Write down necessary and sufficient conditions on the functions $p(z)$ and $q(z)$ for the point $z=0$ to be (i) an ordinary point and (ii) a regular singular point of the equation

$$w^{\prime \prime}+p(z) w^{\prime}+q(z) w=0 .$$

Show that the point $z=\infty$ is an ordinary point if and only if

$$p(z)=2 z^{-1}+z^{-2} P\left(z^{-1}\right), \quad q(z)=z^{-4} Q\left(z^{-1}\right),$$

where $P$ and $Q$ are analytic in a neighbourhood of the origin.

Find the most general equation of the form $(*)$ that has a regular singular point at $z=0$ but no other singular points.