数论与特殊函数（英文版）
作者：吴振斌 著
出版时间：2011年版
内容简介
the purpose of this book is two-fold.firstly， it gives some basics of complex function theory andspecial functions and secondly， it assembles most important resultsin my own research made in the last several years under theguidance of professor s. kanemitsu， known as jin guangzi in china.the author would like to thank mr. y. -l. lu for helping withtypesetting. thus a beginner reader can use this book as a quick introductionto complex analysis and special functions， and an advanced readercan use it as a source book of many research problems. e.g. in thestudy of the euler integral which appeared in a generalization ofjensen's formula， there are possible new results obtained，likewisethe study of catalan's constant and kummer's fourier series wouldbe a rich arsenal for future studies.
目录
Preface
Chapter 1 A quick introduction to complex analysis
1.1 Introduction
1.2 A quick introduction to complex analysis
1.2.1 Complex number system
1.2.2 Cauchy-Riemann equation and inverse functions
1.2.3 A rough description of complex analysis
1.2.4 Power series
1.2.5 Laurent expansion， residues
1.3 Around Jensen's formula
1.4 Partial fraction expansion
1.4.1 Partial fraction expansions for rational functions
1.4.2 Partial fraction expansion for the cotangent function and so its applications
Chapter 2 Elaboration of results of Srivastava and Choi
2.1 Glossary of symbols and formulas
2.2 Around the Hurwitz zeta-function
2.2.1 Applications of Proposition 2.1
2.2.2 Applications of Corollary 2.1
2.3 Euler integrals
2.4 Around the Euler integral
2.5 Around the Catalan constant
2.6 Kummer's Fourier series for the Log Gamma function
Chapter 3 Arithmetic Laurent coefficients
3.1 Introduction
3.2 Proof of results
3.3 Examples
3.4 The Piltz divisor problem
3.5 The partial integral Ik（x）
3.6 Generalized Euler constants and modular relation
Chapter 4 Mikolas results and their applications
4.1 From the Riemann zeta to the Hurwitz zeta
4.2 Introduction and the polylogarithm case
4.3 The derivative case
Chapter 5 Zeta-value relations
5.1 The structural elucidation of Eisenstein's formula
5.2 Proof of results
5.3 The Lipshitz-Lerch transcendent
Chapter 6 Summation formulas of Poisson and of Plana
6.1 The Poisson summation formula
6.2 Theta transformation formula and functional equation
6.3 The Hurwitz-Lerch zeta-function
6.4 Proof of results
Chapter 7 Modular relation and its applications
7.1 Introduction
7.2 The Riesz sum case
7.3 The Diophantine Dirichlet series
7.4 Elucidation of Katsurada's results
7.5 Proof of results
7.6 Modular relations
Bibliography
Index
