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MathGroup Archive 2007

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a function containg MeijerG (limit behavior)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg73292] a function containg MeijerG (limit behavior)
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Fri, 9 Feb 2007 23:40:00 -0500 (EST)

I am interested in the behavior of a function as x1->0 and x1-
>Infinity.
The function is:

f[x1_] := -Log[Abs[x1]] - (1/2)*Sqrt[Pi]*MeijerG[{{0}, {}}, {{0, 0},
{1/2}}, x1^2]

>From the following plot one might take the idea that f[x1] is bounded
at x1=0.

Plot[f[x1], {x1, -1, 1}, PlotPoints -> 4000]

However taking the limit as x1->0 yields to

Limit[f[x1], x1 -> 0]
EulerGamma + I*Interval[{-2*Pi, 2*Pi}] + Log[2]

The limit result coincides with the following output

FunctionExpand[f[x1]]
Limit[%,x1->0]

-2*I*Cosh[2*x1]*Interval[{-Pi, Pi}] + Cosh[2*x1]*(CoshIntegral[2*x1] +
(1/2)*(Log[(-I)*x1] + Log[I*x1]) - Log[x1]) - Log[Abs[x1]] -
Sinh[2*x1]*SinhIntegral[2*x1]
EulerGamma + I*Interval[{-2*Pi, 2*Pi}] + Log[2]

The real part of the last expression is the peak of the function shown
in the previous plot. Indeed

DeleteCases[%, I*(x_)]
N[%]

EulerGamma + Log[2]
1.2703628454614782

Applying a Series expansion over 0 does not make the things more
clear; quite the opposite!

Normal[Series[f[x1], {x1, 0, 3}]]
Limit[%, x1 -> 0]

-Log[Abs[x1]] - (1/2)*Sqrt[Pi]*MeijerG[{{0}, {}}, {{0, 0}, {1/2}}, 0]
-
  (1/2)*Sqrt[Pi]*x1^2*Derivative[0, 0, 1][MeijerG][{{0}, {}}, {{0, 0},
{1/2}}, 0]
Infinity

So, what is really the behavior of the function as x1->0?

Is it real valued and bounded as the Plot command and the following
command indicate?

(f[N[10^(-10*#1), 300]] & ) /@ Range[20]

Does it having an imaginary part with range between -2Pi and 2Pi as
the Limit command indicates or is in reallity be unbounded as the
Series command shows above?

I really appreciate anyone's ideas/response. It is very important
(about a research project) to know whether f[x1] is bounded as x->0 or
not!

Here are some relevant graphs:

(Plot3D[Evaluate[#1[-Log[Abs[x1]] - (1/2)*Sqrt[Pi]*MeijerG[{{0}, {}},
{{0, 0}, {1/2}}, x1^2] /. x1 -> x + I*y]], {x, -2, 2},
    {y, -2, 2}, PlotPoints -> 50, Mesh -> False] & ) /@ {Re, Im}

(ContourPlot[Evaluate[#1[-Log[Abs[x1]] - (1/2)*Sqrt[Pi]*MeijerG[{{0},
{}}, {{0, 0}, {1/2}}, x1^2] /. x1 -> x + I*y]],
    {x, -2, 2}, {y, -2, 2}, PlotPoints -> 50, Contours -> 50,
ContourShading -> False] & ) /@ {Re, Im}

As regards the behavior at infinity version 5.2 simply returns the
limit unevaluated:

Limit[f[x1], x1 -> Infinity]
Limit[-Log[Abs[x1]] - (1/2)*Sqrt[Pi]*MeijerG[{{0}, {}}, {{0, 0},
{1/2}}, x1^2], x1 -> Infinity]

I tried also

Series[f[x1], {x1, Infinity, 5}]

f[1/x1] // FunctionExpand
Limit[%, x1 -> 0]

with failed results.

Can anyone provide with information about the behavior of f[x1] as x1-
>Infinity?

It is the same important to know this thing.

Thanks in advance for anyone's response!

Dimitris



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