Asymptotic integration theory gives a collection of results which provide a thorough description of the asymptotic growth and zero distribution of solutions of (*) f + P (z)f = 0, where P (z) is a polynomial. These results have been used by several authors to find interesting properties of solutions of (*). That said, many people have remarked that the proofs and discussion concerning asymptotic integration theory that are, for example, in E. Hille's 1969 book Lectures on Ordinary Differential

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... quations are difficult to follow. The main purpose of this paper is to make this theory more understandable and accessible by giving complete explanations of the reasoning used to prove the theory and by writing full and clear statements of the results. A considerable part of the presentation and explanation of the material is different from that in Hille's book. J o u r n a l P r e -p r o o f Journal Pre-proof (II) To obtain the results in (I), we need Liouville's transformation and Hille's theory on asymptotic integration, which we discuss in detail in such a way that the explanations will be more accessible to non-expert readers who have some background in complex analysis. Some of the presentations and explanations will be in revised frameworks and will be different from the material in [12, Ch. 7.4]. In the literature, the following two simplified facts are often referenced to [12, Ch. 7.4]: (A) On all the rays in an open sector S(θ j−1 , θ j ) determined by any two consecutive critical rays, each solution f ≡ 0 either blows up on each ray or decays to zero exponentially on each ray. By this we mean, respectively, that for θ ∈ (θ j−1 , θ j ). (B) If a sector W j (ε) = {z : | arg(z) − θ j | < ε} contains infinitely many zeros, then the number of zeros in W j (ε) is asymptotically comparable to r (n+2)/2 . 2 J o u r n a l P r e -p r o o f Journal Pre-proof where the counting function n(r, Λ j,c , 1/f ) refers to only those zeros in Λ j,c satisfying |z| ≤ r and N (r, Λ j,c , 1/f ) is the corresponding integrated counting function. 3 J o u r n a l P r e -p r o o f Journal Pre-proof J o u r n a l P r e -p r o o f Journal Pre-proof − π < arg(1 + (z)) ≤ π. (2.11) 6 J o u r n a l P r e -p r o o f Journal Pre-proof J o u r n a l P r e -p r o o f Journal Pre-proof J o u r n a l P r e -p r o o f Journal Pre-proof J o u r n a l P r e -p r o o f Journal Pre-proof J o u r n a l P r e -p r o o f Journal Pre-proof w + (1 − F (z))w = 0, (3.4) where F (z) satisfies either Hypothesis F + or Hypothesis F − below. 14 J o u r n a l P r e -p r o o f Journal Pre-proof ∞ z |F (t)||dt| exists along the path of integration given by t = z − r, 0 ≤ r < ∞. Moreover, there exists a δ satisfying δ > δ 0 such that lim R→∞ sup z∈D − (δ,R) ∞ z |F (t)||dt| = 0. 15 J o u r n a l P r e -p r o o f Journal Pre-proof for any fixed σ > 0. If n = 0, then the function w 0 (z) = w sin (z) is a solution of the sine equation (3.7), and hence there exists a constant C 0 = C 0 (δ) > 0 such that |w sin (z)| ≤ C 0 for all z ∈ Σ σ . Suppose that there exists a constant C n−1 = C n−1 (δ) > 0 such that |w n−1 (z)| ≤ C n−1 for all z ∈ Σ σ . Then it follows from (3.10) that |w n (z)| ≤ |w sin (z)| + ∞ z | sin(t − z)||F (t)||w n−1 (t)||dt|, 16 J o u r n a l P r e -p r o o f Journal Pre-proof = 1 4 e 2(y−y 0 ) − 2 + e −2(y−y 0 ) + 4 sin 2 (x − x 0 ) = sinh 2 (y − y 0 ) + sin 2 (x − x 0 ). 21 J o u r n a l P r e -p r o o f Journal Pre-proof as ζ → ∞ in G j (δ,R). From Lemma 2.1, we have ζ(z) = 2 n + 2 z (n+2)/2 (1 + o(1)), 23 J o u r n a l P r e -p r o o f Journal Pre-proof J o u r n a l P r e -p r o o f Journal Pre-proof J o u r n a l P r e -p r o o f Journal Pre-proof J o u r n a l P r e -p r o o f Journal Pre-proof J o u r n a l P r e -p r o o f Journal Pre-proof J o u r n a l P r e -p r o o f Journal Pre-proof J o u r n a l P r e -p r o o f Journal Pre-proof