Singularities at Infinity. The definitions of an ordinary point and a regular singular

QUESTION:

Singularities at Infinity. The definitions of an ordinary point and a regular singular point given in the preceding sections apply only if the point x0 is finite. In more advanced work in differential equations, it is often necessary to consider the point at infinity. This is done by making the change of variable = 1/x and studying the resulting equation at = 0. Show that, for the differential equation P( x) y__ + Q( x) y_ + R( x) y = 0, the point at infinity is an ordinary point if 1 P(1/) _2P(1/) Q(1/) 2 _ and R(1/) 4P(1/) have Taylor series expansions about = 0. Show also that the point at infinity is a regular singular point if at least one of the above functions does not have a Taylor series expansion, but both P(1/) _2P(1/) Q(1/) 2 _ and R(1/) 2P(1/) do have such expansions.