SCIENCE / Crystallography

This book is an informative and useful account of complex numbers that includes historical anecdotes, ideas for further research, outlines of theory, and a detailed analysis of the ever-elusory Riemann hypothesis. It bridges any gaps that might exist between the two worlds of lattice sums and number theory. It is designed to provide the reader with an introduction to the theory of complex numbers, ranging from elementary concepts to more advanced circumstances such as contour integration. Complex Numbers provides details of important scientific investigations and mathematical research. It offers a fresh and critical approach to the research-based implication of an intriguing mathematical concept. Detailed coverage includes: -- Riemann's zeta function Ñ an investigation of the non-trivial roots by Euler-Maclaurin summation. -- Basic theory Ñ arithmetic, logarithms, indices, and integration procedures are discussed. -- Lattice simulation Ñ the role of complex numbers in Paul Ewald's important work of the 1920s is analyzed. -- Mangolt's study of the xi function Ñ close attention is given to the derivation of N(T) formulae by contour integration. -- Glossary Ñ over 80 terms included in the text are defined.