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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.

Green's functions. --- Exponential functions. --- Mathematical physics. --- Physical mathematics --- Physics --- Functions, Exponential --- Hyperbolic functions --- Exponents (Algebra) --- Logarithms --- Transcendental functions --- Functions, Green's --- Functions, Induction --- Functions, Source --- Green functions --- Induction functions --- Source functions --- Differential equations --- Potential theory (Mathematics) --- Mathematics

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Eugène Charles Catalan made his famous conjecture – that 8 and 9 are the only two consecutive perfect powers of natural numbers – in 1844 in a letter to the editor of Crelle’s mathematical journal. One hundred and fifty-eight years later, Preda Mihailescu proved it. Catalan’s Conjecture presents this spectacular result in a way that is accessible to the advanced undergraduate. The first few sections of the book require little more than a basic mathematical background and some knowledge of elementary number theory, while later sections involve Galois theory, algebraic number theory and a small amount of commutative algebra. The prerequisites, such as the basic facts from the arithmetic of cyclotomic fields, are all discussed within the text. The author dissects both Mihailescu’s proof and the earlier work it made use of, taking great care to select streamlined and transparent versions of the arguments and to keep the text self-contained. Only in the proof of Thaine’s theorem is a little class field theory used; it is hoped that this application will motivate the interested reader to study the theory further. Beautifully clear and concise, this book will appeal not only to specialists in number theory but to anyone interested in seeing the application of the ideas of algebraic number theory to a famous mathematical problem.

Mathematics. --- Number Theory. --- General Algebraic Systems. --- Mathematics, general. --- Algebra. --- Number theory. --- Mathématiques --- Algèbre --- Théorie des nombres --- Roots, Numerical. --- 511.5 --- Numerical roots --- Cube root --- Exponents (Algebra) --- Square root --- Number study --- Numbers, Theory of --- Algebra --- Diophantine equations --- Catalan, Eugène. --- Mihailescu, Preda. --- Catalan, Eugène Charles, -- 1814-1894. --- Mihăilescu, Preda. --- Number theory --- Roots, Numerical --- Mathematics --- Physical Sciences & Mathematics --- Elementary Mathematics & Arithmetic --- 511.5 Diophantine equations --- Catalan, Eugène Charles --- Catalan, Eugène Charles, --- Mihăilescu, Preda. --- Math --- Science --- Mathematical analysis --- Catalan, Eugène