Singularities at Infinity. The definitions of an ordinary

Chapter 5, Problem 43

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Singularities at Infinity. The definitions of an ordinary point and a regular singular pointgiven in the preceding sections apply only if the point x0 is finite. In more advanced workin differential equations, it is often necessary to consider the point at infinity. This is doneby making the change of variable = 1/x and studying the resulting equation at = 0.Show that, for the differential equationP(x)y+ Q(x)y+ R(x)y = 0,the point at infinity is an ordinary point if1P(1/) 2P(1/) Q(1/)2and R(1/)4P(1/)have Taylor series expansions about = 0. Show also that the point at infinity is a regularsingular point if at least one of the above functions does not have aTaylor series expansion,but bothP(1/) 2P(1/) Q(1/)2and R(1/)2P(1/)do have such expansions.