An Introduction to Analytic Functions

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.

0. Conventions, Set Theory, Number Systems.- 1. The Complex Plane, Relations, Functions.- 2. Boundedness, Convergence, Continuity.- 3. Paths, Integrals, Derivatives.- 4. Connectedness, Convexity, Analyticity.- 5. Triangles, Polygons, Simple Regions.- 6. Extensions, Contours, Elementary Functions.- 7. Power-Series, Residues, Singularities.- 8. Analytic Inverses, Standard Regions, Convergence Continuation.- 9. Extended Complex Plane, Linear-Fractional Transformations, Meromorphic Functions.- 10. Analytic Relations, Analytic Continuation, Functional Boundaries, Branch-Points.- Appendix A. Homotopy Groups.- Appendix B. Automorphic Functions.- Appendix C. Excepted Values and Uniformization.- Appendix D. Well-ordering.- Appendix E. Analytic Surfaces.- Mac Nerney's Theorem Numbering in the original Edition.- Index.