Mathematica: numerical non-evaluation of special functions

[muppet]

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.

 

[muppet]

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.

 

[muppet]

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?

 

[djelovin]

For some choices of parameters MeijerG is not defined! Try posting whole integral.

 

[alphy]

Hello,

Thank you for sharing your experience with Mathematica and special functions. From what you have described, it seems that Mathematica is having trouble evaluating the Meijer-G function numerically. This could be due to the complexity of the function or potential limitations of the software.

In general, special functions can be challenging to evaluate numerically because they often have complex mathematical properties and may require special algorithms or techniques. It is not uncommon for numerical evaluation of special functions to produce unexpected or non-numerical results.

As for the discrepancy between the plot and the PossibleZeroQ test, this could also be due to the numerical evaluation of the function. It is possible that the function is approaching zero at the origin, but due to the limitations of numerical precision, it is not exactly equal to zero and therefore the PossibleZeroQ test returns "False".

In any case, it is always important to carefully analyze and interpret numerical results, especially when dealing with complex functions. I would recommend consulting with a mathematician or expert in special functions for a more in-depth explanation of your specific case.

I hope this helps clarify things for you. Keep exploring and experimenting with Mathematica - it is a powerful tool for scientific research and discovery. Best of luck in your endeavors!