Re: bugs in Mathematica 5.1

*To*: mathgroup at smc.vnet.net*Subject*: [mg54045] Re: [mg54032] bugs in Mathematica 5.1*From*: yehuda ben-shimol <bsyehuda at gmail.com>*Date*: Wed, 9 Feb 2005 09:27:17 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

Hi, I use the same system AND, for Integrate[E^( I*x^2),{x,0,Infinity}] I just got a complaint from Mathematica that the analytic integral cannot converge, so How exactly did you get 0??? The line In[1] in your post however is correct for the Sine and Cosine rpresentation. Note that in In[3] the integrand of the NIntegrate is not identical to the integrand of the analytical Integrate (you changed the - sign into + sign). I changed the (+) to (-) and reevaluated. After evaluation I got the following result: Analytic -> 7.95493 NIntegrate -> 15.4442 This is surprsing since the function is smooth, always positive and always stays under 8, so the result I got from the NIntegrate is the wrong one. I played a little bit with AccuracyGoal, Method and other parameters of NIntegrate, with no success. Does anyone else has an idea?? yehuda On Tue, 8 Feb 2005 05:31:09 -0500 (EST), Gennady Stupakov <stupakov at yahoo.com> wrote: > I tried to post this a few days ago, but it looks like it did not make it. > > Here is a couple of bugs that I found recently in Mathematica 5.1. > > In[13]:={$System, $Version, $MachineType, $ProcessorType} > Out[13]={"Microsoft Windows","5.1 for Microsoft Windows (October 25, 2004)", > "PC", "x86"} > > First, I integrate E^(I*x^2), from 0 to Infinity and get zero, which, of > course, is wrong. > > In[1]:={Integrate[E^(I*x^2), {x, 0, Infinity}], Integrate[Cos[x^2] + > I*Sin[x^2], {x, 0, > Infinity}]} > Out[1]={0, (1/2 + I/2)*Sqrt[Pi/2]} > > Second is a more complicated integral that I recently encounted in my > research. > > In[2]:=Integrate[E^(a*Cos[x] - b*Cos[2*x]), {x, 0, 2*Pi}, > GenerateConditions -> True] > Out[2]=If[Re[a] < Re[b], 2*Pi*BesselI[0, -a + b], Integrate[E^(a*Cos[x] - > b*Cos[2*x]), {x, 0, > 2*Pi},Assumptions -> Re[a] >= Re[b]]] > > Let us check this result comparing it with numerical integration for, say, > b=2 and a=1: > > In[3]:= > b = 2.; > a = 1.; > {Integrate[E^(a*Cos[x] - b*Cos[2*x]), {x, 0, 2*Pi}], NIntegrate[E^(a*Cos[x] > + b*Cos[2*x]), > {x, 0, 2*Pi}]} > Out[5]={7.95493, 20.8711} > > Again, the analytical result is wrong. > > It would be interesting if those bugs are reproduced on other OS and/or > versions of Mathematica. > > Gennady. > >