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Since Laplace's equation is related to the steady state of heat flow problems, I'm guessing that there is a way to get from the heat kernel to the fundamental solution of Laplace's equation by letting $t\to \infty$ and renormalizing somehow. Is this correct? If so, in the $\mathbb{R}^2$ case how do I get from $(1)$ to $(2)$?

$$(1)\hspace5ex\Phi(x,t)=\frac{1}{4\pi t}e^{-\frac{|x|^2}{4t}}$$ $$(2)\hspace5ex\Phi(x)=-\frac{1}{2\pi }\text{Log}|x|$$

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i would read this paper and see the references contained there:


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