Evolution Equations With A Complex Spatial Variable

In Evolution Equations With A Complex Spatial Variable, the author delves into various classes of partial differential equations that incorporate real time variables and complex spatial variables. This exploration includes notable equations such as the heat, Laplace, wave, and Schrödinger equations, among others. Each equation is examined through the lens of complexifying the spatial variable using two distinct methods, providing a rich and thorough understanding of their behaviors and properties.

The first method discussed involves the complexification of the spatial variable within the framework of semigroups of linear operators. This approach reveals how certain geometric properties of the boundary function, such as univalence, starlikeness, convexity, and spirallikeness, are preserved in the solutions. The second method takes a more direct approach, substituting the real spatial variable with a complex one in the corresponding evolution equations, thus allowing for the exploration of both analytic and non-analytic solutions.

For the first time, Evolution Equations With A Complex Spatial Variable presents a comprehensive study of key evolution equations, making it a significant contribution to the field. The author also touches upon potential physical interpretations of these equations, showcasing the versatility and applicability of the methods employed. This book is an essential resource for those interested in the intricate relationship between time and complex spatial variables in mathematical physics.