Tritruncated Solutions For The First Painlevé Equation
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This chapter is devoted to the construction of the tritruncated solutions for the first Painlevé equation, the existence of which being announced in Sect. 2.6. This example will introduce the reader to common reasonings in resurgence theory. We construct a prepared form associated with the first Painlevé equation (Sec 3.1). This prepared ODE has a unique formal solution from which we deduce the existence of truncated solutions by application of the “‘main asymptotic existence theorem”. We then study the Borel-Laplace summability property of the formal solution by various methods (Sect. 3.3). One deduces the existence of the tritruncated solutions for the first Painlevé equation, by Borel-Laplace summation (Sect. 3.4).
KeywordsHolomorphic Function Entire Function Formal Power Series Formal Series Convolution Product
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