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


"Robert Israel" <israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote in message news:rbisrael.20081128060325$7cb6@xxxxxxxxxxxxxxxxxxxx
"Jay R. Yablon" <jyablon@xxxxxxxxxxxx> writes:

Dear Friends,

I submitted a post yesterday where I inquired whether anyone knew of a
closed form solution for the definite integral $ from -oo to +oo which I
will represent here as:

$dx exp [ -Ax^4 -Bx^3 -.5Kx^2 +Jx ]

I have received several opinions across several Usenet groups (including
here) that there is no closed form for this.

I believe there may be a closed form, and that one can do this to any
order and is not limited to only fourth order. Please tell me if you
agree.

The (possible) closed form solution is posted below.


http://jayryablon.files.wordpress.com/2008/11/polynomial-solution-writeup.pdf


Please advise if this is a) correct and b) new, or just a good
"exercise."

No, it is incorrect. Try, for example, A=1, B=0, J=0.
Your proposed form would be sqrt(2 pi/K) which is clearly wrong. For one
thing, it would blow up as K -> 0, while the actual integral (which
is known in closed form, and involves a modified Bessel function of the second
kind) is finite in a neighbourhood of K=0.
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

Interesting, and true.

Might it be that this is correct iff J <> 0? (Especially given that the derivation uses d/dJ throughout which makes no sense for J=0. If J=0 we would need to use d/dK.)

Can you pinpoint a problem in the derivation?

I note that this d/dJ is an approach used regularly in Quantum Field Theory to generate terms for Feynman diagrams, Wick contractions, etc., and that it is always assumed in these settings that J<>0.

Jay.

.

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