Die Riemannfunktion der Differentialgleichungw zζ + [z + ϕ′ (ζ)]w = 0

Ohne ZusammenfassungHerrn Prof. Dr. E.Ledinegg zum 70. Geburtstag gewidmet     

Behavior of the roots of the equationw′1(z)+бw1(z)=0

Some properties of the roots of the equationw′₁(z)+бw₁(z)=0 when the parameter σ is real are studied.     

k-Diskonjugiertheit vonw (n)+p(z)w=0

Ohne Zusammenfassung     

Non-recurrence of exp(z)/z

In this paper, we consider the dynamics of the map z→exp(z)/z on the punctured plane C*=C\{0}. We show that for almost every point z ∈ C*, the ω-limit set of z is equal to {0, ∞}. In particular, the map is not recurrent.     

Nichteuklidischer Nullstellenabstand der Lösungen vonw″+p(z)w=0

Ohne Zusammenfassung     

Regions of Values of the Systems {f(z 1), f(z 2), f'(z 2)} and {f(z 1), f'(z 1), f'(z 1)} on the Class of Typically Real Functions

Let T be the class of functions f(z) = z + a ₂ z ² + . . . that are regular in the unit disk and satisfy the condition Im f(z) Im z > 0 for Im z 0, and let z ₁ and z ₂ be any distinct fixed points in the disk |z| < 1. For the systems of functionals mentioned in the title, the regions of values on T are studied. As a corollary, the regions of values of f'(z ₂) and f'(z ₁) on the subclasses of functions in T with fixed values f (z ₁), f (z ₂) and f (z ₁), f'(z ₁), respectively, are found. Bibliography: 7 titles.     

The Fourier expansion of $$\varvec{\eta (z)\eta (2z)\eta (3z)/\eta (6z)}$$

We compute the Fourier coefficients of the weight one modular form \(\eta (z)\eta (2z)\eta (3z)/\eta (6z)\) in terms of the number of representations of an integer as a sum of two squares. We deduce a relation between this modular form and translates of the modular form \(\eta (z)^4/\eta (2z)^2\). In the last section we use our main result to give an elementary proof of an identity by Victor Kac.     

A note on log (f(z)/z) forf inS

We prove that forf a normalized schlicht function in the disk the boundary values of log |f(z)/z| satisfy a growth condition on subarcs of the unit circle .     

关于φ（z）f（z）f＇（z）的值分布

本文研究有穷圆内的值分布，其中ｆ（２）为非常数亚纯函数，为非零亚纯函数，并对平面上这样的函数，若，得到     

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.