Regularization of integral by substraction

[zetafunction]

given the divergent integral in n-variables

\int_{V} f(q1,q2,...,qn)dq1,dq2,...dqn

my question is if in general one can substract a Polynomial K in the variables q1,q2,...,qn so the integral

\int_{V} (f(q1,q2,...,qn)-K(q1,q2,...,qn))dq1,dq2,...dqn

is FINITE , then it would appear divergent integrals related to \int (q1)^{m}dq1

for positive 'm'

 

[Eynstone]

No, consider the integrand exp(q_1+q_2).

 

[vanhees71]

I guess what the OP is after is the BPHZ renormalization, where divergent integrals appearing in perturbative calculations of one-particle irreducible Green's functions are not regularized in any way but made finite directly by subtracting the integral with certain values of the external momenta (determining the renormalization point). This of course rests on Weinberg's theorem on the asymptotic behavior of such integrals. You find a quite detailed description of this technique in my qft writeup:

http://theorie.physik.uni-giessen.de/~hees/publ/lect.pdf