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MathGroup Archive 2006

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Re: Singularities at end point in integrations...

  • To: mathgroup at smc.vnet.net
  • Subject: [mg64140] Re: [mg64104] Singularities at end point in integrations...
  • From: Pratik Desai <pdesai1 at umbc.edu>
  • Date: Thu, 2 Feb 2006 00:06:50 -0500 (EST)
  • References: <200602010934.EAA23064@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

ashesh wrote:

>Hi,
>
> I Need to perform an integration with poles and zeros in the integrand. Please let me know if there a way in Mathematican that can be used to perform the definite integral 
>    
>    sqrt((x-a)*(x-b)/((x-c)*(x-d))) 
>
>between the limits (c,d), (a,d), (a,b) or (b,c).
>    
>I have read about the routine in quadpack called "dqawse.f" which can perform "integration of functions having algebraico-logarithmic end point singularities".
>
>
>Hope someone can give some leads to solve the above problem.
>
>Thanks in advance.
>Ashesh
>
>  
>
Perhaps using the package <<NumericalMath`CauchyPrincipalValue`
Here is what I tried


In[333]:=
a=-1;
b=1;
c=-2;
d=2;
<<NumericalMath`CauchyPrincipalValue`
Integrate[((x-a)*(x-b)/((x-c)*(x-d)))//ExpandAll//Together//Sqrt,{x,a,b},PrincipalValue->True]//N//Chop
CauchyPrincipalValue[((x-a)*(x-b)/((x-c)*(x-d)))//ExpandAll//Together//Sqrt, 
{x, a,{c},d}]
CauchyPrincipalValue[((x-a)*(x-b)/((x-c)*(x-d)))//ExpandAll//Together//Sqrt, 
{x, a,{c},b}]
CauchyPrincipalValue[((x-a)*(x-b)/((x-c)*(x-d)))//ExpandAll//Together//Sqrt, 
{x, b,{c},d}]
CauchyPrincipalValue[((x-a)*(x-b)/((x-c)*(x-d)))//ExpandAll//Together//Sqrt, 
{x, b,{d},c}]


Out[338]=
0.812598

Out[339]=
0.812589\[InvisibleSpace]+1.34387 \[ImaginaryI]

Out[340]=
0.812598

Out[341]=
0.\[InvisibleSpace]+1.34385 \[ImaginaryI]

Out[342]=
-0.812589-1.34387 \[ImaginaryI]

Hope this helps

Pratik


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