Re: hadamard finite part
- To: mathgroup at smc.vnet.net
- Subject: [mg68074] Re: hadamard finite part
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Fri, 21 Jul 2006 17:36:26 -0400 (EDT)
- Organization: The University of Western Australia
- References: <e9ibvc$52e$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
In article <e9ibvc$52e$1 at smc.vnet.net>, dimmechan at yahoo.com wrote:
> I work in the field of applied mathematics and I am interested in the
> symbolical/numerical integration of integrals in the Hadamard sense
> (that is, the finite part of divergent integrals).
> My integrals are much more complicated but here I use some trivial
> examples to show the point.
Actually, it would be helpful if you could give some of your complicated
examples.
> Next, suppose the integral
>
> Integrate[1/x^2,{x,-1,2}]
In the Hadamard sense, I would compute this as follows:
Assuming[0 < e < 1,
Integrate[1/x^2, {x, -1, -e}] + Integrate[1/x^2, {x, e, 2}] ]
-3/2 + 2/e
followed by
Coefficient[%, e, 0]
-3/2
> My first question now:
> Is it a way to get the finite part of a divergent integral through
> performing numerical integration (e.g. using NIntegrate) in Mathematica?
> I have seen some papers presenting some propper algorithms dealing with
> numerical integration of Hadamard finite part integrals but I cannot
> find any related work in connection with Mathematica.
I expect that the general answer is no. To obtain the Hadamard finite
part one must first locate the singular points, including the
end-points, and then determine the behavior near these points.
> Mathematica 3.0 and 4.0 suceeds in providing this result:
>
> Integrate[1/x,{x,0,2},GenerateConditions->False] (*version 3.0 and
> 4.0*)
>
> Log[2]
>
> However Mathematica 5.1 and 5.2 gives the result
>
> Integrate[1/x,{x,0,2},GenerateConditions->False] (*version 5.1 and
> 5.2*)
>
> Infinity
>
> Why exists this difference?
Mathematica is now more careful. In general, I would not trust
GenerateConditions->False to do Hadamard integration. For simple cases
you can always use indefinite integration.
> I can trust that for divergent integrals
> Integrate[integrand,{x,a,b},GenerateConditions->False]
> provides the desirable result in the Hadamard sense?
I don't understand: you've shown above that it does not?
> Is it a way to get Integrate to give always the finite part of a
> divergent integral?
Yes -- if you locate the singularities first. At
http://physics.uwa.edu.au/pub/Mathematica/MathGroup/Hadamard.nb
you will find a Notebook that implements one class of Hadamard integrals.
> Are there any other alternative methods (such as the implementation of
> aymptotic techniques in mathematica) to get the finite part of a
> divergent integral?
You can use series expansions about the singular points.
> P.S. The finite part of a divergent integral is of great importance in
> the area of applied mathematics.
I think that that is an exaggeration. It is not even mentioned in
Handbook of Applied Mathematics: selected results and methods,
edited by Carl E. Pearson
which is over 1300 pages long.
Cheers,
Paul
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