View source for Fresnel integral

{{Short description|Special function defined by an integral}}
{{Use American English|date = January 2019}}
[[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 {{val|0.977451424}}. If the integrands of {{mvar|S}} and {{mvar|C}} were defined using {{math|{{sfrac|''π''|2}}''t''<sup>2</sup>}} instead of {{math|''t''<sup>2</sup>}}, then the image would be scaled vertically and horizontally (see below).]]

The '''Fresnel integrals''' {{math|''S''(''x'')}} and {{math|''C''(''x'')}}, and their auxiliary functions {{math|''F''(''x'')}} and {{math|''G''(''x'')}} are [[transcendental function]]s named after [[Augustin-Jean Fresnel]] that are used in [[optics]] and are closely related to the [[error function]] ({{math|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 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]] {{tmath|\bigl(S(t), C(t)\bigr)}} 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 {{math|a}} is real and positive; this can be evaluated by closing a contour in the complex plane and applying [[Cauchy's integral theorem]].

== Definition ==
[[image:Fresnel Integrals (Normalised).svg|250px|thumb|
Fresnel integrals with arguments {{math|{{sfrac|''π''|2}}''t''<sup>2</sup>}} instead of {{math|''t''<sup>2</sup>}} converge to {{sfrac|1|2}} instead of {{math|{{sfrac|1|2}}·{{sqrt|{{frac|''π''|2}}}}}}.]]
The Fresnel integrals admit the following [[Maclaurin series]] that converge for all {{mvar|x}}:
<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 tables{{sfn|Abramowitz|Stegun|1983|loc=eqn 7.3.1–7.3.2}}{{sfn|Temme|2010}} use {{math|{{sfrac|''π''|2}}''t''<sup>2</sup>}} instead of {{math|''t''<sup>2</sup>}} for the argument of the integrals defining {{math|''S''(''x'')}} and {{math|''C''(''x'')}}.  This changes their [[Limit of a function#Limits at infinity|limits at infinity]] from {{math|{{sfrac|1|2}}·{{sqrt|{{sfrac|''π''|2}}}}}} to {{sfrac|1|2}}{{sfn|Abramowitz|Stegun|1983|loc=eqn 7.3.20}} and the arc length for the first spiral turn from {{math|{{sqrt|2''π''}}}} to 2 (at {{math|1=''t'' = 2}}). These alternative functions are usually known as '''normalized Fresnel integrals'''.

The Auxiliary functions {{math|''F''(''x'')}} and {{math|''G''(''x'')}} provide monotonic bounds for the Fresnel Integrals:<ref name="atlas">{{cite book |last1=Oldham |first1=Keith B. |last2=Myland |first2=Jan C. |last3=Spanier |first3=Jerome |last4=Myland |first4=Jan |title=An Atlas of functions: with equator, the atlas function calculator |date=2009 |publisher=Springer US Springer e-books |location=New York, NY |isbn=978-0-387-48807-3}}</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 ==
{{Main|Euler spiral}}
[[Image:Cornu Spiral.svg|250px|thumb|
Euler spiral {{math|1=(''x'', ''y'') = (''C''(''t''), ''S''(''t''))}}. The spiral converges to the centre of the holes in the image as {{mvar|t}} tends to positive or negative infinity.]]
[[File:CornuSpiralAnimation.gif|thumb|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 {{math|''S''(''t'')}} against {{math|''C''(''t'')}}. 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 [[infinitesimal]]s {{mvar|dx}} and {{mvar|dy}} 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 (mathematics)|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 {{mvar|t}} is the curve length measured from the origin {{math|(0, 0)}}, and the Euler spiral has [[Infinity|infinite]] length. The vector {{math|(cos(''t''<sup>2</sup>), sin(''t''<sup>2</sup>))}}, where {{math|1=''θ'' = ''t''<sup>2</sup>}}, also expresses the [[unit vector|unit]] [[tangent vector]] along the spiral. Since {{mvar|t}} is the curve length, the curvature {{mvar|κ}} 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 [[track transition curve|transition curve]] in highway and railway engineering: if a vehicle follows the spiral at unit speed, the parameter {{mvar|t}} 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 loop]]s.

== Properties ==
{{math|''C''(''x'')}} and  {{math|''S''(''x'')}} are [[odd function]]s of {{mvar|x}},

<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 {{math|''x'' → ∞}} 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>

[[File:Fresnel S with domain coloring.svg|right|thumb|300px|Complex Fresnel integral {{math|''S''(''z'')}}]]
Using the power series expansions above, the Fresnel integrals can be extended to the domain of [[complex number]]s, where they become  [[entire function]]s of the complex variable {{mvar|z}}.

