Regularized Green's function

  • Thread starter Einj
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Hello everyone,
I would like to know if there is a known, rigorous way to regularize a Green's function in coordinate space. In particular, it is known that the Green's function for a circle of radius R and source located at [itex]\vec x_0[/itex] is given by:
$$
G(\vec x,\vec x_0)=\frac{1}{2\pi}\ln\left[\frac{\left|\vec x-\vec x_0\right|}{\left|\vec x-\frac{R^2}{|\vec x_0|^2}\vec x_0\right|}\frac{R}{|\vec x_0|} \right],
$$
and therefore for [itex]\vec x=\vec x_0[/itex] diverges as [itex]\frac{1}{2\pi}\ln\left|\vec x-\vec x_0\right|[/itex]. Is there any rigorous way of regularizing this function? The most natural way that is coming to my mind is clearly to subtract the divergence by simply defining:
$$
G_R(\vec x,\vec x_0)=G(\vec x,\vec x_0)-\frac{1}{2\pi}\ln\left|\vec x-\vec x_0\right|.
$$

Am I right? Is this rigorous?

Thanks a lot!
 
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Answers and Replies

  • #2
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Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 
  • #3
jasonRF
Science Advisor
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What do you mean by "regularize" it, and why do you want to do it? We usually expect Green's functions to be singular at the source location.
 

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