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Effective computation of the first Lyapunov quantities for a planar differential equation

A. Gasull, R. Prohens (1997)

Applicationes Mathematicae

We take advantage of the complex structure to compute in a short way and without using any computer algebra system the Lyapunov quantities $V_{3}$ and $V_{5}$ for a general smooth planar system.

Ein Vergleichssatz für komplexe lineare Differentialgleichungen.

Horst Herold (1972)

Mathematische Zeitschrift

Eine Bemerkung zur lokalen Werteverteilung der Lösungen linearer Differentialgleichungen.

Otto Knab (1980)

Manuscripta mathematica

Eine Verallgemeinerte Fuchs'sche Theorie.

Klaus Böhmer (1970)

Manuscripta mathematica

Elementary Acceleration and Multisummability I

Jean Martinet, Jean-Pierre Ramis (1990)

Recherche Coopérative sur Programme n°25

Elementary acceleration and multisummability. I

Jean Martinet, Jean-Pierre Ramis (1991)

Annales de l'I.H.P. Physique théorique

Équation de Hill à potentiel méromorphe

Thierry Ramond (1992)

Journées équations aux dérivées partielles

Equation $Z^{'} = A (t) - Z^{2}$ coefficient of which has a small modulus

Miloš Ráb (1971)

Czechoslovak Mathematical Journal

Équations différentielles à points singuliers irréguliers en dimension 2

Claude Sabbah (1993)

Annales de l'institut Fourier

Essential singular points of solutions of an algebraic differential equation.

Tosihusa Kimura, Yasutaka Sibuya (1975)

Journal für die reine und angewandte Mathematik

Exceptional Values of Solutions of Linear Differential Equations.

Norbert Steinmetz (1989)

Mathematische Zeitschrift

Feuilletages de briot et Bouquet.

Francine Diener (1977)

Collectanea Mathematica

Fixed points of the derivative and $k$ -th power of solutions of complex linear differential equations in the unit disc.

Zhang, Guowei, Chen, Ang (2009)

Electronic Journal of Qualitative Theory of Differential Equations [electronic only]

Fonctions multiplicatives et équations différentielles

Jean-Paul Bézivin (1995)

Bulletin de la Société Mathématique de France

Fonctions multisommables

Bernard Malgrange, Jean-Pierre Ramis (1992)

Annales de l'institut Fourier

La notion de multisommabilité intervient dans la théorie des équations différentielles lorsque des exponentielles d’ordres différents se mélangent. Elle a été introduite par J. Écalle et étudié récemment par plusieurs auteurs. On en donne ici une définition simple, qui fait uniquement intervenir des propriétés de décroissance exponentielle.

Fuchsian differential equation for the perimeter generating function of three-choice polygons.

Guttmann, Anthony J., Jensen, Iwan (2005)

Séminaire Lotharingien de Combinatoire [electronic only]

Generalized trigonometric functions in complex domain

Petr Girg, Lukáš Kotrla (2015)

Mathematica Bohemica

We study extension of $p$ -trigonometric functions ${sin}_{p}$ and ${cos}_{p}$ to complex domain. For $p = 4, 6, 8, \dots$ , the function ${sin}_{p}$ satisfies the initial value problem which is equivalent to (*) $- {(u^{'})}^{p - 2} u^{''} - u^{p - 1} = 0, u (0) = 0, u^{'} (0) = 1$ in $ℝ$ . In our recent paper, Girg, Kotrla (2014), we showed that ${sin}_{p} (x)$ is a real analytic function for $p = 4, 6, 8, \dots$ on $(- π_{p} / 2, π_{p} / 2)$ , where $π_{p} / 2 = \int_{0}^{1} {(1 - s^{p})}^{- 1 / p}$ . This allows us to extend ${sin}_{p}$ to complex domain by its Maclaurin series convergent on the disc ${z \in ℂ : | z | < π_{p} / 2}$ . The question is whether this extensions ${sin}_{p} (z)$ satisfies (*) in the sense of differential equations in complex domain. This interesting...

Gevrey series of arithmetic type. I: Purity and duality theorems. (Séries Gevrey de type arithmétique. I: Théorèmes de pureté et de dualité.)

André, Yves (2000)

Annals of Mathematics. Second Series

Gevrey series of arithmetic type. II: Transcendence without transcendence. (Séries Gevrey de type arithmétique. II: Transcendance sans transcendance.)

André, Yves (2000)

Annals of Mathematics. Second Series

Holomorphic foliations in $ℂ ℙ (2)$ having an invariant algebraic curve

Dominique Cerveau, Alcides Lins Neto (1991)

Annales de l'institut Fourier

We give estimations for the degree of separatrices of algebraic foliations in $CP (2)$ .

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