Regulated Integral with Convergence Factor

["pi"mp]

I have a question pertaining to a computation I'm trying to carry out. Without getting too much into the details, I have a finite integral over two variables. One integral vanishes, and one diverges allowing for the finite value. I had to regulate the divergent integral so I introduced a small \epsilon that makes that integral finite. Of course, it diverges as \epsilon \to 0 and that other integral goes to zero under this same limit. Turns out I also had to introduce a convergence factor \delta. I am easily able to get a result in terms of \epsilon and \delta. But now I need to take both those parameters to zero but it appears that my answer will depend on which one I take to zero first and how fast it goes to zero.

What is the protocol here? For example, I get terms like \epsilon \log \epsilon/ \sqrt{\delta} which is zero if I take one to zero first and indeterminate if I take the other first. Any help would be appreciated!

 

[Stephen Tashi]

What is the protocol here?

You haven't made it clear what you are doing. Are you trying to prove the double integral exists and evaluate it? Or are you dealing with a divergent double integral and tying to define some modified version of integration that assigns it a finite value?

 

["pi"mp]

Indeed, sorry, the computation is a bit of a mess and didn't want to write up all the ugly details. I want to show the double integral exists, and evaluate it. I've got an analytic expression for the answer, but it depends on both the parameters described above which both need to be taken to zero. It seems my answer depends on which I take to zero first.