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Re: Evaluating a Meijer G-function

• To: mathgroup at smc.vnet.net
• Subject: [mg69444] Re: Evaluating a Meijer G-function
• From: dimmechan at yahoo.com
• Date: Wed, 13 Sep 2006 04:00:54 -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. More�

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 έγÏ?αÏ?ε:
> 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