ABSTRACT
I. Introduction 192II. Introductory Sources-RecommendedUndergraduate Preparation 193A. Advanced calculus/introduction to analysis 193B. Abstract analysis/topology for analysis 195C. Linear algebra 195D. Picking an area of interest 196III. Real Analysis 197A. Standard real analysis texts 198B. Other useful sources for real analysisincluding classic works 199IV. Complex Analysis 201A. Standard texts 202B. Other useful sources for complex analysisincluding classic works 203V. Functional Analysis 204VI. Harmonic Analysis 206A. Introductory sources 208B. General sources 208C. Wavelets 208D. Harmonic analysis on groups and symmetric spaces 208VII. Classical Analysis: Special Functions,Orthogonal Polynomials 209 191
A. Special functions 210B. Orthogonal polynomials 210VIII. Function Spaces 210IX. Advanced Topics in Complex Analysis 211A. Functions of several complex variables 211B. Riemann surfaces and related topics 212C. Abelian, automorphic,.and elliptic functions 212D. Complex dynamics 212X. Major Journals Publishing Papers in Analysis 212A. Selected journals frequently carrying expository or survey articles on real or complex analysis topics 212B. Selected journals carrying significant numbers of papers on real or complex analysis 213XI. Miscellaneous Sources 214
(Fundamental Theorem of calculus) (Taylor series expansion)
illustrate what analysis is about. The Fundamental Theorem shows how to analyze the function / via its derivative. The Taylor series decomposes the function as an infinite series of simple polynomials. Both formulas involve the idea of limit, both involve the notion of derivative, and the Fundamental Theorem involves the concept of integral. The three concepts-derivative, integral, and, underlying both, the idea of limit-are common to the numerous subdisciplines that constitute analysis. Within the general subject of analysis, real and complex analysis are core areas in the sense that a basic knowledge of these areas is a prerequisite for work at the research level. Furthermore, they should be part of the educational background of any well-educated mathematician.
