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.

Autorentext

Boris Sternin is a full professor at the Department of Applied Mathematics at the RUDN University in Moscow. He received his PhD in 1965 from Moscow State University and his Doctor of Physico-Mathematical Sciences in 1984. His main area is partial differential equations, in particular, Global asymptotic methods and Noncommutative theory of elliptic operators. Prof. Sternin has published over 300 scientific articles and 17 books since 1964. Anton Savin is an associate professor at the Department of Applied Mathematics at the RUDN University in Moscow. He received his PhD in 2000 from Moscow State University and his Doctor of physico-mathematical sciences in 2012. Dr. Savin has published over 70 scientific articles and two books since 1997.

Inhalt

Leray residues.- Ramied integrals.- Asymptotics of ramied integrals.- Ramied Fourier transform.- Properties of ramied Fourier transform.- The Cauchy problem for equations with constant coefficients.- Singularities of the solution of Cauchy problem.- The Cauchy problem for equations with variable coefficients. Leray's uniformization.- Balayage inwards problem.- Mother body problem.- Hints for exercises.