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Re: principal value

  • To: mathgroup at smc.vnet.net
  • Subject: [mg67927] Re: principal value
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Fri, 14 Jul 2006 02:11:33 -0400 (EDT)
  • Organization: The University of Western Australia
  • References: <e95aqh$gin$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <e95aqh$gin$1 at smc.vnet.net>, dimmechan at yahoo.com wrote:

> I have to evaluate the following integral (from an elasticity project)

[For clarity, I have converted the input and output expressions to 
InputForm]

  Integrate[ArcTan[x]/(x - p), {p, 0, 1}, {x, 0, 1}]

> Since p take values in the interval {0,1}, the option PrincipalValue
> must be used.

  Integrate[ArcTan[x]/(x - p), {p, 0, 1}, {x, 0, 1},   
     PrincipalValue -> True]

  (1/32) (32 Catalan + 4 Log[2]^2 - Pi (Pi + Log[16]))

with numerical value

  N[%] 

  0.39539882209500143

> I want also to compute previous numerically. 

So why not use NIntegrate? The syntax

  NIntegrate[f[x], {x, 0, p, 1}]

tests for a singularity at the intermediate point. So try

  NIntegrate[ArcTan[x]/(x - p), {p, 0, 1}, {x, 0, p, 1}]

However, this does not work: NIntegrate gives a message to the effect 
that its singularity handling has failed.

Next, try splitting the integration into two parts:

  eps = 10^(-5);

  NIntegrate[ArcTan[x]/(x - p), {p, 0, 1}, {x, 0, p - eps}] +
  NIntegrate[ArcTan[x]/(x - p), {p, 0, 1}, {x, p + eps, 1}]

  0.39538307925436555

which is a reasonable approximation; it can be shown that the error is 
first order in eps:

  -Pi/2 eps

For this 2D integral you can combine Integrate and NIntegrate as 
follows: first compute the singular integral using Integrate,

  Assuming[0 < x < 1, 
    Integrate[1/(x - p), {p, 0, 1}, PrincipalValue -> True]]

  Log[-(x/(-1 + x))]

then perform the second integration using NIntegrate:

  NIntegrate[ArcTan[x] Log[x/(1 - x)], {x, 0, 1}]

  0.39539882209498356

This "trick" is quite general for singular integrals.

Cheers,
Paul

_______________________________________________________________________
Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)    
AUSTRALIA                               http://physics.uwa.edu.au/~paul


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