复函数论中的经典论题（英文版）
出版时间：2013年版
内容简介
In addition to the correction of typographical errors， the text has been materially changed in three places. The derivation of Stirling's formula in Chapter 2.4， now follows the method of Stieltjes in a more systematic way. The proof of Picard's little theorem in Chapter 10， 2， is carried out following an idea of H. Konig. Finally， in Chapter 11， 4， an inaccuracy has been corrected in the proof of Szego's theorem.
目录
Preface to the Second German Edition
Preface to the First German Edition
Acknowledgments
Advice to the reader
A Infinite Products and Partial Fraction Series
1 Infinite Products of Holomorphic Functions
1. Infinite Products
1. Infinite products of numbers
2. Infinite products of functions
2. Normal Convergence
1. Normal convergence
2. Normally convergent products of holomorphic functions
3. Logarithmic difi.erentiation
3. The Sine Product sin πz =πz ∏∞v=1(1-z2/v2)
1. Standard proof
2. Characterization of the sine by the duplication formula
3. Proof of Euler's formula using Lemma 2
4. Proof of the duplication formula for Euler's product, following Eisenstein
5. On the history of the sine product
4. Euler Partition Products
1. Partitions of natural numbers and Euler products
2. Pentagonal number theorem. Recursion formulas for p(n) and σ(n)
3. Series expansion of ∏∞v=1(1+ qvz) in powers of z
4. On the history of partitions and the pentagonal number theorem
5*. Jacobi's Product Representation of the SeriesJ(z,q):=∑∞v=-∞qv2zv
1. Jacobi's theorem
2. Discussion of Jacobi's theorem
3. On the history of Jacobi's identity
Bibliography
The Gamma Function
1. The Weierstrass Function △(z) = zeγz ∏v≥1(1+z/v)e-z/v
1. The auxiliary function
H(z):= z∏∞v(1+z/v)e-z/v
2. The entire function △(z):=eγzH(z)
2. The Gamma Function
1. Properties of the F-function
2. Historical notes
3. The logarithmic derivative
4. The uniqueness problem
5. Multiplication formulas
6. H(o)lder's theorem
7. The logarithm of the F-function
3. Euler's and Hankel's Integral Representations of Γ(z)
1. Convergence of Euler's integral
2. Euler's theorem
3. The equation
4. Hankel's loop integral
4. Stirling's Formula and Gudermann's Series
1. Stieltjes's definition of the function μ(z)
2. Stirling's formula
3. Growth of |Γ(x+iy)|for |y|→∞
4. Gudermann's series
5. Stirling's series
6. Delicate estimates for the remainder term
7. Binet's integral
8. Lindel(o)f's estimate
5. The Beta Function
1. Proof of Euler's identity
2. Classical proofs of Euler's identity
Bibliography
……
B Mapping Theory
C Selecta
Short Biographies
Symbol Index
Name Index
Subject Index
