The asymptotic behavier of the solution of $ f''(r)+\frac{1}{r}f(r)+a^2 f(r)=0 $.

Solution 1:

Rescale the equation with $\rho = ar$

$$\rho^2f'' + \rho f' + \rho^2 f = 0$$

This is the Bessel differential equation with $n=0$. The homogenous solution is

$$f(\rho) = C_1J_0(\rho)+C_2Y_0(\rho)$$

For a fundamental solution we require that

$$Df = \frac{1}{2\pi r}\delta(r) = \frac{a^2}{2\pi\rho}\delta(\rho)$$

Can you prove from here which $C$ would give you the desired quantity?

Hint: near $0$, $Y_0(z) \sim \frac{2}{\pi}\left(\gamma+\log\frac{z}{2}\right)$ and $J_0(z) \sim 1 - \frac{z^2}{4}$ whereas the homogeneous solutions for the Poisson equation radially were $f = C_1 \log r + C_2$