This book is devoted to the theory of vector-valued functions with one variable. This theory is one of the fundamental tools employed in modern physics, the spectral theory of operators, approximation of analytic operators, analytic mappings between vectors, and vector-valued functions of several variables. The book contains three chapters on the theory of normal functions, Hp-space, and vector-valued functions and their applications. Among the topics dealt with are the properties of complex functions in a complex plane and infinite-dimensional spaces, and the solution of vector-valued integral equations and boundary value problems by complex analysis and functional analysis, which involve methods which can be applied to problems in operations research and control theory. Much original research is included. This volume will be of interest to those whose work involves complex analysis and control theory, and can be recommended as a graduate text in these areas.