Re: Evaluating a Meijer G-function

*To*: mathgroup at smc.vnet.net*Subject*: [mg69446] Re: Evaluating a Meijer G-function*From*: dimmechan at yahoo.com*Date*: Wed, 13 Sep 2006 04:00:58 -0400 (EDT)*References*: <ee64o4$7pj$1@smc.vnet.net>

Hello Raul, ***I believe, by as far I as I know, you cannot establish what you want symbolically. $Version "5.2 for Microsoft Windows (June 20, 2005)" ***(I have converted everything to InputForm). g=MeijerG[{{1/ 2, 1/ 2}, {1}}, {{0, 0, 0}, { }}, 4 / t^2] / (2 Pi); g/.t->1 MeijerG[{{1/2, 1/2}, {1}}, {{0, 0, 0}, {}}, 4]/(2*Pi) FullSimplify[%]//Timing {5.203*Second, MeijerG[{{1/2, 1/2}, {1}}, {{0, 0, 0}, {}}, 4]/(2*Pi)} g/.t->0 -2 Power::infy: Infinite expression 0 encountered. MoreÎ²â?¬Â¦ MeijerG[{{1/2, 1/2}, {1}}, {{0, 0, 0}, {}}, ComplexInfinity]/(2*Pi) Limit[g,t->0]//Timing {13.421*Second, Limit[MeijerG[{{1/2, 1/2}, {1}}, {{0, 0, 0}, {}}, 4/t^2]/(2*Pi), t -> 0]} ***However, you get an idea of what happens at t->0, evaluating numerically a sequence of decreasing small t, 0 < t < 1; it is clear as you said that the value approaches 0 as t -> 0. ***From the following command you get one more evidence that you are right. gplot=Plot[g,{t,0,0.01},DisplayFunction->Identity,PlotPoints->100]; Take[Nest[First,gplot,4],5] {{1.0101010101010101*^-10, 2.635883044016847*^-9}, {0.00009834422199494737, 0.0012102710347072572}, {0.00020559709056817862, 0.0023785596747130104}, {0.0003063257365782754, 0.003421751319955433}, {0.00040319614005049234, 0.004393035231558852}} ***You can also use the function NLimit. ***This loads the package Needs["NumericalMath`NLimit`"] FullDefinition[NLimit] Attributes[NLimit] = {Protected, ReadProtected} Options[NLimit] = {Direction -> Automatic, WorkingPrecision -> MachinePrecision, Scale -> 1, Terms -> 7, Method -> EulerSum, WynnDegree -> 1} Options[NLimit] Information[Evaluate[#[[1]]]]&/@%; {Direction -> Automatic, WorkingPrecision -> MachinePrecision, Scale -> 1, Terms -> 7, Method -> EulerSum, WynnDegree -> 1} Direction is an option for Limit. Limit[expr, x -> x0, Direction -> 1] computes the limit as x approaches x0 from smaller values. Limit[expr, x -> x0, Direction -> -1] computes the limit as x approaches x0 from larger values. Direction -> Automatic uses Direction -> -1 except for limits at Infinity, where it is equivalent to Direction -> 1. InputForm[Attributes[Direction] = {Protected}] WorkingPrecision is an option for various numerical operations which specifies how many digits of precision should be maintained in internal computations. InputForm[Attributes[WorkingPrecision] = {Protected}] Scale is an option of NLimit and ND. It specifies the initial stepsize in the sequence of steps or the radius of the circle of integration for Cauchy's integral formula in ND. Terms is an option of EulerSum, NLimit, and ND. In EulerSum it specifies the number of terms to be included explicitly before the extrapolation process begins. In NLimit and ND it specifies the total number of terms to be used. Method is an option to Solve, related functions, and various numerical functions, which specifies what algorithm to use in evaluating the result. InputForm[Attributes[Method] = {Protected}] WynnDegree is an option to NSum and NProduct that specifies the degree used with Wynn's epsilon algorithm for approximating the limit of a sequence. WynnDegree -> 1 gives Aitken's delta-squared algorithm. InputForm[Attributes[WynnDegree] = {Protected}] ***Here we have NLimit[g,t->0,WorkingPrecision->50,Terms->30]//Timing {4.657*Second, -9.253969619008001038766399803`18.54697098301814*^-30} ***I hope you found helpful my response. Cheers Dimitris Î?/Î? Raul Martinez ÎÎ³Ï?Î±Ï?Îµ: > I have the following special case of Meijer's G-function: > > g = MeijerG[{{1/ 2, 1/ 2}, {1}}, {{0, 0, 0}, { }}, 4 / t^2] / (2 Pi), > where t is real. > > When I evaluate it numerically for a sequence of decreasing small t, > 0 < t < 1, it is clear that the value approaches 0 as t -> 0. > > But neither > > N[g /. t -> 0] > > nor > > Limit[g, t -> 0] > > yields the result that g = 0. > > Can anyone show that g -> 0 as t -> 0? > > I've consulted functions.wolfram.com, mathworld.wolfram.com, and many > other web sites and reference works, to no avail. > > Thanks in advance. > > Raul