=van Snyder |first1=W |title=Algorithm 723: Fresnel integrals |journal=ACM Trans. Math. Softw. |date=December 1993 |volume=19 |issue=4 |pages=452–456 |doi=10.1145/168173.168193 |ref=snyder}}</ref> Boersma developed an approximation with error less than {{val|1.6|e=-9}}.<ref>{{cite journal|first1=J. |last1=Boersma
|title=Computation of Fresnel Integrals
|journal=Math. Comp.
|volume=14
|issue=72
|year=1960
|pages=380
|mr=0121973
|doi=10.1090/S0025-5718-1960-0121973-3 
}}
</ref>

== Applications ==
The Fresnel integrals were originally used in the calculation of the electromagnetic field intensity in an environment where light bends around opaque objects.<ref name=Beatty>{{cite web|last=Beatty|first=Thomas|title=How to evaluate Fresnel Integrals|url=http://www.thomasbeatty.com/MATH%20PAGES/ARCHIVES%20-%20NOTES/Complex%20Variables/How%20to%20evaluate%20Fresnel%20Integrals.pdf|work=FGCU MATH - SUMMER 2013|accessdate=27 July 2013}}</ref> More recently, they have been used in the design of highways and railways, specifically their curvature transition zones, see [[track transition curve]].<ref name=Stewart /> Other applications are roller coasters<ref name=Beatty /> or calculating the transitions on a velodrome track to allow rapid entry to the bends and gradual exit.{{citation needed|date=March 2014}}

== See also ==
* [[Böhmer integral]]
* [[Fresnel zone]]
* [[Track transition curve]]
* [[Euler spiral]]
* [[Zone plate]]
*[[Dirichlet integral]]

== References ==
{{Reflist}}

*{{cite book|first1= A. |last1=van Wijngaarden
|first2= W. L. |last2=Scheen
|title=Table of Fresnel Integrals
|year=1949
|series= Verhandl. Konink. Ned. Akad. Wetenschapen
|volume=19
|number=4
}}
* {{AS ref|7|297}}
*{{cite journal|first1=Roland| last1=Bulirsch
|title=Numerical calculation of the sine, cosine and Fresnel integrals
|year=1967
|volume=9
|number=5
|pages=380–385
|journal=Numer. Math.
|doi=10.1007/BF02162153
}}
*{{cite journal|first1=R. J.|last1=Hangelbroek
|title=Numerical approximation of Fresnel integrals by means of Chebyshev polynomials
|journal=J. Eng. Math.
|year=1967
|volume=1
|number=1
|pages=37–50
|doi=10.1007/BF01793638
|bibcode = 1967JEnMa...1...37H }}
*{{cite web|first1=R. |last1=Nave
|url=http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/cornu.html#c1
|title=The Cornu spiral
|year=2002}} ''(Uses πt²/2 instead of t².)''
*{{dlmf|id=7|title=Error Functions, Dawson’s and Fresnel Integrals|first=N. M. |last=Temme}}
*{{cite journal|first1=Mohammad |last1=Alazah
|title=Computing fresnel integrals via modified trapezium rules
|year=2012
|arxiv=1209.3451
 |doi=10.1007/s00211-014-0627-z
 |volume=128
 |issue=4
|journal=Numerische Mathematik
 |pages=635–661
|bibcode=2012arXiv1209.3451A
}}
*{{cite arxiv|first1 =R. J. | last1=Mathar
|title=Series Expansion of Generalized Fresnel Integrals |year=2012 |eprint=1211.3963
| class=math.CA
}}

==External links==
*[http://www.netlib.org/cephes/ Cephes], [[Free and open-source software|free/open-source]] C++/C code to compute Fresnel integrals among other special functions. Used in [[SciPy]] and [[ALGLIB]].
*[http://ab-initio.mit.edu/Faddeeva Faddeeva Package], [[Free and open-source software|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.
* {{springer|title=Fresnel integrals|id=p/f041720}}
*{{cite web |url=http://fy.chalmers.se/LISEBERG/eng/loop_pe.html |title=Roller Coaster Loop Shapes |accessdate=2008-08-13 |url-status=dead |archiveurl=https://web.archive.org/web/20080923190052/http://fy.chalmers.se/LISEBERG/eng/loop_pe.html |archivedate=September 23, 2008 }}
*{{mathworld|title=Fresnel Integrals|urlname=FresnelIntegrals}}
*{{mathworld|title=Cornu Spiral|urlname=CornuSpiral}}
{{Nonelementary Integral}}
[[Category:Integral calculus]]
[[Category:Spirals]]
[[Category:Optics]]
[[Category:Special functions]]
[[Category:Special hypergeometric functions]]
[[Category:Analytic functions]]