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