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 analy sis 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 conse quences 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.
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.
| Reihe/Serie |
Modern Birkhäuser Classics
|
| Zusatzinfo |
XI, 228 p. |
| Verlagsort |
Secaucus |
| Sprache |
englisch |
| Maße |
155 x 235 mm |
| Themenwelt
|
Mathematik / Informatik ► Mathematik ► Analysis |
| ISBN-10 |
0-8176-4918-2 / 0817649182 |
| ISBN-13 |
978-0-8176-4918-0 / 9780817649180 |
| Zustand |
Neuware |
