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

On Sun, 30 Nov 2008 08:59:17 -0800 (PST), Dono <sa_ge@xxxxxxxxxxx>
wrote:
[...]

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.

Not hard to do, the exponential makes things easy.


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.

Term by term integration of a convergent series does not always
promise a convergent result.



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

No.

The final polar integral is over the unit circle. Whether or not there
is a singular point at infinity does not matter as I'm not extending
the contour out to infinity.

The essential singularity at z=0 is enough to ruin any possible
application of this method. That's my conclusion: wrong, but for a
subtle reason.
.

Relevant Pages