Einj
- 464
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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!
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!