An Introduction to Multicomplex SPates and FunctionsA rather pretty little book, written in the form of a text but more likely to be read simply for pleasure, in which the author (Professor Emeritus of Mathematics at the U. of Kansas) explores the analog of the theory of functions of a complex variable which comes into being when the complexes are re |
Contents
FOREWORD Olga Taussky Todd PREFACE Chapter 1 THE BICOMPLEX SPACE 1 Introduction | 1 |
A Linear Space | 2 |
A Banach Space 4 Multiplication | 3 |
Fractions and Quotients | 5 |
The Idempotent Representation | 8 |
Two Principal Ideals | 10 |
The Auxiliary Complex Spaces | 15 |
The Discus | 21 |
Holomorphic Functions and Their Inverses | 60 |
INTEGRALS AND HOLOMORPHIC FUNCTIONS 30 Introduction | 60 |
The Fundamental Theorem of the Integral Calculus | 56 |
A Special Case | 74 |
Existence of Primitives | 36 |
The General Case | 38 |
Integrals Independent of the Path | 39 |
Integrals and the Idempotent Representation | 38 |
FUNCTIONS DEFINED BY BICOMPLEX POWER SERIES 10 Introduction 11 Limits of Sequences 12 Infinite Series | 23 |
Power Series 14 Functions Represented by Power Series | 32 |
Holomorphic Functions of a Bicomplex Variable | 36 |
Algebras of Holomorphic Functions | 41 |
Elementary Functions | 79 |
The Logarithm Function | 46 |
DERIVATIVES AND HOLOMORPHIC FUNCTIONS 19 Introduction | 50 |
Derivatives and the Stolz Condition | 52 |
Differentiability Implies the Strong Stolz Condition 22 The Weak Stolz Condition Implies Differentiability | 55 |
Necessary Conditions | 54 |
Sufficient Conditions | 57 |
Holomorphic and Differentiable Functions | 60 |
The Calculus of Derivatives | 60 |
The Taylor Series of a Holomorphic Function | 60 |
Isomorphic Bicomplex Algebras and CauchyRiemann Matrices | 60 |
Cauchys Integral Theorem and the Idempotent Representation | 39 |
Cauchys Integral Formula | 40 |
Taylor Series | 41 |
Sequences of Holomorphic Functions Chapter 5 GENERALIZATIONS TO HIGHER DIMENSIONS 43 Introduction | 42 |
The Spaces ℂn | 44 |
The Idempotent Representation | 43 |
Singular Elements CauchyRiemann Matrices | 46 |
Power Series and Holomorphic Functions in ℂn | 47 |
Derivatives of Functions in ℂn | 48 |
Integrals and Their Applications | 49 |
EPILOGUE | 104 |
BIBLIOGRAPHY | 106 |
| 107 | |
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Common terms and phrases
1+in Banach algebra BH(X bicomplex numbers bicomplex power series C1 and C2 cartesian set Cauchy sequence Cauchy–Riemann differential equations Cauchy–Riemann equations Cauchy–Riemann matrix Cauchy’s integral theorem closed curve coefficients complete complex variable continuous derivative continuous function converges absolutely Corollary corresponding cosh Definition denote differentiable function discus domain in C2 elements in G2 Exercise exists exponential function f satisfies fis a holomorphic following theorem formula function f fundamental theorem holomorphic functions hypothesis idempotent representation ilz2 inequality infinite series integral calculus inverse ioz2 isomorphic iſz2 iyz2 Lemma Let f limx linear log(z mapping multicomplex multiplication neighborhood nonsingular polygonal curve polynomial proof of Theorem properties Prove the following represented respectively satisfies the strong satisfies the weak satisfy the Cauchy–Riemann series converges Show that f sinh star-shaped statements strong Stolz condition Taylor series THEOREM Let theory of functions true values weak Stolz condition X1 and X2 zero