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A remark on Schröder's equation: Formal and analytic linearization of iterative roots of the power series f(z)=z ; Tomaschek, Jörg in Monatshefte für Mathematik (in press) We study Schröder’s equation (i.e. the problem of linearization) for local analytic functions F with F (0)=0, F(0)=1, F(0) a root of 1. While Schröder’s equation in this case need not have even a formal ... [more ▼] We study Schröder’s equation (i.e. the problem of linearization) for local analytic functions F with F (0)=0, F(0)=1, F(0) a root of 1. While Schröder’s equation in this case need not have even a formal solution, we show that if F is formally linearizable, then it can also be linearized by an invertible local analytic transformation. On the other hand, there exist also divergent series solutions of Schröder’s equation in this situation. We give some applications of our results to iterative functional equations, functional-differential equations and iteration groups. [less ▲] Detailed reference viewed: 113 (14 UL)Generalized Dhombres equations in the complex domain a survey Tomaschek, Jörg ; in ESAIM:Proceedings and Surveys (2014, November) In this survey paper we present the Main Theorems related to the generalized Dhombres equation in the complex domain. We discuss the local as well as the global theory. This includes formal, local ... [more ▼] In this survey paper we present the Main Theorems related to the generalized Dhombres equation in the complex domain. We discuss the local as well as the global theory. This includes formal, local analytic, entire and meromorphic solutions. [less ▲] Detailed reference viewed: 75 (0 UL)Associative Formal Power Series in Two Indeterminates ; ; et al in Semigroup Forum (2014), 88(3), 529-540 Investigating the associativity equation for formal power series in two variables we show that the transcendental associative formal power series are of order one or two and that they can be represented ... [more ▼] Investigating the associativity equation for formal power series in two variables we show that the transcendental associative formal power series are of order one or two and that they can be represented by an invertible formal power series in one variable. We also discuss the convergence of associative formal power series. [less ▲] Detailed reference viewed: 103 (5 UL)Local solutions of the generalized Dhombres functional equation in a neighbourhood of infinity Tomaschek, Jörg ; in ESAIM: Proceedings and Surveys (2014), 46 Detailed reference viewed: 79 (0 UL)On the characterization of generalized Dhombres equations having non constant local analytic or formal solutions ; Tomaschek, Jörg in Annales Universitatis Scientiarium Budapestinensis de Rolando Eötvös Nominatae Sectio Computatorica (2013), 41 We discuss solvability conditions of the generalized Dhombres functional equation in the complex domain. Our focus lies on the so-called 'infinity' case, but also the $z_0$--case is investigated. That ... [more ▼] We discuss solvability conditions of the generalized Dhombres functional equation in the complex domain. Our focus lies on the so-called 'infinity' case, but also the $z_0$--case is investigated. That means that we consider solutions of a generalized Dhombres equation with initial value $f(\infty)=w_0$, $w_0 \neq 0$, or $f(\infty)=\infty$, or $f(z_0)=1$ for $z_0 \neq 0, \infty$. For both situations we give a characterization of the generalized Dhombres equations which are solvable. [less ▲] Detailed reference viewed: 74 (2 UL)On a functional-differential equation of A. F. Beardon and functional-differential equations of Briot-Bouquet type ; Tomaschek, Jörg in Computational Methods and Function Theory (2013), 13(3), 383-395 We look for solutions of the functional-differential equation f(φ(z))=a(z)f(z)f′(z) for given series φ and a . We show that all formal solutions f of this equation are local analytic. In order to do this ... [more ▼] We look for solutions of the functional-differential equation f(φ(z))=a(z)f(z)f′(z) for given series φ and a . We show that all formal solutions f of this equation are local analytic. In order to do this we transform the equation to a special type of Briot–Bouquet differential equation. [less ▲] Detailed reference viewed: 94 (2 UL)Some remarks to the formal and local theory of the generalized Dhombres functional equation ; Tomaschek, Jörg in Results in Mathematics [=RM] (2013), 63(1-2), 377-395 We are looking for local analytic respectively formal solutions of the generalized Dhombres functional equation $f(zf(z))=\varphi(f(z))$ in the complex domain. First we give two proofs of the existence ... [more ▼] We are looking for local analytic respectively formal solutions of the generalized Dhombres functional equation $f(zf(z))=\varphi(f(z))$ in the complex domain. First we give two proofs of the existence theorem about solutions $f$ with $f(0) = w_0$ and $w_0 \in \mathbb{C}^\star \setminus \mathbb{E}$ where $\mathbb{E}$ denotes the group of complex roots of $1$. Afterwards we represent solutions $f$ by means of infinite products where we use on the one hand the canonical convergence of complex analysis, on the other hand we show how solutions converge with respect to the weak topology. In this section we also study solutions where the initial value $z_0$ is different from zero. [less ▲] Detailed reference viewed: 80 (2 UL)Formal solutions of the generalized Dhombres functional equation with value one at zero Tomaschek, Jörg ; in Aequationes Mathematicae (2012), 83(1), 117-126 We study formal solutions f of the generalized Dhombres functional equation f(zf(z))=φ(f(z)). Unlike in the situation where f(0) = w 0 and w0∈C∖E where E denotes the complex roots of 1, which were already ... [more ▼] We study formal solutions f of the generalized Dhombres functional equation f(zf(z))=φ(f(z)). Unlike in the situation where f(0) = w 0 and w0∈C∖E where E denotes the complex roots of 1, which were already discussed, we investigate solutions f where f(0) = 1. To obtain solutions in this case we use new methods which differ from the already existing ones [less ▲] Detailed reference viewed: 86 (0 UL) |
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