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Re: A bug of Integrate[] in Mathematica 4.1 (and 4.0)

Rino Bandiera wrote:

> > Thank you for the report. Integrate[] appears to be behaving
> erroneously in
> > all of the cases you present. I have notified the the developers of
> the
> > problems. Unfortunately, there does not appear to be a workaround.
> > Unfortunately, there are no bug lists available, and I will not be
> informed
> > of the status of the bug until it is fixed. Most likely, a patch will
> not be
> > made available.

I think this behaviour of Wolfram Research should be made public
somewhere on the web! Why do these guys not care about BUGS? A software
developer should fix his bugs and make this bug fixes available to the
users of his product. 

Another annoying bug in Integrate is its handling of branch cuts in real
integrals. A "nice" example is the area of a half circle. You may try if
there has something changed in version 4.1 compared to 3.0 or 4.0.
Reporting this bug they did not admit that it is one at all because
Integrate would do "complex integrals". After my reply that then there
should be a possibility to define the path of integration in the complex
plane I never got a convincing answer ;-(. So here is my example:

Correct is the following:

In[1]:= Integrate[Sqrt[r^2-x^2],{x,-r,r},Assumptions->{r>0}]

        Pi r
Out[1]= -----   (correct area of a half circle!)

Now doing the same integral with a mathematically identical integrand:

In[1]:= Integrate[Sqrt[(r-x)(r+x)],{x,-r,r},Assumptions->{r>0}]

        Pi r Sqrt[r ]
Out[1]= -------------

In[2]:= Simplify[%,r>0]

        Pi r
Out[2]= ----- (WRONG factor 1/2)

The reason is that Mathematica simply puts the boundaries which are on
the branch points of the integrand in the indefinite integrals which are
along branchcuts without proper I epsilon-descriptions (well known in
quantum theory):

In[3]:=  Integrate[Sqrt[(r-x)(r+x)],x]

        x Sqrt[(r - x) (r + x)]
Out[3]= ----------------------- - 
      2                                 Sqrt[r - x] x
     r  Sqrt[(r - x) (r + x)] ArcTan[--------------------]
                                     (-r + x) Sqrt[r + x]
>    -----------------------------------------------------
                   2 Sqrt[r - x] Sqrt[r + x]

Hendrik van Hees		Phone:  ++49 6159 71-2751
c/o GSI-Darmstadt SB3 3.183	Fax:    ++49 6159 71-2990
Planckstr. 1			mailto:h.vanhees at
D-64291 Darmstadt

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