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Introduction to the Ecalle theory. (English) Zbl 0805.40007
Tournier, E. (ed.), Computer algebra and differential equations. Selected papers of the third biennial workshop on computer algebra and differential equations held Marseille, France, June 1-5, 1992 (CADE-92). Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 193, 59-101 (1994).
The paper gives an overview of the so-called resurgence theory based on the work of J. Ecalle. The first section is a brief introduction to resurgence. A central role plays the algebra of simple resurgent functions, a subalgebra of the multiplicative algebra of formal power series $$\mathbb{C}[[ x^{-1} ]]$$. A resurgent function $$\varphi$$ is a formal power series $\varphi(x)= \sum^ \infty_{\nu=0} {{a_ \nu} \over {x^ \nu}}$ such that the Borel transform $B\varphi(\xi)= \sum^ \infty_{\nu=1} {{a_{\nu+1}} \over {\nu!}} \xi^ \nu$ has positive radius of convergence and has in a certain sense only a few singularities which is essential to apply Laplace transformation. After having defined this algebra the notation of resurgent symbols is introduced by an analysis of the Stokes phenomena. The second section is devoted to the study of resurgence of some particular differential equations of the form ${\partial \over {\partial z}}\Psi= F(z,x,\psi)$ where $$x$$ is a (nonzero) complex parameter and $$F$$ is a certain analytic function. An example is the Riccati equation ${\partial\over {\partial z}} \psi- x\psi=- {1\over z} - {1\over {z^ 2}} \psi^ 2,$ where the two concepts of “equational resurgence” and “quantum resurgence” are compared.
For the entire collection see [Zbl 0785.00038].
Reviewer: J.Müller (Trier)

##### MSC:
 40G10 Abel, Borel and power series methods 40C10 Integral methods for summability 30H05 Spaces of bounded analytic functions of one complex variable