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

*To*: mathgroup at smc.vnet.net*Subject*: [mg27364] Re: A bug of Integrate[] in Mathematica 4.1 (and 4.0)*From*: Hendrik van Hees <h.vanhees at gsi.de>*Date*: Thu, 22 Feb 2001 02:25:09 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

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}] 2 Pi r Out[1]= ----- (correct area of a half circle!) 2 Now doing the same integral with a mathematically identical integrand: In[1]:= Integrate[Sqrt[(r-x)(r+x)],{x,-r,r},Assumptions->{r>0}] 2 Pi r Sqrt[r ] Out[1]= ------------- 4 In[2]:= Simplify[%,r>0] 2 Pi r Out[2]= ----- (WRONG factor 1/2) 4 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 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 gsi.de D-64291 Darmstadt http://theory.gsi.de/~vanhees/index.html