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Peter Spellucci
science forum Guru

Joined: 29 Apr 2005
Posts: 702

Posted: Wed Jul 12, 2006 9:19 pm    Post subject: Re: integrand numerical singular near boundary

In article <44B53BD2.4A2DBC6A@axelvogt.de>,
Axel Vogt <test3@axelvogt.de> writes:
 Quote: What is an appropriate method to (automatically) care for something like Int(t/(1-z*t),t = 0 .. 1), z = 1-1e-8*I close to the boundary? I do not mean the specific form above (only: to integrate over an interval). If the singularity 1/z is off the bounds one can pass to another path to reach 0 and 1. But if 1/z is close contour integrals do not help me much. Is there some standard method to 'zoom in around the boundary'?

integrals over infinite intervals into ones over [0,1] with singularity
at the boundary and integrates this with adaptive refinement at the
boundaries using gauss and gauss kronrod nodes (interior nodes)
try it here:
hth
peter
Robert B. Israel
science forum Guru

Joined: 24 Mar 2005
Posts: 2151

Posted: Wed Jul 12, 2006 9:19 pm    Post subject: Re: integrand numerical singular near boundary

In article <44B53BD2.4A2DBC6A@axelvogt.de>,
Axel Vogt <test3@axelvogt.de> wrote:
 Quote: What is an appropriate method to (automatically) care for something like Int(t/(1-z*t),t = 0 .. 1), z = 1-1e-8*I close to the boundary? I do not mean the specific form above (only: to integrate over an interval). If the singularity 1/z is off the bounds one can pass to another path to reach 0 and 1. But if 1/z is close contour integrals do not help me much. Is there some standard method to 'zoom in around the boundary'?

One thing you might try is subtracting off the singular part and
integrating it separately.
Thus the singular part of your integrand t/(1-z t) at t=1/z is
-1/z^2 (t-1/z)^(-1), and int_0^1 (t-1/z)^(-1) dt = ln(1-z).
After subtracting the singular part, you have a nice function
with no singularity at t=1/z.

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Axel Vogt

Joined: 03 May 2005
Posts: 93

 Posted: Wed Jul 12, 2006 6:13 pm    Post subject: integrand numerical singular near boundary What is an appropriate method to (automatically) care for something like Int(t/(1-z*t),t = 0 .. 1), z = 1-1e-8*I close to the boundary? I do not mean the specific form above (only: to integrate over an interval). If the singularity 1/z is off the bounds one can pass to another path to reach 0 and 1. But if 1/z is close contour integrals do not help me much. Is there some standard method to 'zoom in around the boundary'?

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