Re: Integrable singularity

From: David Wilkinson (david_at_wilkinson6337.freeserve.co.uk)
Date: 01/29/05

```Date: Sat, 29 Jan 2005 20:14:13 +0000

```

> 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.

Relevant Pages

• Re: Trouble with function dblquad
... I've got an annoying error with the function dblquad. ... Warning: Maximum function count exceeded; singularity likely. ... 'Maximum function count exceeded' indicates that the integrand ...
(comp.soft-sys.matlab)
• Re: "Ambiguous" Borel summation?
... alternating divergent series and this is what makes them called "non- ... at least if the singularity is a simple pole. ... where the integrand has a simple pole at t = 1/z. ... one may be resolvable as mentioned, though it may lead to ambiguity, ...
(sci.math)
• Re: "Ambiguous" Borel summation?
... at least if the singularity is a simple pole. ... where the integrand has a simple pole at t = 1/z. ... one may be resolvable as mentioned, though it may lead to ambiguity, ...
(sci.math)
• Re: Trouble with function dblquad
... I've got an annoying error with the function dblquad. ... singularity likely. ... A nonintegrable singularity is possible. ... 'Maximum function count exceeded' indicates that the integrand has ...
(comp.soft-sys.matlab)
• Re: Third Draft, Possible Closed Form Solution to Gaussian-Based Integrals with High-Order F
... essential singularity at z=0. ... The Gaussian integral has an integrand of the form exp. ... the otherwise tame Laurent series for the Gaussian function! ... Actually that's how residues at infinity are defined, ...
(sci.physics.relativity)