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