There are incredibly rich connections between classical analysis and number theory. For instance, analytic number theory contains many examples of asymptotic expressions derived from estimates for analytic functions, such as in the proof of the Prime Number Theorem. In combinatorial number theory, exact formulas for number-theoretic quantities are derived from relations between analytic functions. Elliptic functions, especially theta functions, are an important class of such functions in this context, which had been made clear already in Jacobi's Fundamenta nova.
The general aim of this book is to provide a modern approach to number theory through a blending of complementary algebraic and analytic perspectives, emphasizing harmonic analysis on topological groups. The more particular goal is to cover John Tate's visionary thesis, giving virtually all of the necessary analytic details and topological preliminaries--technical prerequisites that are often foreign to the typical, more algebraically inclined number theorist.
This ninth edition has been revised to ensure that it provides mathematically precise, succinct and readable engineering/science oriented calculus material. It features a visual presentation, designed to encourage learning; revised exercises to ensure clarity, balance and relevance; and clear commentary on the difficult subject of critical multivariable calculus topics.
Partial Differential Equations: Sources and Solutions
Offering a welcome balance between rigor and ease of comprehension, this book presents full coverage of the analytic (and accurate) method for solving PDEs -- in a manner that is both decipherable to engineers and physically insightful for mathematicians. By exploring the eigenfunction expansion method based on physical principles instead of abstract analyses, it makes the analytic approach understandable, visualizable, and straightforward to implement. Contains tabulations and derivations of all known eigenfunction expansions.
Diffusions, Superdiffusions and Partial Differential Equations
Interactions between the theory of partial differential equations of elliptic and parabolic types and the theory of stochastic processes are beneficial for both probability theory and analysis. At the beginning, mostly analytic results were used by probabilists.
