Use the Green’s function obtained in problem 2 to obtain the leading term in the asymptotic expansion for $ \phi(x) $ as $ \lvert x \rvert \to \infty $, where $ \phi $ satisfies $ \triangledown^2 \phi - \lambda^2 \phi = S(x) $ in $ R^3 $, and $ S(x) = 1 $ if $ \lvert x \rvert < a $ and $ S(x) = 0 $ if $ \lvert x \rvert \geq a $.
Ans: $\phi =-\frac{e^{-\lambda r}}{\lambda^2 r}\left(a \cosh(\lambda a) - \frac{1}{\lambda}\sinh(\lambda a) \right)$
I'm very new to Green's function, so I'm trying to learn.
Some steps I took are $$\triangledown^2G - \lambda ^2 G = \delta(x -x')$$
where $ G $ is the Green's function. Maybe convert to spherical coordinates?
We need to compute $ \int\int\int_V G(x, x')S(x')dV $
I would really appreciate it if someone would explain the steps in detail.