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|{{frac|''π''|2}}}}}}.</ref> [[File:Fresnel C with domain coloring.svg|right|thumb|300px|Complex Fresnel integral {{math|''C''(''z'')}}]]

<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 {{math|''x''}} approaches infinity===
The integrals defining {{math|''C''(''x'')}} and {{math|''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 {{mvar|x}} 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>

{{Collapse top|title=Proof of the formula}}
[[Image:Fresnel Integral Contour.svg|right|250px|thumb|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 [[Integration using parametric derivatives|parametric integration]] is described for example in {{harvnb|Zajta|Goel|1989}}.</ref> uses a [[contour integral]] of the function <math display="block"> e^{-z^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|1=''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|''γ''<sub>2</sub>}} tends to {{math|0}}
<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 {{math|1=''z'' = ''Re<sup>it</sup>''}} were used and [[Jordan's inequality]] was utilised for the second inequality. The integral along the real axis {{math|''γ''<sub>1</sub>}} 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 {{math|''γ''<sub>3</sub>}} 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 {{mvar|t}} ranges from 0 to {{math|+∞}}. Note that the square of this expression is just {{math|+''it''<sup>2</sup>}}. 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 {{math|''e''<sup>−''it''<sup>2</sup></sup>}} 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 {{math|0''i''}} 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 {{math|''I<sub>C</sub>''}} and {{math|''I<sub>S</sub>''}}:
<math display="block">\begin{align}
I_C + I_S & = \sqrt{\frac{\pi}{2}}, \\
I_C - I_S & = 0.
\end{align}</math>

Solving this for {{math|''I<sub>C</sub>''}} and {{math|''I<sub>S</sub>''}} gives the desired result.
{{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 function]]{{sfn|Mathar|2012}}
<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 {{math|1=''m'' = 0}}, 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 {{math|{{!}}''a''{{!}} > 1}} and the right-hand side being its analytical extension to the whole plane less where lie the poles of {{math|''Γ''(''a''<sup>−1</sup>)}}.

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.{{sfn|Temme|2010|loc=§7.12(ii)}} Continued fraction methods may also be used.{{sfn|Press|Teukolsky|Vetterling|Flannery|2007}}

For computation to particular target precision, other approximations have been developed. Cody{{sfn|Cody|1968}} 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.{{sfn|van Snyder|1993}} Boersma developed an approximation with error less than {{val|1.6|e=-9}}.{{sfn|Boersma|1960}}

== Applications ==
The Fresnel integrals were originally used in the calculation of the electromagnetic field intensity in an environment where light bends around opaque objects.{{sfn|Beatty|2013}} More recently, they have been used in the design of highways and railways, specifically their curvature transition zones, see [[track transition curve]].{{sfn|Stewart|2008|page=383}} Other applications are [[rollercoaster]]s{{sfn|Beatty|2013}} or calculating the transitions on a [[velodrome]] track to allow rapid entry to the bends and gradual exit.{{citation needed|date=March 2014}}

== Gallery ==
<gallery>
File: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
File: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
File: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
File: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
</gallery>

== See also ==
{{Portal|Mathematics}}
{{Div col|colwidth=25em}}
* [[Böhmer integral]]
* [[Fresnel zone]]
* [[Track transition curve]]
* [[Euler spiral]]
* [[Zone plate]]
* [[Dirichlet integral]]
{{Div col end}}

== Notes ==
{{Reflist}}

== References ==
{{sfn whitelist |CITEREFAbramowitzStegun1983 |CITEREFTemme2010}}
* {{AS ref|7}}<!--Abramowitz & Stegun 1983-->
*{{cite journal
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| s2cid=13934493
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*{{cite web
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 | url=https://www.thomasbeatty.com/MATH%20PAGES/ARCHIVES%20-%20NOTES/Complex%20Variables/How%20to%20evaluate%20Fresnel%20Integrals.pdf
 | work=FGCU Math - Summer 2013
 | year=2013
 | archive-url=https://web.archive.org/web/20230409062958/https://www.thomasbeatty.com/MATH%20PAGES/ARCHIVES%20-%20NOTES/Complex%20Variables/How%20to%20evaluate%20Fresnel%20Integrals.pdf
 | access-date=27 July 2013
| archive-date=9 April 2023
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*{{cite journal
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 | doi-access=free
}}
*{{cite journal
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| s2cid=121794086
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*{{cite journal
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| doi-access=free
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*{{cite journal
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| s2cid=122271446
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*{{cite arXiv
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}}
*{{cite web
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 | year=2002
}} (Uses {{math|{{sfrac|''π''|2}}''t''<sup>2</sup>}} instead of {{math|''t''<sup>2</sup>}}.)
*{{cite book
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 | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=297
 | access-date=2011-08-09
 | archive-date=2011-08-11
 | archive-url=https://web.archive.org/web/20110811154417/http://apps.nrbook.com/empanel/index.html#pg=297
 | url-status=dead
 }}
*{{cite journal
 | last1=van Snyder
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 | journal=ACM Trans. Math. Softw.
 | year=1993
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| s2cid=12346795
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*{{cite book
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 | year=2008
 | publisher=Cengage Learning EMEA
 | isbn=978-0-495-38273-7
}}
*{{dlmf <!--Temme 2010-->
 | id=7
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 | title=Error Functions, Dawson’s and Fresnel Integrals
}}
*{{cite book
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 | volume=19
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}}
*{{cite journal
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| doi=10.1080/0025570X.1989.11977462
 }}

==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 |access-date=2008-08-13 |url-status=dead |archive-url=https://web.archive.org/web/20080923190052/http://fy.chalmers.se/LISEBERG/eng/loop_pe.html |archive-date=September 23, 2008 }}
*{{mathworld|title=Fresnel Integrals|urlname=FresnelIntegrals}}
*{{mathworld|title=Cornu Spiral|urlname=CornuSpiral}}
{{Nonelementary Integral}}

[[Category:Integral calculus]]
[[Category:Spirals]]
[[Category:Physical optics]]
[[Category:Special functions]]
[[Category:Special hypergeometric functions]]
[[Category:Analytic functions]]
[[Category:Diffraction]]