Several Complex Variables V: Complex Analysis in Partial Differential Equations and Mathematical Physics

In this part, we present a survey of mean-periodicity phenomena which arise in connection with classical questions in complex analysis, partial differential equations, and more generally, convolution equations. A common feature of the problem we shall consider is the fact that their solutions depend on tech- niques and ideas from complex analysis. One finds in this way a remarkable and fruitful interplay between mean-periodicity and complex analysis. This is exactly what this part will try to explore. It is probably appropriate to stress the classical flavor of all of our treat- ment. Even though we shall frequently refer to recent results and the latest theories (such as algebmic analysis, or the theory of Bernstein-Sato polyno- mials), it is important to observe that the roots of probably all the problems we discuss here are classical in spirit, since that is the approach we use. For instance, most of Chap. 2 is devoted to far-reaching generalizations of a result dating back to Euler, and it is soon discovered that the key tool for such gen- eralizations was first introduced by Jacobi! As the reader will soon discover, similar arguments can be made for each of the subsequent chapters. Before we give a complete description of our work on a chapter-by-chapter basis, let us make a remark about the list of references. It is quite hard (maybe even impossible) to provide a complete list of references on such a vast topic.