An Introduction to Multicomplex SPates and Functions

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CRC Press, Oct 23, 1990 - Mathematics - 424 pages
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A 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
 

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Contents

THE BICOMPLEX SPACE
1
A Linear Space
3
A Banach Space
4
Multiplication
6
Fractions and Quotients
13
The Idempotent Representation
18
Two Principal Ideals
25
The Auxiliary Complex Spaces
34
The Calculus of Derivatives
179
The Taylor Series of a Holomorphic Function
184
Isomorphic Bicomplex Algebras and CauchyRiemann Matrices
187
Holomorphic Functions and Their Inverses
194
INTEGRALS AND HOLOMORPHIC FUNCTIONS
202
Curves in C2
204
Integrals of Functions with Values in C2
210
The Fundamental Theorem of the Integral Calculus
222

The Discus
44
FUNCTIONS DEFINED BY BICOMPLEX POWER SERIES
53
Limits of Sequences
55
Infinite Series
57
Power Series
61
Functions Represented by Power Series
72
Holomorphic Functions of a Bicomplex Variable
83
Algebras of Holomorphic Functions
94
Elementary Functions
104
The Logarithm Function
122
DERIVATIVES AND HOLOMORPHIC FUNCTIONS
131
Derivatives and the Stolz Condition
134
Differentiability Implies the Strong Stolz Condition
143
The Weak Stolz Condition Implies Differentiability
148
Necessary Conditions
155
Sufficient Conditions
164
Holomorphic and Differentiable Functions
175
A Special Case
235
Existence of Primitives
240
The General Case
244
Integrals Independent of the Path
248
Integrals and the Idempotent Representation
255
Cauchys Integral Theorem and the Idempotent Representation
263
Cauchys Integral Formula
276
Taylor Series
291
Sequences of Holomorphic Functions
301
GENERALIZATIONS TO HIGHER
306
The Idempotent Representation
313
Singular Elements CauchyRiemann Matrices
322
Power Series and Holomorphic Functions in Cn
343
Derivatives of Functions in Cn
354
Integrals and Their Applications
373
EPILOGUE
395
Copyright

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