Paper 2, Section II, B

(i) The function $y(z)$ satisfies the equation

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

Give the definitions of the terms ordinary point, singular point, and regular singular point for this equation.

(ii) For the equation

$4 z y^{\prime \prime}+2 y^{\prime}+y=0,$

classify the point $z=0$ according to the definitions you gave in (i), and find the series solutions about $z=0$. Identify these solutions in closed form.

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