## 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 |

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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 |

### Common terms and phrases

Banach algebra BH(X bicomplex numbers bicomplex power series C-Co cartesian set Cauchy sequence Cauchy-Riemann differential equations Cauchy-Riemann equations Cauchy-Riemann matrix Cauchy's integral formula Cauchy's integral theorem closed curve complex variable continuous derivative corresponding cosh Definition denote determinant differentiable function discus domain in C2 elements in C2 equal establish Exercise exists exponential function f(zl following theorem functions in Cn fundamental theorem Furthermore holomorphic function hypothesis idempotent representation ilz2 ilz2)el inC2 inequality infinite series integral calculus inverse isomorphic Lemma linear mapping multicomplex multiplication neighborhood nonsingular point in Xl polygonal curve polynomial power series proof of Theorem properties Prove the following result satisfies the strong satisfies the uniform satisfies the weak satisfy the Cauchy-Riemann segment sequence of subdivisions singular elements sinh star-shaped domain star-shaped with respect statements strong Stolz condition Taylor series THEOREM Let theory of functions uniform strong Stolz weak Stolz condition Xl and X2 zero zl+i2z2