Re: Integrable singularity
From: David Wilkinson (david_at_wilkinson6337.freeserve.co.uk)
Date: 01/29/05
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Date: Sat, 29 Jan 2005 20:14:13 +0000
Madhusudan Singh wrote:
> Hi
>
> I wish to numerically integrate a real function with an integrable
> singularity.
>
> What are the recommended methods for accomplishing the above ?
>
> Thanks.
This is one way of doing it. Suppose the integrand is singular at x=x0
within the range of integration and can be put in the form
f(x)/(x-x0) where f(x) is non-singular.
Replace 1/(x-x0) with whatever type of integrable singularity you have
as it is just an example.
The trick then us to replace the integrand by
(f(x)-f(x0))/(x-x0) + f(x0)/(x-x0)
so you now have two integrals. The first integral is probably no longer
singular because the integrand tends to df/dx(x0) as x tends to x0 and
can be integrated numerically to any accuracy required. The second
integral can be integrated exactly as the singularity is integrable
using the Cauchy principle value.
Hope this helps.
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