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Note: this is fixed in version 9.

When I perform the double integral in Mathematica, 

Integrate[(x (1 - x))^z (y (1 - y))^z, {x, 0, 1}, {y, 0, 1}]

which should give

$$B(z+1,z+1)^2 = \frac{\Gamma(z+1)^4}{\Gamma\left(2(z+1)\right)^2}$$

where $B(x,y)$ is the Beta function and $\Gamma(z)$ is the Gamma function because the integral is a product of two Beta functions, I instead get this ratio of Gamma functions times the extra factor $(-1)^{2z}$. What is going on here? To make matters stranger, if I do the integral instead using two nested calls to Integrate (one to integrate out $x$ and one to integrate out $y$), I get the ratio of Gamma functions without the incorrect extra factor.

Szabolcs
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Mike
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  • After playing around and breaking the problem up, it looks like a legitimate issue with Integrate. Send the example in to support@wolfram.com. – Searke Jul 19 '12 at 20:12
  • Are you sure about the validity of the expression you gave over the entire complex plane? It seems that the $ (-1)^{2z} $ factor comes from a resultant of the Gammas. – gpap Jul 19 '12 at 22:03
  • @gpap, that's not the issue: try running Assuming[Re[z]>0 && Im[z]==0, Integrate[(x (1 - x))^z (y (1 - y))^z, {x, 0, 1}, {y, 0, 1}]. $(-1)^{2z}$ should definitely not be there. – Mike Jul 19 '12 at 22:12
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    Definitely a bug. Hoping the fix creates no trouble of its own. – Daniel Lichtblau Jul 19 '12 at 23:06
  • @Mike Please post that as an answer and accept when the system lets you (so the question doesn't appear unanswered). – Szabolcs Jul 21 '12 at 15:28

1 Answers1

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Wolfram Research has confirmed that this is a bug. From an email I just received: Integrate does appear to be giving an incorrect result, and I have forwarded the example to our developers.

Mike
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  • They always say the same about Integrate. Look e.g. here http://mathematica.stackexchange.com/questions/8161/mathematica-gives-wrong-answer-for-integral and here http://mathematica.stackexchange.com/questions/3568/bug-in-integrate-for-mathematica and read the comments here : http://stackoverflow.com/questions/8353441/can-you-explain-this-mathematica-assumptions-behaviour – Artes Jul 24 '12 at 19:33
  • @Artes Yes, it's quite annoying that they just wave you off with the statement that Integrate is buggy rather than giving a general sense of the reasons why it might not be working so this could at least be avoided in the future. – Mike Jul 25 '12 at 22:32