Introduction to Complex Theory of Differential Equations

Description

This book discusses the complex theory of differential equations or more precisely, the theory of differential equations on complex-analytic manifolds. Although the theory of differential equations on real manifolds is well known – it is described in thousands of papers and its usefulness requires no comments or explanations – to date specialists on differential equations have not focused on the complex theory of partial differential equations. However, as well as being remarkably beautiful, this theory can be used to solve a number of problems in real theory, for instance, the Poincaré balayage problem and the mother body problem in geophysics. The monograph does not require readers to be familiar with advanced notions in complex analysis, differential equations, or topology. With its numerous examples and exercises, it appeals to advanced undergraduate and graduate students, and also to researchers wanting to familiarize themselves with the subject. Front Matter....Pages i-ix Leray residues....Pages 1-10 Ramified integrals....Pages 11-29 Asymptotics of ramified integrals....Pages 31-42 Ramified Fourier transform....Pages 43-52 Properties of the ramified Fourier transform....Pages 53-59 The Cauchy problem for equations with constant coefficients....Pages 61-68 Singularities of the solution of the Cauchy problem....Pages 69-79 The Cauchy problem for equations with variable coefficients. Leray’s uniformization....Pages 81-93 Balayage inwards problem....Pages 95-107 Mother body problem....Pages 109-128 Back Matter....Pages 129-138