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Re: Integrating Feynman integrals in mathematica
*To*: mathgroup at smc.vnet.net
*Subject*: [mg50874] Re: Integrating Feynman integrals in mathematica
*From*: pabird at supanet.com (Xman)
*Date*: Sat, 25 Sep 2004 01:55:15 -0400 (EDT)
*References*: <ciu5iv$81d$1@smc.vnet.net> <cj0n33$lc6$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
"Steve Luttrell" <steve_usenet at _removemefirst_luttrell.org.uk> wrote in message news:<cj0n33$lc6$1 at smc.vnet.net>...
> Unless I have missed the point of your question, this is standard material
> covered in textbooks on quantum field theory. I have Itzykson + Zuber's
> tome, and I find your first integral discussed in section 1-3-1 on Green
> functions. Your second integral is related to your first integral by the
> trick of differentiating w.r.t. an auxiliary parameter (i.e. make the
> replacement m^2-->a m^2 in your first integral, differentiate w.r.t. a, then
> set a-->1).
>
Thanks for your input. Yes, the integrals are from QFT but the books
on QFT I have read usually leave the integrals as they are or give an
approximation to them only. So I want to find a *simple* series
solution to them which I believe is simplest in the form above. I am
sure I have seen something similar somewhere before unsfortunately
can't remember where! I was hoping that someone more knowledgable in
Mathematica or QFT than me would be able to confirm what I susupect
which is that the second integral can be written as:
Sum[ ((m^2/4)^n |x.x|^n / n!^2)*( log|x.x| - Sum[1/p ,p=1..n])
,n=0..Infinity]
From which, for example, I can instantly see the function is smooth
and not wave-like.
Such a series would also, it seems, converge very rapidly and
accuratley which would be very useful. I also believe that in the form
P(m|x|)log(|x|)-Q(m|x|), the function P is the BesselI[0,x] function
from Mathematica but I don't know whether Q is a well-known function.
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