The coefficients $p(z)$ and $q(z)$ of the differential equation

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

are analytic in the punctured disc $0<|z|<R$, and $w_{1}(z)$ and $w_{2}(z)$ are linearly independent solutions in the neighbourhood of the point $z_{0}$ in the disc. By considering the effect of analytically continuing $w_{1}$ and $w_{2}$, show that the equation $(*)$ has a non-trivial solution of the form

$$w(z)=z^{\sigma} \sum_{n=-\infty}^{\infty} c_{n} z^{n}$$