I have previously seen the Green's function for Laplace's equation in two spatial dimensions determined using the method of images. Since then, I have learned some more Fourier analysis and have attempted this problem using Fourier machinery. I'm still in the process of learning, and would therefore appreciate if someone can point out if there's anything wrong with my work.

Let $\mathcal{L} = \Delta = \partial_{x}^{2} + \partial_{y}^{2}$. Green's function must satisfy $\mathcal{L}G(x, y; x_{0}, y_{0}) = \delta((x,y) - (x_{0}, y_{0}))$.

Before taking the Fourier transform, we note that the Fourier transform of $\mathcal{L}$ can be written as the polynomial $-(\xi_{1}^{2} + \xi_{2}^{2})$, where $(\xi_{1}, \xi_{2})$ live in the range of the Fourier transform. Thus, if we let $z = (x,y)$, $z_{0} = (x_{0}, y_{0})$ and $\xi = (\xi_{1}, \xi_{2})$:

$$\mathcal{F} \left[ \mathcal{L}G(z ; z_{0})\right] = \mathcal{F}[\delta(z - z_{0})]$$

which I have reduced to:

$$\mathcal{F}(\mathcal{L})(\xi) \mathcal{F}(G)(\xi; z_{0}) = e^{-i \xi \cdot z_{0}} $$

and so we can take the inverse Fourier transform to find the Green's function:

$$G(z, z_{0}) = \frac{1}{(2 \pi)^{2}} \int_{\mathbb{R}^{2}} \frac{e^{-i \xi \cdot z_{0}}}{-(\xi_{1}^{2} + \xi_{2}^{2})} e^{i \xi \cdot z} d\xi = \frac{1}{4 \pi^{2}} \int_{\mathbb{R}^{2}} \frac{e^{-i \xi \cdot (z_{0} - z)}}{-(\xi_{1}^{2} + \xi_{2}^{2})} d\xi $$

I suppose that if I use this method, it's okay to learn the solution in integral form? Or is there a straightforward way to compute this integral?


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