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