Let gooby do teh hoemwerk: Green function of the two-dimensional Laplace operator

Problem

Determine the Green function of the two dimensional Laplace operator

\Delta_2 = \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}

Solution

Recall that the Green function G has to satisfy

\Delta G(\vec x) = \delta^{(2)}(\vec x)

As an ansatz, we assume that G depends only on the magnitude of \vec x, i.e. G(\vec x) = G(|\vec x|=\rho).  Then let us use Gauss theorem with a “volume” D(\rho) being a disk with radius \rho.
Then

1 = \int_{D(\rho)}d^2x'\, \vec{\nabla}^{\,\prime}\cdot\vec{\nabla}^{\,\prime} G(\rho') = \oint_{\partial D(\rho)} d\vec{\sigma}^{\,\prime}\cdot\vec{\nabla}^{\,\prime} G(\rho')\,.

Since the boundary of the disk is the circle and \vec{\nabla}^{\,\prime} G(\rho') = G'(\rho')\hat{\rho}^{\,\prime}, one obtains

G'(\rho) = \frac{1}{2\pi\rho}

from which follows that G(\rho) is given by (up to functions that lie in the kernel of the two-dimensional Laplace operator)

G(\rho) = \frac{\log(\rho/\lambda)}{2\pi}\,,

where \lambda is some length scale in order to make the logarithm well-defined.

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About goobypl5

pizza baker, autodidact, particle physicist
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