en>Klnasy at 09:29, 21 January 2014

2014-01-21T09:29:02Z

← Older revision		Revision as of 11:29, 21 January 2014
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	~~Andrew Simcox~~ is the ~~name his parents gave him~~ and ~~he completely loves this title~~. ~~The favorite hobby for him and his kids~~ is ~~to perform lacross and he would never give it up~~. For ~~years he~~'~~s been residing~~ in ~~Alaska and he doesn't plan on changing it~~. ~~Distributing production has been his occupation for some time~~.<br><br>~~my web site~~ ... ~~free online tarot card readings~~ ([http://~~sharefinder~~.~~org~~/~~doku~~.~~php~~?id=~~profile_gerardostearns please click sharefinder~~.org])		In [[mathematics]], the '''Mittag-Leffler function''' ''E''<sub>''α'',''β''</sub> is a [[special function]], a [[complex number\|complex]] [[function (mathematics)\|function]] which depends on two complex parameters ''α'' and ''β''. It may be defined by the following [[series (mathematics)\|series]] when the real part of α is strictly positive:

			:<math>E_{\alpha, \beta} (z) = \sum_{k=0}^\infty \frac{z^k}{\Gamma(\alpha k + \beta)}.</math>

			In the case ''α'' and ''β'' are real and positive, the series converges for all values of the argument ''z'', so the Mittag-Leffler function is an [[entire function]]. This function is named after [[Gösta Mittag-Leffler]]. This class of functions are important in the theory of the [[fractional calculus]].

			For ''α'' > 0, the Mittag-Leffler function ''E''<sub>''α'',1</sub> is an entire function of order 1/''α'', and is in some sense the simplest entire function of its order.

			==Special cases==
			[[Exponential function]]:
			:<math>E_{1,1}(z) = \sum_{k=0}^\infty \frac{z^k}{\Gamma (k + 1)} = \sum_{k=0}^\infty \frac{z^k}{k!} = \exp(z).</math>

			[[Error function]]:
			:<math>E_{1/2,1}(z) = \exp(z^2)\operatorname{erfc}(-z).</math>

			Sum of a [[geometric progression]]:
			:<math>E_{0,1}(z) = \frac{1}{1-z}.</math>

			[[Hyperbolic cosine]]:
			:<math>E_{2,1}(z) = \cosh(\sqrt{z}).</math>

			==Mittag-Leffler's integral representation==
			:<math>E_{\alpha,\beta}(z)=\frac{1}{2\pi i}\int_C \frac{t^{\alpha-\beta}e^t}{t^\alpha-z} \, dt</math>
			where the contour ''C'' starts and ends at −∞ and circles around the singularities and branch points of the integrand.

			==See also==

			*[[Mittag-Leffler summation]]
			*[[Mittag-leffler distribution]]

			==References==
			*{{dlmf\|first=F. W. J. \|last=Olver\|authorlink1=Frank W. J. Olver\|first2=L. C. \|last2=Maximon\|id=10.46}}
			* {{cite book\|title=Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications\|series=Mathematics in Science and Engineering\|author=Igor Podlubny\|publisher=Academic Press\|year=1998\|isbn=0-12-558840-2\|chapter=chapter 1}}
			* {{cite book\|author=Kai Diethelm\|title=The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type\|location=Heidelberg and New York\|publisher=Springer-Verlag\|year=2010\|series=Lecture notes in mathematics\|isbn=978-3-642-14573-5\|chapter=chapter 4}}

			== External links ==
			* [http://mathworld.wolfram.com/Mittag-LefflerFunction.html Mittag-Leffler function on MathWorld]
			* [http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=8738&objectType=FILE Mittag-Leffler function: MATLAB code]
			* [http://arxiv.org/abs/0707.2582 Mittag-Leffler and stable random numbers: Continuous-time random walks and stochastic solution of space-time fractional diffusion equations]
			* [http://www.sciencedirect.com/science/article/pii/S0378375810001837 Parameter estimation for fractional Poisson processes]

			{{PlanetMath attribution\|id=6594\|title=Mittag-Leffler function}}

			[[Category:Special functions]]
			[[Category:Analytic functions]]

83.248.147.232: Undid revision 458497959 by 79.70.238.107 (talk)

2011-11-15T00:08:04Z

Undid revision 458497959 by 79.70.238.107 (talk)

New page

Andrew Simcox is the name his parents gave him and he completely loves this title. The favorite hobby for him and his kids is to perform lacross and he would never give it up. For years he's been residing in Alaska and he doesn't plan on changing it. Distributing production has been his occupation for some time.<br><br>my web site ... free online tarot card readings ([http://sharefinder.org/doku.php?id=profile_gerardostearns please click sharefinder.org])

Dittography - Revision history

en>Klnasy at 09:29, 21 January 2014

83.248.147.232: Undid revision 458497959 by 79.70.238.107 (talk)