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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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