Mathematica: numerical non-evaluation of special functions

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Discussion Overview

The discussion centers around the numerical evaluation of special functions, specifically Meijer-G functions, in Mathematica. Participants are exploring issues related to obtaining numerical values from symbolic results, particularly in the context of integrals.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant reports difficulties in obtaining numerical evaluations for sums of Meijer-G functions, noting that Mathematica returns a symbolic form instead of a numerical value.
  • Another participant mentions encountering an internal precision limit error when trying to create a table of results, suggesting that the evaluation may be sensitive to the precision settings in Mathematica.
  • A participant reflects on the behavior of the function at zero, indicating that while a plot suggests it vanishes at the origin, the PossibleZeroQ function returns "False," raising questions about the function's behavior.
  • There is a suggestion that for certain parameter choices, Meijer-G functions may not be defined, prompting a request for the integral that produced the problematic expression.

Areas of Agreement / Disagreement

Participants express uncertainty about the behavior of Meijer-G functions in specific contexts, and there is no consensus on the underlying reasons for the numerical evaluation issues or the implications of the results obtained.

Contextual Notes

Participants note limitations related to Mathematica's handling of precision and the definition of Meijer-G functions under certain parameters, but these aspects remain unresolved.

muppet
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Hi all,
I've been getting Mathematica to do some integrals for me, which are typically returning sums of Meijer-G functions. When I try and obtain numerical values for these sums, some of my results have contained terms which Mathematica has refused to evaluate numerically; an example is
MeijerG[{{}, {}}, {{1, 7/6, 4/3, 4/3, 3/2, 5/3, 11/6}, {0, 5/6, 7/6, 4/3, 3/2, 5/3, 11/6}}, 0]

Inside a N[], Mathematica just spits back
N[%]= MeijerG[{{}, {}}, {{1., 1.16667, 1.33333, 1.33333, 1.5, 1.66667, 1.83333}, {0., 0.833333, 1.16667, 1.33333, 1.5, 1.66667, 1.83333}}, 0.]

A plot of the (real and imaginary parts of the) function
MeijerG[{{}, {}}, {{1, 7/6, 4/3, 4/3, 3/2, 5/3, 11/6}, {0, 5/6, 7/6, 4/3, 3/2, 5/3, 11/6}}, x]
suggests that it vanishes at the origin, and its limit as x->0 is zero. But the test PossibleZeroQ, applied to this function at zero argument, yields the result "False".

If anyone could explain to me what's going on here, I'd be grateful.
 
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Update: when I tried to create a table of these results, I got an error message I didn't get yesterday:
N::meprec: Internal precision limit $ MaxExtraPrecision=50`. reached whilst evaluating MeijerG[...].

(Why it would helpfully tell me this today but not yesterday I have no idea, but never mind.)

It looks as if this value is indeed zero, which somehow upsets Mathematica's idea of precision, so I'm recreating my table now using the command Quiet[N[MeijerG[...]]]. I'll be back if that doesn't work, but otherwise thanks to all who read this.
 
In my previous post I forgot that the problematic expression was the result of doing an integral, so the output to my table was still given as a number plus this complicated way of writing zero; it also gives the absolute value of each entry formally in terms of this abomination, rather than as a number. Does anyone know a way of preventing this from happening?
 
For some choices of parameters MeijerG is not defined! Try posting whole integral.
 

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