Multidimensional numerical integration problem
- To: mathgroup at smc.vnet.net
- Subject: [mg38506] Multidimensional numerical integration problem
- From: "Andi Tröster" <troester at ap.univie.ac.at>
- Date: Fri, 20 Dec 2002 04:27:42 -0500 (EST)
- Organization: Vienna University, Austria
- Sender: owner-wri-mathgroup at wolfram.com
hi,
i appologize in advance if the following sounds a little naive;
however, I have tried very hard to find a solution to
the following problem, without success, and I would appreciate
any hint towords a solution gladly:
i need to evaluate two mutidimensional integrals of the following form
as functions of a parameter m: (i give their form in TeX style to
make the problem more clear, and since a direct mathematica code
would not be useful anyway):
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$
I_1(m):=\int_{|p|<1} d^3p \int_{|q|<1} d^3q \int_{|r|<1} d^3r
\delta^3(p+q+r)
1/(p^2+m) 1/(q^2+m) 1/(r^2+m)
$$
$$
I_2(m):=\int_{|p|<1} d^3p\int_{|q|<1} d^3q\int_{|r|<1} d^3r
\delta^3(p+q+r+s)
1/(p^2+m) 1/(q^2+m) 1/(r^2+m) 1/(s^2+m)
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%`%%%
The dirac delta functions can of course be used to eliminate on of
the tree-dimensional integrals, in which case one is left with a
6 - and a 9-dimensional integral, respectively.
(To physicists: These integrals coorespond to certain Feynman diagrams in a
threedimensional
theory with mass m evaluated using a momentum cutoff at modulus 1)
I have tried various analytical simplifications (like i.e. Schwinger
parametrization,
variable substitutions, partial integration etc.)
but I cannot come up with any form that allows a numerical evaluation for
m small, which is the case I am interested in.
Any ideas?
regards,
Andy