Georges Valiron

Bio: Georges Valiron is an academic researcher. The author has an hindex of 12, co-authored 24 publications receiving 1075 citations.

Sur la dérivée des fonctions algébroïdes

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Iteration of meromorphic functions

Abstract: This paper attempts to describe some of the results obtained in the iteration theory of transcendental meromorphic functions, not excluding the case of entire functions. The reader is not expected to be familiar with the iteration theory of rational functions. On the other hand, some aspects where the transcendental case is analogous to the rational case are treated rather briefly here. For example, we introduce the different types of components of the Fatou set that occur in the iteration of rational functions but omit a detailed description of these types. Instead, we concentrate on the types of components that are special to transcendental functions (Baker domains and wandering domains).

Difference analogue of the Lemma on the Logarithmic Derivative with applications to difference equations

Abstract: The Lemma on the Logarithmic Derivative of a meromorphic function has many applications in the study of meromorphic functions and ordinary differential equations. In this paper, a difference analogue of the Logarithmic Derivative Lemma is presented and then applied to prove a number of results on meromorphic solutions of complex difference equations. These results include a difference analogue of the Clunie Lemma, as well as other results on the value distribution of solutions.

Nevanlinna theory for the difference operator

Abstract: Certain estimates involving the derivative f 7→ fof a meromorphic function play key roles in the construction and applications of classical Nevanlinna theory. The purpose of this study is to extend the usual Nevan- linna theory to a theory for the exact difference f 7→ �f = f(z + c) − f(z). An a-point of a meromorphic function f is said to be c-paired at z ∈ C if f(z) = a = f(z+c) for a fixed constant c ∈ C. In this paper the distribution of paired points of finite-order meromorphic functions is studied. One of the main results is an analogue of the second main theorem of Nevanlinna the- ory, where the usual ramification term is replaced by a quantity expressed in terms of the number of paired points of f. Corollaries of the theorem include analogues of the Nevanlinna defect relation, Picard's theorem and Nevanlinna's five value theorem. Applications to difference equations are discussed, and a number of examples illustrating the use and sharpness of the results are given.

Nonlinear Electrodynamics: Lagrangians and Equations of Motion

Abstract: After a brief discussion of well‐known classical fields we formulate two principles: When the field equations are hyperbolic, particles move along rays like disturbances of the field; the waves associated with stable particles are exceptional. This means that these waves will not transform into shock waves. Both principles are applied to nonlinear electrodynamics. The starting point of the theory is a Lagrangian which is an arbitrary nonlinear function of the two electromagnetic invariants. We obtain the laws of propagation of photons and of charged particles, along with an anisotropic propagation of the wavefronts. The general ``exceptional'' Lagrangian is found. It reduces to the Lagrangian of Born and Infeld when some constant (probably simply connected with the Planck constant) vanishes. A nonsymmetric tensor is introduced in analogy to the Born‐Infeld theory, and finally, electromagnetic waves are compared with those of Einstein‐Schrodinger theory.