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.

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

```

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