Limits and principal value integrals
- To: mathgroup at smc.vnet.net
- Subject: [mg16510] Limits and principal value integrals
- From: p_mclean at postoffice.utas.edu.au (Patrick McLean)
- Date: Tue, 16 Mar 1999 03:59:55 -0500
- Organization: Maths Dept, University of Tasmania
- Sender: owner-wri-mathgroup at wolfram.com
I would like to use mathematica to implement the Cauchy singular integral
operator ie for a given function f, find the function
/ 1
Sf(x) = | f(y)/(x-y) dy -1<x<1
/ -1
and where the integral is a principal value integral.
My first attempt would be to take f to be identically 1:
In[1]:= Integrate[1/(x-y),{y,-1,1},PrincipalValue->True,Assumptions->-1<x<1]
1
Out[1]= If[x > 0, -Log[1 - x] + Log[1 + x], Integrate[-----, {y, -1, 1}]]
x - y
which is a litle annoying...
Then I would try:
In[2]:= Integrate[
y Sqrt[1-y^2]/(x-y),{y,-1,1}
,PrincipalValue->True,Assumptions->-1<x<1]
2 -2 2
Out[2]= If[x > 0, (-(x (Pi + 4 x - 2 Pi x + 2 Pi Sqrt[1 - x ] x -
-2 2 1
> 4 Sqrt[1 - x ] x ArcSin[-])) +
x
3/2 1 1 3 5 1 1 3 2
> 2 Pi MeijerG[{{-, 1}, {-, -, -}}, {{-, 1, 1}, {-, -}}, x ]) / (4 x)
2 4 4 2 2 4 4
2
y Sqrt[1 - y ]
> , Integrate[--------------, {y, -1, 1}]]
x - y
which is not very satisfactory at all since the answer should be x^2 -1/2...
I would like some comment on what is going on, in particular,
How to use the options Assumptions and PrincipalValue properly.
How mathematica does principal value integrals (and limits).
Any references to mathematica and principal value integral (or limits)
Thanks heaps and heaps
Patrick