This book investigates several classes of partial differential equations of real time variable and complex spatial variables, including the heat, Laplace, wave, telegraph, Burgers, Black-Merton-Scholes, Schroedinger and Korteweg-de Vries equations.The complexification of the spatial variable is done by two different methods. The first method is that of complexifying the spatial variable in the corresponding semigroups of operators. In this case, the solutions are studied within the context of the theory of semigroups of linear operators. It is also interesting to observe that these solutions preserve some geometric properties of the boundary function, like the univalence, starlikeness, convexity and spirallikeness. The second method is that of complexifying the spatial variable directly in the corresponding evolution equation from the real case. More precisely, the real spatial variable is replaced by a complex spatial variable in the corresponding evolution equation and then analytic and non-analytic solutions are sought.For the first time in the book literature, we aim to give a comprehensive study of the most important evolution equations of real time variable and complex spatial variables. In some cases, potential physical interpretations are presented. The generality of the methods used allows the study of evolution equations of spatial variables in general domains of the complex plane.

This volume contains proceedings from the AMS conference on Applied Analysis held at LSU (Baton Rouge) in April 1996. Topics include partial differential equations, spectral theory, functional analysis and operator theory, complex analysis, numerical analysis and related mathematics. Applications include quantum theory, fluid dynamics, control theory and abstract issues, such as well-posedness, asymptotics, and more. The book presents the scope and depth of the conference and its lectures. The state-of-the-art surveys by Jerry Bona and Fritz Gesztesy contain topics of wide interest. There have been a number of good conferences on related topics, yet this volume offers readers a unique, varied viewpoint. The scope of the material in the book will benefit readers approaching the work from diverse perspectives. It will serve those seeking motivational scientific problems, those interested in techniques and subspecialties and those looking for current results in the field.