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What is the fundamental solution of the linear operator $L = \nabla^2 u - k^2 u$ on $\mathbb{R}^2$, with the constraint that the solution goes to zero at infinity? I've figured out that $u(r) = aK_0(kr)$, where $K_0$ is the zeroth modified Bessel function, but cannot find $a$.

I understand that what I need to do is integrate the equation on both sides, but I cannot estimate how $LK_0(kr)$ behaves near the origin!

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I guess you are working in 2D? – Fabian Sep 7 '12 at 21:49
Yes, excuse me! In 2D indeed. – Pinkwater Sep 7 '12 at 22:14
up vote 1 down vote accepted

The modified Bessel function behaves like $$K_0(x) \sim - \log(x) $$ for $x \to 0$.

If you know the fundamental solution of the 2D-Laplace equation, then you can find the normalization constant by noting that for $x\to0$ the term proportional to $k^2$ becomes unimportant such that $u(r)$ should approach the fundamental solution of the Laplace equation in this case.

\emph{Hint:} (otherwise) 8 The fundamental solution has the property that $$1=\int_{B_\epsilon(0)} \!d^2r\,L\,u(r).$$ Apply Gauss theorem and use the asymptotic form of $K_0(x)$ given above to determine $a$.

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