mm-889 === Subject: a function containg MeijerG (limit behavior) The function is: f[x1_] := -Log[Abs[x1]] - (1/2)*Sqrt[Pi]*MeijerG[{{0}, {}}, {{0, 0}, {1/2}}, x1^2] at x1=0. EulerGamma + I*Interval[{-2*Pi, 2*Pi}] + Log[2] The limit result coincides with the following output FunctionExpand[f[x1]] -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}]] -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 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 not! Here are some relevant graphs: (Plot3D[Evaluate[#1[-Log[Abs[x1]] - (1/2)*Sqrt[Pi]*MeijerG[{{0}, {}}, (ContourPlot[Evaluate[#1[-Log[Abs[x1]] - (1/2)*Sqrt[Pi]*MeijerG[{{0}, limit unevaluated: Limit[-Log[Abs[x1]] - (1/2)*Sqrt[Pi]*MeijerG[{{0}, {}}, {{0, 0}, I tried also Series[f[x1], {x1, Infinity, 5}] f[1/x1] // FunctionExpand with failed results. Can anyone provide with information about the behavior of f[x1] as x1- It is the same important to know this thing. Dimitris === Subject: limit behavior, part 2 A graphical examination shows that the function in question (- Log[Abs[x1]] - (1/2)*Sqrt[Pi]*MeijerG[{{0}, {}}, {{0, 0}, {1/2}}, but I still look for a symbolic verification if possibly! Plot[{-Log[Abs[x1]], -Log[Abs[x1]] - (1/2)*Sqrt[Pi]*MeijerG[{{0}, {}}, Dimitris === Subject: Re: record intermediate steps Oooops! I meant Can you provide me with some examples of application of the trap method of your own? that is,examples instead of advances (I really don't know what I was Dimitris === Subject: Array reference help please I've managed to get this code working but I'm stumped on a good way to refer to the entries in the data when plotting. The only thing I can get to work is to refer to the first of each pair as Part[data[[i]],1] and the second as Part[data[[i]],2]. And more humiliating, the second one I have to spell out twice. I tried all kinds of substitutions but failed. I know you gurus here know how to make this code look better and ë±=100;(* decay parameter*) (* first parameter is pulse amplitude, second is time of occurrence - later to follow Poisson *) data=Table[Random[],{40},{2}]; li=Length[data]; Plot[Sum[ Part[data[[i]],1] UnitStep[t-Part[data[[i]],2]] Exp[-ë± (t-Part[data[[i]],2])], {i,li} ] ,{t,0,1}] === Subject: GraphicsToolbox - 3D Programming Package The latest free 3D programming packaged for interactive Mathematica graphics is available at: http://www.grtoolbox.com/ What do you think? === Subject: Re: Integrating SphericalHarmonicY Such integrals are _not_ trivial for general values of n, l, m. The correct approach is essentially the approach you use in physics (and mathematics) if you are doing the calculation by hand. To compute integrals such as Integrate[LegendreP[m, x] x^n, {x, -1, 1}] (where it is implicit that n and m are non-negative integers), or to show that Integrate[LegendreP[m, x] LegendreP[n, x], {x, -1, 1}] vanishes for n != m, one can use the generating function (1 - 2 x t + t^2)^(1/2) == Sum[t^n LegendreP[n, x], {n, 0, Infinity}] and interchange the order of summation and integration. As an exercise, show that Integrate[(LegendreP[n + 1, x] - LegendreP[n - 1, x])/(1 - x)^(3/2), {x, -1, 1}] == -4 Sqrt[2] for n = 1, 2, 3, ... (see The Mathematica Journal 7 (2)). Paul _______________________________________________________________________ Paul Abbott Phone: 61 8 6488 2734 School of Physics, M013 Fax: +61 8 6488 1014 The University of Western Australia (CRICOS Provider No 00126G) AUSTRALIA http://physics.uwa.edu.au/~paul === Subject: Re: How to parse result from Reduce[ ] function Hi Anton, you e.g. use Cases. Assume that your output is in t: will give you the part belonging to p=. Daniel === Subject: Re: Unwrap list of angles Here are a couple of MathGroup posts on this topic. As author of the first one, I'd say go with the second. http://forums.wolfram.com/mathgroup/archive/1998/May/msg00107.html http://forums.wolfram.com/mathgroup/archive/1998/May/msg00105.html Daniel Lichtblau Wolfram Research === Subject: Re: Unwrap list of angles I ended up writing a version of this a while ago. Here's the definition I use. The first is for a list of one dimension; it's used in the second definition (since I usually deal with lists of two dimensions where the second element in the ordered pairs is the phase. UnwrapPhase[list_ /; Length[Dimensions[list]] == 1] := Module[{newTest, lC = Mod[ ListCorrelate[{-1, 1}, list], Pi, -Pi/2]}, newTest[1] = list[[1]]; newTest[i_] := newTest[i] = newTest[i - 1] + lC[[i - 1]]; newTest[#] & /@ Range[Length[list]]] UnwrapPhase[list:{{_,_}..