Complex Numbers

1st Edition

Lattice Simulation and Zeta Function Applications

Authors: S C Roy
Paperback ISBN: 9781904275251
eBook ISBN: 9780857099426
Imprint: Woodhead Publishing
Published Date: 1st July 2007
Page Count: 144
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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

Details

No. of pages:
144
Language:
English
Copyright:
© Woodhead Publishing 2007
Published:
Imprint:
Woodhead Publishing
eBook ISBN:
9780857099426
Paperback ISBN:
9781904275251

About the Author

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.

Reviews

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