Gumbel distribution

2014-01-19T03:37:21Z

75.172.95.102: Referencing an earlier original paper by Gumbel.

[[image:Fresnel Integrals (Unnormalised).svg|250px|thumb|
''S''(''x'') and ''C''(''x'') The maximum of ''C''(''x'') is about 0.977451424. If π''t''²/2 were used instead of ''t''², then the image would be scaled vertically and horizontally (see below).]]

'''Fresnel integrals''', ''S''(''x'') and ''C''(''x''), are two [[transcendental function]]s named after [[Augustin-Jean Fresnel]] that are used in [[optics]], which are closely related to the [[error function]] (erf). They arise in the description of [[near and far field|far field]] [[Fresnel diffraction]] phenomena, and are defined through the following [[integral]] representations:

<math>S(x)=\int_0^x \sin(t^2)\,\mathrm{d}t,\quad C(x)=\int_0^x \cos(t^2)\,\mathrm{d}t.</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|last=Stewart|first=James|title=Essential Calculus|year=2007|publisher=Thomson Brooks/Cole|location=Belmont, Calif.|isbn=0-495-01442-7|page=230}}</ref>

== Definition ==
The Fresnel integrals admit the following [[power series expansion]]s that converge for all ''x'':

[[image:Fresnel Integrals (Normalised).svg|250px|thumb|
Normalised Fresnel integrals, ''S''(''x'') and ''C''(''x''). In these curves, the argument of the trigonometric function is π''t''2/2, as opposed to just ''t''2 as above.]]

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

Some authors, including [[Abramowitz and Stegun]], (eqs 7.3.1 – 7.3.2) use <math>\frac{\pi}{2}t^2</math> for the argument of the integrals defining ''S''(''x'') and ''C''(''x''). To get these functions, multiply the above integrals by <math>\sqrt{\frac{2}{\pi}}</math> and multiply the argument ''x'' by <math>\sqrt{\frac{\pi}{2}}</math>.

== Euler spiral ==
{{Main|Euler spiral}}
[[Image:Cornu Spiral.svg|250px|thumb|
Euler spiral (''x'', ''y'') = (''C''(''t''), ''S''(''t'')). The spiral converges to the centre of the holes in the image as ''t'' 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 ''S''(''t'') against ''C''(''t''). 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 infinitesimals ''dx'' and ''dy'' are thus:
: <math> \mathrm{d}x = C'(t)\mathrm{d}t = \cos(t^2) \mathrm{d}t \,</math>

: <math> \mathrm{d}y = S'(t)\mathrm{d}t = \sin(t^2) \mathrm{d}t \,</math>

Thus the length of the spiral measured from the origin can be expressed as:

: <math>L = \int_0^{t_0} {\sqrt {\mathrm{d}x^2 + \mathrm{d}y^2}} = \int_0^{t_0}{\mathrm{d}t} = t_0 </math>

That is, the parameter {{math| t}} is the curve length measured from the origin (0,0) and the Euler spiral has [[Infinity|infinite]] length. The vector {{math| [cos(t²), sin(t²)]}} also expresses the [[unit vector|unit]] [[tangent vector]] along the spiral, giving θ = {{math| t²}}. Since t is the curve length, the curvature, <math>\kappa</math> can be expressed as:
: <math> \kappa = \tfrac {1}{R} = \tfrac {\mathrm{d}\theta}{\mathrm{d}t} = 2t </math>

And the rate of change of curvature with respect to the curve length is:
: <math>\tfrac{\mathrm{d}\kappa}{\mathrm{d}t}=\tfrac {\mathrm{d}^2\theta}{\mathrm{d}t^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}} 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''.
* 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>

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

* ''C'' and ''S'' are [[entire function]]s.
* 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\,\mathrm{d}t = \int_{0}^{\infty} \sin t^2\,\mathrm{d}t = \frac{\sqrt{2\pi}}{4} = \sqrt{\frac{\pi}{8}}.</math>

=== Evaluation ===
[[Image:Fresnel Integral Contour.svg|right|250px|thumb|The sector contour used to calculate the limits of the Fresnel integrals]]
The limits of ''C'' and ''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^{-\frac{1}{2}t^2}</math>

around the boundary of the [[Circular sector|sector]]-shaped region in the [[complex plane]] formed by the positive ''x''-axis, the half-line ''y'' = ''x'', ''x'' ≥ 0, and the circle of radius ''R'' centered at the origin.

As ''R'' goes to infinity, the integral along the circular arc tends to 0, the integral along the real axis tends to the [[Gaussian integral]]

:<math> \int_{0}^{\infty} e^{-\frac{1}{2}t^2}\mathrm{d}t =
\sqrt{\frac{\pi}{2}}, </math>

and after routine transformations, the integral along the bisector of the first quadrant can be related to the limit of the Fresnel integrals.

== Generalization ==
The integral
<math>\int x^m \exp(ix^n)\mathrm{d}x = \int\sum_{l=0}^\infty\frac{i^lx^{m+nl}}{l!}\mathrm{d}x
= \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]]

<math>\int x^m \exp(ix^n)\mathrm{d}x =\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(\frac{m+1}{n},-ix^n),</math>

which reduces to Fresnel integrals if real or imaginary parts are taken:

<math>\int x^m\sin(x^n)\mathrm{d}x = \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(\frac{m+1}{n})
e^{i\pi(m+1)/(2n)} x^{-m+1}</math>,

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

For ''m=0'', the imaginary part of this equation in particular is
<math>\int_0^\infty\sin(x^a)\ \mathrm{d}x = \Gamma\left(1+\frac{1}{a}\right)\sin(\frac{\pi}{2a})</math>

with the left-hand side converging for ''a>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)\mathrm{d}x = 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>.

== Applications ==
The Fresnel integrals were originally used in the calculation of the field intensity in an environment related to the bending of light 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<ref name=Stewart /> and roller coasters.<ref name=Beatty />

== See also ==
* [[Augustin-Jean Fresnel]]
* [[Fresnel zone]]
* [[Track transition curve]]
* [[Euler spiral]]
* [[Zone plate]]

== 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
}}
*{{cite journal|first1=J. |last1=Boersma
|title=Computation of Fresnel Integrals
|journal=Math. Comp.
|volume=14
|year=1960
|pages=380-380
|mr=0121973
|doi=10.1090/S0025-5718-1960-0121973-3
}}
* {{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 }}
*{{Citation | 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 | publication-place=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}}
*{{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 arxiv|first1=Mohammad |last1=Alazah
|title=Computing fresnel integrals via modified trapezium rules
|year=2012
|eprint=1209.3451
}}
*{{cite arxiv|first1 =R. J. | last1=Mathar
|title=Series Expansion of Generalized Fresnel Integrals |year=2012 |eprint=1211.3963
}}

==External links==
*[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}} {{Dead link|date=September 2010|bot=H3llBot}}
*{{mathworld|title=Fresnel Integrals|urlname=FresnelIntegrals}}
*{{mathworld|title=Cornu Spiral|urlname=CornuSpiral}}

[[Category:Integral calculus]]
[[Category:Spirals]]
[[Category:Optics]]
[[Category:Special functions]]
[[Category:Special hypergeometric functions]]
[[Category:Analytic functions]]

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Gumbel distribution