What is the result of this double integral?

  • Thread starter arroy_0205
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  • #1
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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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  • #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?
 
  • #3
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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.
 
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D H
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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.
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?
 
  • #5
HallsofIvy
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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?
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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