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.
>