# What is the result of this double integral?

1. Jun 23, 2008

### arroy_0205

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

2. Jun 23, 2008

### D H

Staff Emeritus
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?

3. Jun 23, 2008

### 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.

Last edited: Jun 23, 2008
4. Jun 23, 2008

### D H

Staff Emeritus
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?

5. Jun 23, 2008

### HallsofIvy

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