Table of Contents

Preface.- Outline.- 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.

Reviews

From the reviews: The first chapter deals with a beautiful presentation of special functions... The third chapter covers elliptic and modular functions...in much more detail, and from a different point of view, than one can find in standard introductory books... For [the] subjects that are omitted, the author has suggested some excellent references for the reader who wants to go through these topics. The book is read easily and with great interest. It can be recommended to both students as a textbook and to mathematicians and physicists as a useful reference. ---Mathematical Reviews Mainly original papers are cited to suppoert the historical remarks. The book is well readable. ---ZentralblattMATH This is an unusual textbook, incorporating material showing how classical function theory can be used. The general scheme is to show the reader how things were developed without following the traditional approach of most books on functional theory. This book can be recommended to those who like to see applications of the theory taught in "classical courses". ---EMS "The intended audience for this book is anyone who has taken a calculus course, who knows or is willing to believe the elementary theorems of real analysis given in the appendix, and who wants to learn the classical theory of analytic functions. ! The book is easy to read and with great interest. It can be recommended to both students as a textbook and to mathematicians and physicists as a useful reference." (Vasily A. Chernecky, Zentralblatt MATH, Vol. 1188, 2010)