Complex Numbers

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. Stephen Roy assumes no detailed mathematical knowledge on the part of the reader and provides a fascinating description of the use of this fundamental idea within the two subject areas of lattice simulation and number theory. Complex Numbers offers a fresh and critical approach to research-based implementation of the mathematical concept of imaginary numbers. Detailed coverage includes:* Riemann's zeta function: an investigation of the non-trivial roots by Euler-Maclaurin summation.* Basic theory: logarithms, indices, arithmetic and integration procedures are described.* Lattice simulation: the role of complex numbers in Paul Ewald's important work of the I 920s is analysed.* Mangoldt's study of the xi function: close attention is given to the derivation of N(T) formulae by contour integration.* Analytical calculations: used extensively to illustrate important theoretical aspects.* Glossary: over 80 terms included in the text are defined. * Offers a fresh and critical approach to the research-based implication of complex numbers* Includes historical anecdotes, ideas for further research, outlines of theory and a detailed analysis of the Riemann hypothesis* Bridges any gaps that might exist between the two worlds of lattice sums and number theory