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I have a function defined as follows: $f(x,y)= \dfrac{x^2-y^2}{\left(x^2+y^2\right)^2}$, if $(x,y)\neq (0, 0)$ and $f(x,y)=0$ if $(x,y)=(0,0)$. Now, $$\int_0^1\int_0^1 f(x,y)\,\text{d}x~\text{d}y=-\frac{\pi}{4}$$ and $$\int_0^1\int_0^1 f(x,y)\,\text{d}y~\text{d}x=\frac{\pi}{4}.$$ The question I have is this: why does this not contradict Fubini's theorem?

thanks.

  • 5
    Read a statement of Fubini's theorem. Go through its hypotheses. Work out which is not satisfied.2011-11-24
  • 15
    For crying out loud... the analysis of this **exact** function is on the wikipedia page... http://en.wikipedia.org/wiki/Fubini%27s_theorem#Rearranging_a_conditionally_convergent_iterated_integral2011-11-24
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    @TheChaz: For crying out loud...did you read my question? I asked why Fubini's theorem is NOT refuted, and not why it doesn't hold?2011-11-24
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    @Chris What do you mean by "Fubini's theorem is NOT refuted"? The theorem is not refuted because it is not even applicable for this example. And that is because the premises or hypotheses of the theorem are not met.2011-11-24
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    @Srivatsan: Well that was the question asked. Moreover, at the linked page given by The Chaz, after, giving the example, they said that...Fubini's theorem, says that...so if the conditions are not satisfied, why are they applying the it?2011-11-24
  • 0
    Why the downvotes? What is wrong with the question?2011-11-24
  • 5
    I didn't downvote. [Fubini's theorem](http://en.wikipedia.org/wiki/Fubini's_theorem) says: *if* $\int_{A \times B} |f(x,y)|\,d(x,y)\lt\infty$ *then* the double integral $\int_{A \times B} f(x,y)\,d(x,y)$ is equal to the iterated integrals you write. Here you have two iterated integrals that aren't equal, so you can conclude from Fubini's theorem that the double integral isn't finite. And indeed, as Michael's answer shows, the first double integral I mentioned is infinite. No contradiction here, but a cautionary example showing that *some* hypotheses are necessary to get a Fubini-like theorem.2011-11-24
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    To supplement t.b.'s answer, let's note that the theorem is a conditional statement "if P, then Q". P is about finiteness; Q is about integrals being equal. The contra positive states (again, roughly) that if the integrals aren't equal, then there was an infinity to start out with! Also, I haven't voted on this question.2011-11-24
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    @All: Thanks very much for taking the time to make me sort out my confusion.2011-11-24

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