# Bicomplex Holomorphic Functions the Algebra, Geometry and Analysis of Bicomplex Numbers by M. Elena Luna Elizarraras, Michael Shapiro, Daniele C. Struppa and Adrian Vajiac

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### Introduction

The best-known extension of the field of complex numbers to the four-dimensional setting is the skew field of quaternions, introduced by W.R. Hamilton in 1844, [36], [37]. Quaternions arise by considering three imaginary units, i, j, k that anticommuting and such that ij = k. The beauty of the theory of quaternions is that they form a field, where all the customary operations can be accomplished. Their blemish, if one can use this word, is the loss of commutativity. While from a purely algebraic point of view, the lack of commutativity is not such a terrible problem, it does create many difficulties when one tries to extend to quaternions the fecund theory of holomorphic functions of one complex variable. Within this context, one should at least point out that several successful theories exist for holomorphicity in the quaternionic setting. Among those the notion of Fueter regularity (see for example Fueter’s own work [27], or [97] for a modern treatment), and the theory of slice regular functions, originally introduced in [30], and fully developed in [31]. References [97] and [31] contain various quaternionic analogues of the bicomplex results presented in this book

### Contents

Introduction 1

1 The Bicomplex Numbers 5

1.1 Definition of bicomplex numbers . . . . . . . . . . . . . . . . . . . 5

1.2 Versatility of different writings of bicomplex numbers . . . . . . . . 7

1.3 Conjugations of bicomplex numbers . . . . . . . . . . . . . . . . . 8

1.4 Moduli of bicomplex numbers . . . . . . . . . . . . . . . . . . . . . 9

1.4.1 The Euclidean norm of a bicomplex number . . . . . . . . . 11

1.5 Invertibility and zero-divisors in BC . . . . . . . . . . . . . . . . . 12

1.6 Idempotent representations of bicomplex numbers . . . . . . . . . 15

1.7 Hyperbolic numbers inside bicomplex numbers . . . . . . . . . . . 20

1.7.1 The idempotent representation of hyperbolic numbers . . . 23

1.8 The Euclidean norm and the product of bicomplex numbers . . . . 25

2 Algebraic Structures of the Set of Bicomplex Numbers 29

2.1 The ring of bicomplex numbers . . . . . . . . . . . . . . . . . . . . 29

2.2 Linear spaces and modules in BC . . . . . . . . . . . . . . . . . . . 30

2.3 Algebra structures in BC . . . . . . . . . . . . . . . . . . . . . . . 33

2.4 Matrix representations of bicomplex numbers . . . . . . . . . . . . 35

2.5 Bilinear forms and inner products . . . . . . . . . . . . . . . . . . . 37

2.6 A partial order on the set of hyperbolic numbers . . . . . . . . . . 41

2.6.1 Definition of the partial order . . . . . . . . . . . . . . . . . 41

2.6.2 Properties of the partial order . . . . . . . . . . . . . . . . . 42

2.6.3 D-bounded subsets in D. . . . . . . . . . . . . . . . . . . . . 44

2.7 The hyperbolic norm on BC . . . . . . . . . . . . . . . . . . . . . . 47

2.7.1 Multiplicative groups of hyperbolic and bicomplex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3 Geometry and Trigonometric Representations of Bicomplex Numbers 51

3.1 Drawing and thinking in R4 . . . . . . . . . . . . . . . . . . . . . . 52

3.2 Trigonometric representation in complex terms . . . . . . . . . . . 57

3.3 Trigonometric representation in hyperbolic terms . . . . . . . . . . 62

3.3.1 Algebraic properties of the trigonometric representation of

bicomplex numbers in hyperbolic terms . . . . . . . . . . . 65

3.3.2 A geometric interpretation of the hyperbolic trigonometric representation. . . . . . . . . . . . . . . . . . . . . . . . . . 68

4 Lines and curves in BC 73

4.1 Straight lines in BC . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.1.1 Real lines in the complex plane . . . . . . . . . . . . . . . . 73

4.1.2 Real lines in BC . . . . . . . . . . . . . . . . . . . . . . . . 77

4.1.3 Complex lines in BC . . . . . . . . . . . . . . . . . . . . . . 77

4.1.4 Parametric representation of complex lines . . . . . . . . . 78

4.1.5 More properties of complex lines . . . . . . . . . . . . . . . 81

4.1.6 Slope of complex lines . . . . . . . . . . . . . . . . . . . . . 83

4.1.7 Complex lines and complex arguments of bicomplex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.2 Hyperbolic lines in BC . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.2.1 Parametric representation of hyperbolic lines . . . . . . . . 91

