Re: Third Draft, Possible Closed Form Solution to Gaussian-Based Integrals with High-Order Field Interactions


"Dono" <sa_ge@xxxxxxxxxxx> wrote in message news:bcce17c7-87e8-4165-8bb5-2019c3e68ef6@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
.. . .

Jay,

Basically you have two tasks in order to complete the proof:

1. Establish the conditions under which you can exchange the Integral
of the Derivative with the Derivative of the Integral (this is the
"Theorem of Integrals Function of Parameter" I was refering to). In
effect , you ned to prove the convergence of the integral before you
can exchange the order of integration and derivation.

2. Extend the "Theorem of Integrals Function of Parameter" to include
infinite series of functions. This one is going to be tougher, unless
you can find it in some book. The way it stands right now, you are
applying the mechanisms without having a formal proof allowing you to
do so.

PS: I emailed Eric what I thought might be wrong with his proof. The
residue computation is missing a singularity point at the infinity.

Thanks for bearing with me Dono.

Below I have posted a link to an excerpt from Zee's book where he applies a very similar approach to what I have done here:

http://jayryablon.files.wordpress.com/2008/11/zee-baby-problem.pdf

What would be your comments about this use of "Theorem of Integrals Function of Parameter"? Can you pinpoint how what he does is different from what I do, if anything? Has he left something out? Is it implied?

In short, I don't see how what I am doing is really any different form what he is doing -- in fact, all that is different is that I also have a third-order term -- and I'd like to understand the difference, if any.

Jay.

.

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