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Elsevier

Complex Numbers

Lattice Simulation and Zeta Function Applications

  • 1st Edition - July 1, 2007
  • Latest edition
  • Author: S C Roy
  • Language: English

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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Description

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.

Key features

  • 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

Readership

Mathematicans

Table of contents

  • 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

Review quotes

"The reader will not be disappointed. —Zentralblatt MATH

"Roy applies his expertise both in the subject and in teaching in this digestible treatment."—SciTech News

"Offers a fresh and critical approach to research-based implementation of the mathematical concept of imaginary numbers."—Mathematical Reviews

Product details

  • Edition: 1
  • Latest edition
  • Published: July 1, 2007
  • Language: English

About the author

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.

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