The Bessel equation of order $n$ is

$$z^{2} y^{\prime \prime}+z y^{\prime}+\left(z^{2}-n^{2}\right) y=0 .$$

Here, $n$ is taken to be an integer, with $n \geqslant 0$. The transformation $w(z)=z^{\frac{1}{2}} y(z)$ converts (1) to the form

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

where

$$q(z)=1-\frac{\left(n^{2}-\frac{1}{4}\right)}{z^{2}}$$

Find two linearly independent solutions of the form

$$w=e^{s z} \sum_{k=0}^{\infty} c_{k} z^{\rho-k}$$

where $c_{k}$ are constants, with $c_{0} \neq 0$, and $s$ and $\rho$ are to be determined. Find recurrence relationships for the $c_{k}$.

Find the first two terms of two linearly independent Liouville-Green solutions of (2) for $w(z)$ valid in a neighbourhood of $z=\infty$. Relate these solutions to those of the form (3).