- #1

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## Main Question or Discussion Point

Hello everyone,

I would like to know if there is a known,

$$

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!

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!