Complex Numbers

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