This book presents a functional approach to the construction use and approximation of Green's functions and their associated ordered exponentials. After a brief historical introduction the author discusses new solutions to problems involving particle production in crossed laser fields and non-constant electric fields. Applications to problems in potential theory and quantum field theory are covered along with approximations for the treatment of color fluctuations in high-energy QCD scattering and a model for summing classes of eikonal graphs in high-energy scattering problems. The book also presents a variant of the Fradkin representation which suggests a new non-perturbative approximation scheme and provides a qualitative measure of the error involved in each such approximation. Covering the basics as well as more advanced applications this book is suitable for graduate students and researchers in a wide range of fields including quantum field theory fluid dynamics and applied mathematics. Language: English