An Introduction to Analytic Functions - With Theoretical Implications

Description

When first published in 1959, this book was the basis of a two-semester course in complex analysis for upper undergraduate and graduate students. J. S. Mac Nerney was a proponent of the Socratic, or “do-it-yourself” method of learning mathematics, in which students are encouraged to engage in mathematical problem solving, including theorems at every level which are often regarded as “too difficult” for students to prove for themselves. Accordingly, Mac Nerney provides no proofs. What he does instead is to compose and arrange the investigation in his own unique style, so that a contextual proof is always available to the persistent student who enjoys a challenge. The central idea is to empower students by allowing them to discover and rely on their own mathematical abilities. This text may be used in a variety of settings, including: the usual classroom or seminar, but with the teacher acting mainly as a moderator while the students present their discoveries, a small-group setting in which the students present their discoveries to each other, and independent study. The Editors, William E. Kaufman (who was Mac Nerney’s last PhD student) and Ryan C. Schwiebert, have composed the original typed Work into LaTeX ; they have updated the notation, terminology, and some of the prose for modern usage, but the organization of content has been strictly preserved. About this Book, some new exercises, and an index have also been added. John Sheridan Mac Nerney was a student of H. S. Wall. Both of their teaching styles included elements derived from the Moore Method.They both taught by posing problems ranging in difficulty from those one would expect in the usual lectures and texts to others which many might presume to be too difficult for students to solve for themselves. William E. Kaufman received his PhD from the University of Houston in 1979 under J. S. Mac Nerney. His primary areas of research are Hilbert space operator theory and the structure of Banach spaces. He worked on mathematical software for the first Space Shuttle at the Johnson Space Center. He is also interested in functional analysis in general, topological vector spaces, and is currently actively pursuing problems in the theory of nonseparable Banach spaces. Ryan C. Schwiebert received his PhD from Ohio University in 2011 under Sergio López-Permouth and Gregory Oman. His areas of research are in the theory of rings and modules. He has an ongoing interest in applications of abstract algebra to other fields and the creation of software to enhance progress in mathematical research. Preface by the Editors......Page 6 Preface by the Author......Page 7 What Type of Text Is This, and From Whence Did It Come?......Page 9 Suggestions for Using This Book......Page 10 Chapter Notes......Page 11 Acknowledgements......Page 13 Contents......Page 14 About the Author......Page 15 0 Conventions, Set Theory, Number Systems......Page 16 1 The Complex Plane, Relations, Functions......Page 20 2 Boundedness, Convergence, Continuity......Page 26 Optional Project......Page 30 3 Paths, Integrals, Derivatives......Page 31 4 Connectedness, Convexity, Analyticity......Page 38 5 Triangles, Polygons, Simple Regions......Page 42 6 Extensions, Contours, Elementary Functions......Page 47 7 Power-Series, Residues, Singularities......Page 53 8 Analytic Inverses, Standard Regions, Convergence Continuation......Page 58 9 Extended Complex Plane, Linear-Fractional Transformations, Meromorphic Functions......Page 64 10 Analytic Relations, Analytic Continuation, Functional Boundaries, Branch-Points......Page 69 A Homotopy Groups......Page 74 B Automorphic Functions......Page 77 C Excepted Values and Uniformization......Page 80 D Well-ordering......Page 84 E Analytic Surfaces......Page 86 Selected Works by Mac Nerney......Page 89 Mac Nerney's theorem numbering in the original edition......Page 90 Index......Page 93