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What is the result of this double integral?

  1. Jun 23, 2008 #1
    Consider the double integral
    [tex]
    \int_{-\infty}^{\infty}dx f(x) \, \int_{-\infty}^{\infty}dy g(y)
    [/tex]
    The first one gives 0 the second one gives infinity (diverges). Then how to express the result of the integral? Is it 0 or infinity or neither (indeterminate)? Any other comments about the integration?
     
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  3. Jun 23, 2008 #2

    D H

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    What you wrote isn't so much a double integral as it is a product of two integrals. What rule from basic calculus can be used to resolve things that tend to [itex]0\cdot\infty[/itex] as some parameter tends to zero or infinity?
     
  4. Jun 23, 2008 #3
    It was really wrong to call that a double integral.

    L'Hospital rule come to my mind as the answer to your question but that is applicable in calculating limit problems. This is case different. I do not know of any method applicable here.

    Also there is no "single parameter" in the problem which gives rise to [tex]0\cdot\infty[/tex] form.
     
    Last edited: Jun 23, 2008
  5. Jun 23, 2008 #4

    D H

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    The improper definite integral
    [tex]\int_{-\infty}^{\infty}f(x)\,dx[/tex]
    is shorthand for
    [tex]\lim_{L\to\infty}\int_{-L}^{L}f(x)\,dx[/tex]
    So, how is this case any different?
     
  6. Jun 23, 2008 #5

    HallsofIvy

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    NO! that is the "Cauchy Principal Value". The correct definition is
    [tex]\lim_{A\to\infty}\lim_{B\to\infty}\int_{B}^{A}f(x)\,dx[/tex]
     
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