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Example of non-integrable partial derivative

  1. Jun 13, 2010 #1
    Can you give an example of a function [itex]f:X\times Y\to\mathbb{R}[/itex], where [itex]X,Y\subset\mathbb{R}[/itex], such that the integral

    [tex]
    \int\limits_Y f(x,y) dy
    [/tex]

    converges for all [itex]x\in X[/itex], the partial derivative

    [tex]
    \partial_x f(x,y)
    [/tex]

    exists for all [itex](x,y)\in X\times Y[/itex], and the integral

    [tex]
    \int\limits_Y \partial_x f(x,y) dy
    [/tex]

    diverges at least for some [itex]x\in X[/itex]?
     
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  3. Jun 13, 2010 #2

    Hurkyl

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    Have you tried working backwards? Rather than guess at what f has to be to give a [itex]\partial_x f[/itex] with the desired property, instead choose [itex]\partial_x f[/itex] first.
     
  4. Jun 13, 2010 #3
    If I choose [itex]\partial_x f[/itex] so that it cannot be integrated with respect to [itex]y[/itex], then I easily get a function [itex]f[/itex] which cannot be integrated with respect to [itex]y[/itex] either. It looks like a difficult task, either way you try it.
     
  5. Jun 13, 2010 #4

    Ben Niehoff

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    I have a few infinite families of solutions. Do you want me to just post them or let you figure them out?

    Hint: try [itex]X, Y = \left[ 1, \infty \right)[/itex]
     
  6. Jun 13, 2010 #5
    Feel free to post your example functions. I don't mind if you take, the right to the feel of discovery, away from me :smile:

    I might try to obtain the feel of proving somebody wrong, when I check what's wrong with your example functions :wink:
     
  7. Jun 13, 2010 #6

    Ben Niehoff

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    OK, the first one I thought of was

    [tex]f(x,y) = \frac{1}{y^2} x^{-y}[/tex]

    where X and Y are both the interval [1, infinity).

    [tex]\int_1^\infty f(x,y) \; dy[/tex]

    converges for all x in X, but

    [tex]\int_1^\infty \partial_x f(x,y) \; dy = \int_1^\infty \frac{-1}{y} x^{-y-1} \; dy[/tex]

    diverges for x = 1.
     
  8. Jun 14, 2010 #7
    I see. Very nice.
     
  9. Jun 14, 2010 #8

    Ben Niehoff

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    And here is one where the sets X and Y are the entire real line:

    [tex]f(x,y) = \frac{\sin{xy}}{y^2 + a^2}[/tex]

    Then

    [tex]\int_R \partial_x f(x,y) \; dy = \int_R \frac{y \cos{xy}}{y^2 + a^2} \; dy[/tex]

    fails to converge for x = 0.
     
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