= Solution
For the static point source, take $A=0$ and $\alpha=0$. The <scalar> equation becomes
$$
(\nabla^2-m^2)\phi=-c^2\mu_0q\,\delta(x).
$$
Its spatial Fourier transform satisfies
$$
(|k|^2+m^2)\widehat\phi(k)=c^2\mu_0q.
$$
The stated inverse transform therefore gives
$$
\boxed{\phi(r)=\lambda\frac{e^{-mr}}r,\qquad A=0,}
$$
where $\lambda$ absorbs the transform normalization and source strength. This is the <Yukawa potential>: a nonzero vector-field mass screens the interaction beyond the range $m^{-1}$. As $m\to0$, it tends to the long-range Coulomb potential $\lambda/r$. The same <limit> removes the mass term that broke gauge invariance in part (b).
Back to article page
