I read the following line from Weinberg's Lectures in Quantum Mechanics (pg 34):
As long as $V(r)$ is not extremely singular at $r=0$, the wave function $\psi$ must be a smooth function of the Cartesian components $x_i$ near $x=0$, in the sense that it can be expressed as a power series in these components.
What does he mean by the potential $V(r)$ being 'not extremely singular'? I thought of the electrostatic potential of a charge $Q$ placed at $r=0$, which is $V(r)=\frac{kQ}{r}$. This potential is infinite at $r=0$. Is this considered singular or extremely singular?
Also, why is the wave function $\psi$ a smooth function when $V(r)$ is not extremely singular? I only understand that when $V(r')=\infty$, the wave function is $0$ at the point $r=r'$, like at the barrier of an infinite square well.
Or more generally, what are the requirements for $\psi$ to be a smooth function and why?
