Asymptotic integration theory for f′′+P(z)f=0

Asymptotic integration theory gives a collection of results which provide thorough description the asymptotic growth and zero distribution solutions (*) f ′ + P ( z ) = 0 , where is polynomial. These have been used by several authors to find interesting properties (*). That said, many people remarked that proofs discussion concerning are, for example, in E. Hille’s 1969 book Lectures on Ordinary Differential Equations are difficult follow. The main purpose this paper make more understandable accessible giving complete explanations reasoning prove writing full clear statements results. A considerable part presentation explanation material different from book.