Complex Numbers

Lattice Simulation and Zeta Function Applications
Erscheinungsjahr 2007
ISBN: 978-1-904275-25-1
Verlag: Elsevier Science

Buch, Englisch, 144 Seiten, Format (B × H): 159 mm x 234 mm, Gewicht: 222 g

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

Autoren/Hrsg.

Roy, S C

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Reelle Analysis

Inhaltsverzeichnis

- Dedication
- About our Author
- Author's Preface - Background
- Important features
- Acknowledgements
- DEPENDENCE CHART

- Notations
- 1. Introduction - 1.1 COMPLEX NUMBERS
- 1.2 SCOPE OF THE TEXT
- 1.3 G. F. B. RIEMANN AND THE ZETA FUNCTION
- 1.4 STUDIES OF THE XI FUNCTION BY H. VON MANGOLDT
- 1.5 RECENT WORK ON THE ZETA FUNCTION
- 1.6 P. P. EWALD AND LATTICE SUMMATION

- 2. Theory - 2.1 COMPLEX NUMBER ARITHMETIC
- 2.2 ARGAND DIAGRAMS
- 2.3 EULER IDENTITIES
- 2.4 POWERS AND LOGARITHMS
- 2.5 THE HYPERBOLIC FUNCTION
- 2.6 INTEGRATION PROCEDURES USED IN CHAPTERS 3 & 4
- 2.7 STANDARD INTEGRATION WITH COMPLEX NUMBERS
- 2.8 LINE AND CONTOUR INTEGRATION

- 3. The Riemann Zeta Function - 3.1 INTRODUCTION
- 3.2 THE FUNCTIONAL EQUATION
- 3.3 CONTOUR INTEGRATION PROCEDURES LEADING TO N(T)
- 3.4 A NEW STRATEGY FOR THE EVALUATION OF N(T) BASED ON VON MANGOLDT'S METHOD
- 3.5 COMPUTATIONAL EXAMINATION OF ?(s)
- 3.6 CONCLUSION AND FURTHER WORK

- 4. Ewald Lattice Summation - 4.1 COMPUTER SIMULATION OF IONIC SOLIDS
- 4.2 CONVERGENCE OF LATTICE WAVES WITH ATOMIC POSITION
- 4.3 VECTOR POTENTIAL CONVERGENCE WITH ATOMIC POSITION
- 4.4 DISCUSSION AND FINAL ANALYSIS OF THE EWALD METHOD
- 4.5 CONCLUSION AND FURTHER WORK
- APPENDIX 1
- APPENDIX 2

- Bibliography
- Glossary
- Index

Über Autor(innen)

Roy, S C
Dr. Stephen Campbell Roy from the green and pleasant Scottish town of Maybole in Ayreshire, received his secondary education at the Carrick Academy, and then studied chemistry at Heriot-Watt University, Edinburgh where he was awarded a BSc (Hons.) in 1991. Moving to St Andrews University, Fife he studied electro-chemistry and in 1994 was awarded his PhD. He then moved to Newcastle University for work in postdoctoral research until 1997. Then to Manchester University as a temporary Lecturer in Chemistry to teach electrochemistry and computer modelling to undergraduates.