What is the result of this double integral?

[arroy_0205]

Consider the double integral
<br /> \int_{-\infty}^{\infty}dx f(x) \, \int_{-\infty}^{\infty}dy g(y) <br />
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?

 

[D H]

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 0\cdot\infty as some parameter tends to zero or infinity?

 

[arroy_0205]

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 0\cdot\infty form.

 

[D H]

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
\int_{-\infty}^{\infty}f(x)\,dx
is shorthand for
\lim_{L\to\infty}\int_{-L}^{L}f(x)\,dx
So, how is this case any different?

 

[HallsofIvy]

The improper definite integral
\int_{-\infty}^{\infty}f(x)\,dx
is shorthand for
\lim_{L\to\infty}\int_{-L}^{L}f(x)\,dx
So, how is this case any different?

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