Foundations of Analysis Over Surreal Number Fields

Norman L. Alling (Eds.)0444702261, 9780444702265, 9780080872520

In this volume, a tower of surreal number fields is defined, each being a real-closed field having a canonical formal power series structure and many other higher order properties. Formal versions of such theorems as the Implicit Function Theorem hold over such fields. The Main Theorem states that every formal power series in a finite number of variables over a surreal field has a positive radius of hyper-convergence within which it may be evaluated. Analytic functions of several surreal and surcomplex variables can then be defined and studied. Some first results in the one variable case are derived. A primer on Conway’s field of surreal numbers is also given.

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Table of contents :
Content:
Edited by
Page iii

Copyright page
Page iv

Preface
Pages vii-x
Norman L. Alling

Chapter 0 Introduction
Pages 1-11

Chapter 1 Preliminaries
Pages 13-84

Chapter 2 The ξ-Topology
Pages 85-108

Chapter 3 The ξ-Topology on Affine n-Space
Pages 109-116

Chapter 4 Introduction to the Surreal Number Field No
Pages 117-190

Chapter 5 The Surreal Fields ξNo, and Related Topics
Pages 191-205

Chapter 6 The Valuation Theory of Ordered Fields, Applied to No and ξNo
Pages 207-253

Chapter 7 Power Series: Formal and Hyper-Convergent
Pages 255-331

Chapter 8 A Primer on Analytic Functions of a Surreal Variable
Pages 333-351

Bibliography
Pages 353-358

Index
Pages 359-373