bugs in Mathematica 5.1

• To: mathgroup at smc.vnet.net
• Subject: [mg54032] bugs in Mathematica 5.1
• From: "Gennady Stupakov" <stupakov at yahoo.com>
• Date: Tue, 8 Feb 2005 05:31:09 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

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.

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