4.2.2 More properties of hyperbolic lines . . . . . . . . . . . . . . 92

4.3 Hyperbolic and Complex Curves in BC . . . . . . . . . . . . . . . . 95

4.3.1 Hyperbolic curves . . . . . . . . . . . . . . . . . . . . . . . 95

4.3.2 Hyperbolic tangent lines to a hyperbolic curve . . . . . . . 97

4.3.3 Hyperbolic angle between hyperbolic curves . . . . . . . . . 97

4.3.4 Complex curves . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.4 Bicomplex spheres and balls of hyperbolic radius . . . . . . . . . . 101

4.5 Multiplicative groups of bicomplex spheres . . . . . . . . . . . . . . 102

5 Limits and Continuity 107

5.1 Bicomplex sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.2 The Euclidean topology on BC . . . . . . . . . . . . . . . . . . . . 110

5.3 Bicomplex functions . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6 Elementary Bicomplex Functions 113

6.1 Polynomials of a bicomplex variable . . . . . . . . . . . . . . . . . 113

6.1.1 Complex and real polynomials. . . . . . . . . . . . . . . . . 113

6.1.2 Bicomplex polynomials . . . . . . . . . . . . . . . . . . . . 114

6.2 Exponential functions . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.2.1 The real and complex exponential functions . . . . . . . . . 118

6.2.2 The bicomplex exponential function . . . . . . . . . . . . . 119

6.3 Trigonometric and hyperbolic functions of a bicomplex variable . . 123

6.3.1 Complex Trigonometric Functions . . . . . . . . . . . . . . 123

6.3.2 Bicomplex Trigonometric Functions . . . . . . . . . . . . . 124

6.3.3 Hyperbolic functions of a bicomplex variable . . . . . . . . 127

6.4 Bicomplex radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.5 The bicomplex logarithm . . . . . . . . . . . . . . . . . . . . . . . 128

6.5.1 The real and complex logarithmic functions. . . . . . . . . . 128

6.5.2 The logarithm of a bicomplex number . . . . . . . . . . . . 129

6.6 On bicomplex inverse trigonometric functions . . . . . . . . . . . . 131

6.7 The exponential representations of bicomplex numbers . . . . . . . 131

7 Bicomplex Derivability and Differentiability 135

7.1 Different kinds of partial derivatives . . . . . . . . . . . . . . . . . 135

7.2 The bicomplex derivative and the bicomplex derivability . . . . . . 137

7.3 Partial derivatives of bicomplex derivable functions . . . . . . . . . 144

7.4 Interplay between real differentiability and derivability of

bicomplex functions . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7.4.1 Real differentiability in complex and hyperbolic terms. . . . 152

7.4.2 Real differentiability in bicomplex terms . . . . . . . . . . . 156

7.5 Bicomplex holomorphy versus holomorphy in two (complex or

hyperbolic) variables . . . . . . . . . . . . . . . . . . . . . . . . . . 159

7.6 Bicomplex holomorphy: the idempotent representation . . . . . . . 162

7.7 Cartesian versus idempotent representations in BC-holomorphy. 167

8 Some Properties of Bicomplex Holomorphic Functions 179

8.1 Zeros of bicomplex holomorphic functions . . . . . . . . . . . . . . 179

8.2 When bicomplex holomorphic functions reduce to constants . . . . 181

8.3 Relations among bicomplex, complex and hyperbolic holomorphies 185

8.4 Bicomplex anti-holomorphies . . . . . . . . . . . . . . . . . . . . . 186

8.5 Geometric interpretation of the derivative . . . . . . . . . . . . . . 188

8.6 Bicomplex Riemann Mapping Theorem . . . . . . . . . . . . . . . 190

9 Second Order Complex and Hyperbolic Differential Operators 193

9.1 Holomorphic functions in C and harmonic functions in R2 . . . . . 193

9.2 Complex and hyperbolic Laplacians . . . . . . . . . . . . . . . . . 194

9.3 Complex and hyperbolic wave operators . . . . . . . . . . . . . . . 197

9.4 Conjugate (complex and hyperbolic) harmonic functions . . . . . . 198

10 Sequences and Series of Bicomplex Functions 201

10.1 Series of bicomplex numbers . . . . . . . . . . . . . . . . . . . . . . 201

10.2 General properties of sequences and series of functions . . . . . . . 202

10.3 Convergent series of bicomplex functions . . . . . . . . . . . . . . . 204

10.4 Bicomplex power series . . . . . . . . . . . . . . . . . . . . . . . . . 205

10.5 Bicomplex Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . 208

11 Integral Formulas and Theorems 211

11.1 Stokes’ formula compatible with the bicomplex Cauchy–Riemann

operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

11.2 Bicomplex Borel–Pompeiu formula . . . . . . . . . . . . . . . . . . 214

Bibliography 219

Index 22