Pseudo-Singular Integrals

*To*: mathgroup at yoda.physics.unc.edu (Mathematica User's Group)*Subject*: Pseudo-Singular Integrals*From*: Keith Clay <clay at galileo.phys.washington.edu>*Date*: Fri, 20 May 1994 12:00:37 -0700 (PDT)

This is more of a problem of numerical analysis in genereal than of anything specific to MMA, but I was wondering if anyone has specific tools for MMA may to help me deal with it. I have succesfully used Mathematica to analytically integrate out 3 of the variables in some very messy 5 dimensional expressions. The results are functions of two variables (w and y) which must be integrated over y from 0 to 1. These expressions are typically 100 kB of ASCII code, so translation to another language is difficult and would not necessarily solve the problem. Since the expressions as functions of y cannot be integrated in closed form, the use of MMA's NIntegrate is the clear choice. The problem is that these functions are very nearly singular. A fictitious but representative (although overly simplified) example might look like: -PolyLog[2,-1/y] (1 - y) y 1 - y ----------------- + Log[---------] PolyLog[2, -(-----)] + (Less Singular) 4 w + y y (w + y) If I NIntegrate this with w set equal to 1, I have no problem. But if I probe the region where w->0 (try w=10^-n for n = 4,5,6,7...), NIntegrate grinds for hours and then fails miserably. The reason is clearly that the first term (roughly Log[y]^2/y^4 for 1 >> y > w ) appears numerically divergent until one probes the region y << w. Is there any way to help MMA with this? I am looking for three or more significant figures. Thus far, the use of clever variable changes has not done the trick, even when they enlarge the region where y < w, presumably because the Jacobians then become very nearly singular. Thanks is advance for any help you can give me. Sincerely, Keith ------------------------------------------------------------------------ Keith Clay Department of Physics, FM-15 (clay at galileo.phys.washington.edu) University of Washington ( -or- clay at phys.washington.edu ) Seattle, WA 98195