The theory and applications of infinite products of complex variables

The purpose of the study is to provide, in a single source, a comprehensive presentation of the classical theory of complex infinite products along with both an extensive collection of examples illustrating the theory and a selection of the numerous applications of infinite products to other areas of mathematics. The theoretical development includes a thorough presentation of the topics of infinite products of complex numbers and of analytic functions with special attention devoted to the concepts of convergence, conditional convergence, absolute convergence, uniform convergence and divergence of such products. The numerical examples serve to illustrate the relationships that exist, or that fail to exist, between the different types of convergence. Also part of the theoretical development are the representation theorems of Weierstrass and Hadamard as applied to entire functions, and the work of Blaschke on the representation of functions analytic on the open unit disk. Certain additional results, such as a theorem of Laguerre, are presented as examples of the application of the representation theorems to methods of proof in complex analysis. The theory is applied to obtain the infinite product representations of the functions sin((pi)z), cos((pi)z), 1/(1-z), and the Riemann Zeta Function (zeta)(z). As an additional application, the infinite product form of the Zeta Function is utilized to establish the divergence of the series (SIGMA) 1/p(,n), where p(,n) is the n-th prime number. The Eulerian Gamma Function is presented in both the infinite product form and the integral form, and certain standard formulas, such as (GAMMA)(z+1) = z(.)(GAMMA)(z), are derived by means of the infinite product forms. Further applications of infinite products are presented in the material on representing constants as infinite products. The formulas of Wallis and Vieta on the representation of (pi) as infinite products are verified, and the theorem of Cantor concerning the representation of any real number larger than one as a unique infinite product is presented and some examples are evaluated. Also, a technique to rapidly evaluate certain specific form roots via computer evaluation of infinite products is given along with a calculation of(' )SQRT.(3. Infinite product theory and methods are applied to the field of number theory through the presentation of certain topics taken from the theory of partitions, including a theorem of Euler stating the equality of the number of partitions of an integer n into either odd parts or distinct parts. Also presented are the results of Ramanujan concerning the evaluation of a certain continued fraction and the proof by infinite product methods of the congruence relation p(5k+4) (TBOND) 0 (mod 5) valid for non-negative integers k.