} ]:= Transpose[ { list[[All, 1]], UnwrapPhase[list[[All,-1]]] } ] Hope that helps! C.O. -- Curtis Osterhoudt gardyloo@mail.remove_this.wsu.and_this.edu PGP Key ID: 0x088E6D7A Please avoid sending me Word or PowerPoint attachments See http://www.gnu.org/philosophy/no-word-attachments.html === Subject: Re: Unwrap list of angles Do you mean confine then in a certain range (as [0, Pi])? A dirty way is: Table[Random[Real, 100], {i, 1, 100}]; Cos[%]; ArcCos[%] Hao === Subject: Publicon style sheets Are there any sources of additional style sheets for Publicon - Michael === Subject: Re: Would someone confirm they also see this problem? ... ... It doesn't appear to be a browser problem, IE and Opera show identical results. If I don't use a browser at all and inspect the generated gif file with an assortment graphics programs I see the same results. Anyone have a really big display with lots of pixels under Windows XP and can check whether this this persists at higher resolutions? Wolfram support said they thought it vanished at higher resolutions but I'm wondering whether that was just that they didn't set ImageSize large enough to force the result well beyond the size of their screen. === Subject: FindRoot Precision Problem I'm running Mathematica 5.1 and have been struggling with this problem for the last few days... I'm trying to use FindRoot to do a simultaneous solve on five equations involving five unknown variables. Unfortunately for me, I continued getting the Encountered a Singular Jacobian error not far from my guess values. In my attempts at debugging, I noticed the following peculiarity (I'm giving a very simplified example below): If one of my equations is: x1+x2 = = 0 then FindRoot is able to converge successfully. However, if I change the equation to: x1*100+x2*100 = = 0 I get the Singular Jacobian error. It is this latter equation type that I need to be able to solve (i.e. the one with larger orders of magnitude.) It is for this reason that I feel my problem is related to precision and accuracy. I've tried playing around with the precision and accuracy, but to no avail. A couple of things I've tried: changing my PrecisionGoal, AccuracyGoal, and WorkingPrecision parameters, as well as globally increasing my precision using $MinPrecision. My search of the math group archive pulled this up written by Daniel Lichtblau: http://forums.wolfram.com/mathgroup/archive/1998/Mar/msg00336.html This is hauntingly similar to my problem, and it is solved here using the GroebnerBasis function. But, in my situation, my equations are nonlinear...I have some exponentials in my equations, and the GroebnerBasis function reminds me of that by producing the error, Equations appear to be solved in an essentially non-algebraic way. Any help would be immensely appreciated! I've included the actual five equations in a Mathematica notebook file at the URL below, along with my FindRoot call and guesses. I've tried to clean the equations up a little bit using Simplify before making this notebook file. http://people.virginia.edu/~sdk5n/math_group.nb -- ______________________ Scott D. Kasen Graduate Research Assistant Department of Materials Science and Engineering University of Virginia P.O. Box 400745 116 Engineers Way Charlottesville, VA 22904-4745 email:sdk5n@virginia.edu voice: 434-982-5675 fax: 434-982-5660 === Subject: Re: Fitting data to a curve In[1]:= <