Complex Analysis: Fundamentals of the Classical Theory of Functions
All modem introductions to complex analysis follow, more or less explicitly, the pattern laid down in Whittaker and Watson [75]. In "part I'' we find the foundational material, the basic definitions and theorems. In "part II" we find the examples and applications. Slowly we begin to understand why we read part I. Historically this is an anachronism. Pedagogically it is a disaster. Part II in fact predates part I, so clearly it can be taught first. Why should the student have to wade through hundreds of pages before finding out what the subject is good for? In teaching complex analysis this way, we risk more than just boredom. Beginning with a series of unmotivated definitions gives a misleading impression of complex analysis in particular and of mathematics in general. The classical theory of analytic functions did not arise from the idle speculation of bored mathematicians on the possible consequences of an arbitrary set of definitions; it was the natural, even inevitable, consequence of the practical need to answer questions about specific examples. In standard texts, after hundreds of pages of theorems about generic analytic functions with only the rational and trigonometric functions as examples, students inevitably begin to believe that the purpose of complex analysis is to produce more such theorems. We require introductory com­ plex analysis courses of our undergraduates and graduates because it is useful both within mathematics and beyond.
1139947712
Complex Analysis: Fundamentals of the Classical Theory of Functions
All modem introductions to complex analysis follow, more or less explicitly, the pattern laid down in Whittaker and Watson [75]. In "part I'' we find the foundational material, the basic definitions and theorems. In "part II" we find the examples and applications. Slowly we begin to understand why we read part I. Historically this is an anachronism. Pedagogically it is a disaster. Part II in fact predates part I, so clearly it can be taught first. Why should the student have to wade through hundreds of pages before finding out what the subject is good for? In teaching complex analysis this way, we risk more than just boredom. Beginning with a series of unmotivated definitions gives a misleading impression of complex analysis in particular and of mathematics in general. The classical theory of analytic functions did not arise from the idle speculation of bored mathematicians on the possible consequences of an arbitrary set of definitions; it was the natural, even inevitable, consequence of the practical need to answer questions about specific examples. In standard texts, after hundreds of pages of theorems about generic analytic functions with only the rational and trigonometric functions as examples, students inevitably begin to believe that the purpose of complex analysis is to produce more such theorems. We require introductory com­ plex analysis courses of our undergraduates and graduates because it is useful both within mathematics and beyond.
54.99 In Stock
Complex Analysis: Fundamentals of the Classical Theory of Functions

Complex Analysis: Fundamentals of the Classical Theory of Functions

by John Stalker
Complex Analysis: Fundamentals of the Classical Theory of Functions

Complex Analysis: Fundamentals of the Classical Theory of Functions

by John Stalker

Overview

All modem introductions to complex analysis follow, more or less explicitly, the pattern laid down in Whittaker and Watson [75]. In "part I'' we find the foundational material, the basic definitions and theorems. In "part II" we find the examples and applications. Slowly we begin to understand why we read part I. Historically this is an anachronism. Pedagogically it is a disaster. Part II in fact predates part I, so clearly it can be taught first. Why should the student have to wade through hundreds of pages before finding out what the subject is good for? In teaching complex analysis this way, we risk more than just boredom. Beginning with a series of unmotivated definitions gives a misleading impression of complex analysis in particular and of mathematics in general. The classical theory of analytic functions did not arise from the idle speculation of bored mathematicians on the possible consequences of an arbitrary set of definitions; it was the natural, even inevitable, consequence of the practical need to answer questions about specific examples. In standard texts, after hundreds of pages of theorems about generic analytic functions with only the rational and trigonometric functions as examples, students inevitably begin to believe that the purpose of complex analysis is to produce more such theorems. We require introductory com­ plex analysis courses of our undergraduates and graduates because it is useful both within mathematics and beyond.

Product Details

ISBN-13: 9780817649180
Publisher: Birkhäuser Boston
Publication date: 10/16/2009
Series: Modern Birkhäuser Classics
Edition description: 1st ed. 1998. 2nd printing 2009. Softcover reprint of the original 1st ed. 1998
Pages: 228
Product dimensions: 6.10(w) x 9.25(h) x 0.36(d)

Table of Contents

Preface. Outline. 1. Special Functions. 1.1 The Gamma Function. 1.2 The Distribution of Primes I. 1.3 Stirling's Series. 1.4 The Beta Integral. 1.5 The Whittaker Function. 1.6 The Hypergeometric Function. 1.7 Euler-MacLaurin Summation. 1.8 The Zeta Function. 1.9 The Distribution of Primes II.- 2 Analytic Functions. 2.1 Contour Integration. 2.2 Analytic Functions. 2.3 The Cauchy Integral Formula. 2.4 Power Series and Rigidity. 2.5 The Distribution of Primes III. 2.6 Meromorphic Functions. 2.7 Bernoulli Polynomials Revisited. 2.8 Mellin-Barnes Integrals I. 2.9 Mellin-Barnes Integrals II.- 3 Elliptic and Modular Functions. 3.1 Theta Functions. 3.2 Eisenstein Series. 3.3 Lattices. 3.4 Elliptic Functions. 3.5 Complex Multiplication. 3.6 Quadratic Reciprocity. 3.7 Biquadratic Reciprocity.- A Quick Review of Real Analysis.- Bibliography. Index.
From the B&N Reads Blog
Page 1 of

Customer Reviews