Addendum to a note on the zeros of solutions of w″+P(z)w=0 where P is a polynomial

In the original paper [1], it was shown that the zeros of solutions of w″ + P(z)w = 0, where P(z) is a polynomial of degree n ≥ 1, must approach certain rays. This was proved by first obtaining asymptotic formulas for a fundamental set of solutions in sectors, and then using them to derive estimates on the rate at which the nearby zeros approach the ray. The estimates derived in [1] for the rate of approach were rough estimates which were sufficient to prove the main result but simple enough to avoid unnecessary complications in the proof. The present note is intended to give the best estimate which can be derived from the asymptotic formulas for the rate of approach of the zeros. The main reason for deriving these estimates is that they show that for many equations (e.g., the Titchmarsh equation) the rate of approach is actually much faster than that indicated by the rough estimate in [1]. In fact, we show that the estimate dramatically improves whenever P(z) has the property that the translate P(z ? c) which eliminates the term of degree n ? 1 also eliminates the term of degree n ? 2.     

Some properties of solutions ofu″+a(t) f (u)=0

Monatshefte für Mathematik -     

微分方程f″+e-zf′+Q(z)f = 0解的增长性

研究二阶微分方程f″+e^-zf′+Q(z)f = 0解的增长性,其中Q是级为1的整函数,当Q(z)=h(z)e^bz, h(z)是非零多项式,b≠-1是复常数,上面方程的每个解有无穷级且超级为1. 改进了已有的结果.     

关于复微分方程f″+Af′+Bf=0解的增长性(2)

考虑微分方程f″+Af′+Bf=0,其中A(z),B(z)都是亚纯函数.如果A(z)有一个有穷亏值,当赋予B(z)某些条件时,上述方程的每一个非零解具有无穷级.这些结果将伍鹏程和朱军前期的工作扩展到B(z)的级等于1/2的情形.     

Regular functions f(z) for which zf'(z) is θ-spiral-like of order α

Siberian Mathematical Journal -     

The structure of the Mitchell order—I

We isolate here a wide class of well-founded orders called tame orders, and show that each such order of cardinality at most κ can be realized as the Mitchell order on a measurable cardinal κ, from a consistency assumption weaker than o(κ) = κ⁺.     

微分方程f″ e~(az)f′ Q(z)f=F(z)的复振荡

该文研究了线性微分方程f″ eazf′ Q( z) f=F( z)的复振荡问题,其中Q( z)、F( z) ( 0 )是整函数,且σ( Q) =1 ,σ( F) < ∞,Q( z) =h( z) ebz,h( z)是多项式,b≠- 1是复常数,那么上述线性微分方程的所有解f( z)满足λ( f) =λ( f) =σ( f) =∞,　λ2 ( f) =λ2 ( f) =σ2 ( f) =1 .至多除去两个例外复数a及一个可能的有穷级例外解f0 ( z) .     

The Value Regions of the Systems {f(z 1), f′(z 1)} and {f(z 1), f(z 2)} in the Class of Typically Real Functions

In the class T consisting of regular and typically real functions in the disk |x| < 1, the value regions of the system {f(z ₁), f(z ₁)} and {f(z ₁), f(z ₂)} are found for fixed z ₁ and z ₂. As an application, the value regions of f(z ₁) and f(z ₂) are found for f T with fixed value f(z ₁). Bibliography: 11 titles.     

方程y″+py′+qy=f(x)的一个积分形式特解

现行国内使用较广的高等数学教材中 ,求解二阶常系数线性微分方程 ,按照解的结构理论 ,需要求非齐方程的一个特解 ,那么怎样求特解呢 ?教材中仅对 f( x)为多项式、三角函数 ( sinβx或cosβx)、指数函数 eλx及它们的乘积等几种情况分别作了讨论 ,得出特解具有的形式特征 ,在实际求解中再用待定常数法解决之。这种处理方法仅解决了一些特殊类型 ,在理论上有很大的局限性 ,如后文例 3给出的方程并未在其讨论之列 ;另一方面 ,各种情况的结论冗长 ,推导过程复杂。现介绍用特解公式教学 ,学生易学易懂 ,便于记忆 ,收到了较好的教学效果。现将其…     

关于方程f″+Af=F(z)的复振荡

本文讨论了当A(z)为多项式,F(z)为具有无穷多个零点的整函数时,微分方程 f″+A(z)f=F(z)的解f(z)的复振荡的性质.     

Compositions of analytic functions of the form Fn(z) = Fn−1(fn(z)), fn(z) → f(z)

Frequently, in applications, a function is iterated in order to determine its fixed point, which represents the solution of some problem. In the variation of iteration presented in this paper fixed points serve a different purpose. The sequence {F_n(z)} is studied, where F₁(z) = f₁(z) and F_n(z) = F_n−1(f_n(z)), with f_n → f. Many infinite arithmetic expansions exhibit this form, and the fixed point, α, of f may be used as a modifying factor (z = α) to influence the convergence behaviour of these expansions. Thus one employs, rather than seeks the fixed point of the function f.     

Explicit solutions of the equation –div z = f under the constraint ∥z ∥∞ = 1

We consider the eigenvalue problem for the 1-Laplace operator and the corresponding variational problem. Since the underlying functionals are not differentiable, the derivation of the Euler-Lagrange equation is highly nontrivial and involves methods of nonsmooth analysis. A remarkable fact is, that there are infinitely many functions f for which the equation –div z = f has to be satisfied. The purpose of the present work is the investigation of this equation. We show that in a square 2-dimensional domain for each right hand side f there are infinitely many vector fields z satisfying the Euler-Lagrange equation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The hierarchy of higher order solutions of the derivative nonlinear Schrödinger equation

In this paper, we provide a simple method to generate higher order position solutions and rogue wave solutions for the derivative nonlinear Schrödinger equation. The formulae of these higher order solutions are given in terms of determinants. The dynamics and structures of solutions generated by this method are studied.     

On the order of growth of generalized eigenfunctions of the Sturm-Liouville operator. The Shnol’ theorem

For the generalized eigenfunctions corresponding to a Sturm-Liouville operator on a semiaxis, growth estimates are obtained. In the paper, in contrast to the known Shnol’ theorem, the dependence of the growth order on the behavior at infinity of the potential is studied. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 46–51, January, 2000.     

The growth of solutions of <Emphasis Type="Italic">f</Emphasis>″ + e<Superscript>-z</Superscript><Emphasis Type="Italic">f</Emphasis>′+ Q(z)<Emphasis Type="Italic">f</Emphasis> = 0 where the order (Q) = 1

This paper investigates the growth of solutions of the equation f″ + e^-z f′ + Q(z)f = 0 where the order (Q) = 1. When Q(z) = h(z)e^bz, h(z) is nonzero polynomial, b ≠ - 1 is a complex constant, every solution of the above equation has infinite order and the hyper-order 1. We improve the results of M. Frei, M. Ozawa, G. Gundersen and J. K. Langley.     

Zeros of the partial sums of cos (z) and sin (z). I

We extend results of Szeg (1924) and Kappert (1996) on the location of the zeros of the normalized partial sums of cos(z) and sin(z), and their rates of convergence to the associated Szeg curves.     

Rational solutions of the Diophantine equations $$f(x)^2 \pm f(y)^2=z^2$$ f ( x ) 2 ± f ( y ) 2 = z 2

Periodica Mathematica Hungarica - We discuss the rational solutions of the Diophantine equations $$f(x)^2 \pm f(y)^2=z^2$$ . This problem can be solved either by the theory of elliptic curves or by...