State the conditions for a point $z=z_{0}$ to be a regular singular point of a linear second-order homogeneous ordinary differential equation in the complex plane.

Find all singular points of the Bessel equation

$$z^{2} y^{\prime \prime}(z)+z y^{\prime}(z)+\left(z^{2}-\frac{1}{4}\right) y(z)=0$$

and determine whether they are regular or irregular.

By writing $y(z)=f(z) / \sqrt{z}$, find two linearly independent solutions of $(*)$. Comment on the relationship of your solutions to the nature of the singular points.