arXiv subjects
Publications and source records attributed to Norbert Steinmetz.
In the present paper it is discussed to which extend conformal mappings of special circular triangles, which are thoroughly examined by M. Bonk using analytic and classical special functions methods, can be defined and investigated geometrically. This is in analogy to the elliptic modular function, which admits an analytic and also a geometric definition.
The 4IM+1CM problem is determining all pairs (f,g) of meromorphic functions in the complex plane that are not Moebius transformations of each other and share five pairs of complex values, one of them counting multiplicities. It is shown that every solution to this problem parametrizes an algebraic curve of genus zero and low degree depending on five parameters only, which enables handing over the problem to computer algebra specialists.
The purpose of this paper is to determine the main properties of Laplace contour integrals $$Λ(z)=\frac1{2πi}\int_\CCϕ_L(t)e^{-zt}\,dt,$$ that solve linear differential equations $$L[w](z):=w^{(n)}+\sum_{j=0}^{n-1}(a_j+b_jz)w^{(j)}=0.$$ This concerns, in particular, the order of growth, asymptotic expansions, the Phragmén-Lindelöf indicator, the distribution of zeros, the existence of sub-normal and polynomial solutions, and the corresponding Nevanlinna functions.
In this paper we discuss the dynamical structure of the rational family $(f_t)$ given by $$f_t(z)=tz^{m}\Big(\frac{1-z}{1+z}\Big)^{n}\quad(m\ge 2,~t\ne 0).$$ Each map $f_t$ has two super-attracting immediate basins and two free critical points. We prove that for $0<|t|\le 1$ and $|t|\ge 1$, either of these basins is completely invariant and at least one of the free critical points is inactive. Based on this separation we draw a detailed picture the structure of the dynamical and the parameter plane.
We utilise recent results about the transcendental solutions to Riccati differential equations to provide a comprehensive description of the nature of the transcendental solutions to algebraic first order differential equations of genus zero.
We utilise a recent approach via the so-called re-scaling method to derive a unified and comprehensive theory of the solutions to Painleve's differential equations (I), (II) and (IV), with emphasis on the most elaborate equation (IV).
Based on the so-called re-scaling method, we will give a detailed description of the solutions to the Hamiltonian system (\ref{Hsystem}) below, which was discovered only recently by Kecker, and is strongly related to Painleve's fourth differential equation. In particular, the problem to determine those fourth Painleve transcendents with positive Nevanlinna deficiency $δ(0,w)$, is completely resolved.
We determine all pairs $(f,g)$ of meromorphic functions that share four pairs of values $(a_ν,b_ν)$, $1\leν\le 4$, and a fifth pair $(a_5,b_5)$ under some mild additional condition.
In a recent paper, Aimo Hinkkanen and Ilpo Laine proved that the transcendental solutions to Painleve's second differential equation w"=a+zw+w^3 have either order of growth 3 or else 3/2. We complete this result by proving that there exist no sub-normal solutions (order of growth 3/2) other than the so-called Airy solutions.
We discuss families of meromorphic functions $f_h$ obtained from single functions $f$ by the re-scaling process $f_h(z)=h^{-α}f(h+h^{-β}z)$ generalising Yosida's process $f_h(z)=f(h+z)$. The main objective is to obtain information on the value distribution of the generating functions $f$ themselves. Among the most prominent generalised Yosida functions are first, second and fourth Painlevé transcendents. The Yosida class contains all limit functions of generalised Yosida functions--the Yosida class is universal.
In this paper we discuss the dynamics as well as the structure of the parameter space of the one-parameter family of rational maps $\ds f_t(z)=-\frac{t}{4}\frac{(z^{2}-2)^{2}}{z^{2}-1}$ with free critical orbit $\pm\sqrt{2}\xrightarrow{(2)}0\xrightarrow{(4)}t\xrightarrow{(1)}...$. In particular it is shown that for any escape parameter $t$ the boundary of the basin at infinity $\A_t$ is either a Cantor set, a curve with infinitely many complementary components, or else a Jordan curve. In the latter case the Julia set is a Sierpiński curve.
The aim of this paper is to describe the origin, first solutions, further progress, the state of art, and a new ansatz in the treatment of a problem dating back to the 1920's, which still has not found a satisfactory solution and deserves to be better known.
It is proved that any family of analytic functions with spherical derivative uniformly bounded away from zero ist normal.