Complex Numbers

Lattice Simulation and Zeta Function Applications

Häftad, Engelska, 2007

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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.

Produktinformation

Utgivningsdatum2007-07-01
Mått159 x 234 x 8 mm
Vikt230 g
FormatHäftad
SpråkEngelska
Antal sidor144
FörlagElsevier Science & Technology
ISBN9781904275251

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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.

DedicationAbout our AuthorAuthor’s Preface BackgroundImportant featuresAcknowledgementsDEPENDENCE CHARTNotations1. Introduction 1.1 COMPLEX NUMBERS1.2 SCOPE OF THE TEXT1.3 G. F. B. RIEMANN AND THE ZETA FUNCTION1.4 STUDIES OF THE XI FUNCTION BY H. VON MANGOLDT1.5 RECENT WORK ON THE ZETA FUNCTION1.6 P. P. EWALD AND LATTICE SUMMATION2. Theory 2.1 COMPLEX NUMBER ARITHMETIC2.2 ARGAND DIAGRAMS2.3 EULER IDENTITIES2.4 POWERS AND LOGARITHMS2.5 THE HYPERBOLIC FUNCTION2.6 INTEGRATION PROCEDURES USED IN CHAPTERS 3 & 42.7 STANDARD INTEGRATION WITH COMPLEX NUMBERS2.8 LINE AND CONTOUR INTEGRATION3. The Riemann Zeta Function 3.1 INTRODUCTION3.2 THE FUNCTIONAL EQUATION3.3 CONTOUR INTEGRATION PROCEDURES LEADING TO N(T)3.4 A NEW STRATEGY FOR THE EVALUATION OF N(T) BASED ON VON MANGOLDT’S METHOD3.5 COMPUTATIONAL EXAMINATION OF ζ(s)3.6 CONCLUSION AND FURTHER WORK4. Ewald Lattice Summation 4.1 COMPUTER SIMULATION OF IONIC SOLIDS4.2 CONVERGENCE OF LATTICE WAVES WITH ATOMIC POSITION4.3 VECTOR POTENTIAL CONVERGENCE WITH ATOMIC POSITION4.4 DISCUSSION AND FINAL ANALYSIS OF THE EWALD METHOD4.5 CONCLUSION AND FURTHER WORKAPPENDIX 1APPENDIX 2BibliographyGlossaryIndex