integrable singularities
- To: mathgroup at smc.vnet.net
- Subject: [mg9764] integrable singularities
- From: "-don." <dexter at csd.uwm.edu>
- Date: Tue, 25 Nov 1997 00:07:13 -0500
- Organization: University of Wisconsin - Milwaukee
- Sender: owner-wri-mathgroup at wolfram.com
Hello,
I'm trying to perform some integrals of functions which contain
(integrable) singularities. This can be done on 1-dim (line) integrals
by:
1) Integrate[f,{x,x1,x2},PrincipalValue -> True] if the integral has a
"nice" solution.
2) NIntegrate[f,{x,x1,x0,x2}] where x0 is the singularity 3)
CauchyPrincipalValue[f,{x,x1,{x0},x2}] where x0 is the singularity.
Unfortunately, my case is 2 (and will eventually be three) dimensional,
and it doesn't have a "nice" solution. I've tried:
1b) Integrate [f,{x,x1,x2},{y,y1,y2},PrincipalValue -> True] which
returns itself since there's no "nice" solution.
2b) NIntegrate[f,{x,x1,x0,x2},{y,y1,y0,y2}], and 3b)
CauchyPrincipalValue[f,{x,x1,{x0},x2},{y,y1,{y0},y2}]
with the result that 3b complains about the limit lists and then passes
everything to NIntegrate -- effectively doing the same thing as 2b. 2b
kind of works, but I assume it's logically wrong since this removes ALL
points x=x0 and ALL points y=y0 rather than just the singularity at
(x,y)=(x0,y0). 2b also complains about convergence before spitting out
its answer so I'm not sure the result is good.
specifically, for those who've read this far, my f(x,y) is the Hankel
function of second kind, order zero. H2[n_,z_] := BesselJ[n,z] - I
BesselY[n,z]. here n=0 and z is the distance between two points on the
surface, one of which is the singularity. Thus, I'm after:
Integral[H2[0,Distance[{r,theta},{r0,theta0}]],{r,0,R},{theta,0,2 Pi}]
where {r0,theta0} is the singularity point and {r,theta} vary over a
disc containing that singularity point.
The obvious answer is to break the integral up into bits and use the
small argument approximation to the Hankel function for the bit with
the singularity, which I've done. I'm both a) trying to use the power
of Mathematica to do it more elegantly, and b) trying to verify the
results of doing it in small bits.
Suggestions? (if this hasn't been clear enough, let me know...it's hard
to describe this stuff in text). responses here or to
dexter at csd.uwm.edu would be hugely appreciated.
thanx,
-don.