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Complex Numbers
Lattice Simulation and Zeta Function Applications
- 1st Edition - July 1, 2007
- Author: S C Roy
- Language: English
- Paperback ISBN:9 7 8 - 1 - 9 0 4 2 7 5 - 2 5 - 1
- eBook ISBN:9 7 8 - 0 - 8 5 7 0 9 - 9 4 2 - 6
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… Read more
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Request a sales quoteAn 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
Mathematicans
- 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 ζ(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
- No. of pages: 148
- Language: English
- Edition: 1
- Published: July 1, 2007
- Imprint: Woodhead Publishing
- Paperback ISBN: 9781904275251
- eBook ISBN: 9780857099426
SR
S C Roy
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.