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=11-z2v2
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 pn and
σn
3. Series expansion of ∏∞v=11+ qvz in powers of z
4. On the history of partitions and the pentagonal number
theorem
5*. Jacobi''s Product Representation of the
SeriesJz,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≥11+zve-zv
1. The auxiliary function
Hz:= z∏∞v1+zve-zv
2. The entire function △z:=eγzHz
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. Holder''s theorem
7. The logarithm of the F-function
3. Euler''s and Hankel''s Integral Representations of Γz
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 μz
2. Stirling''s formula
3. Growth of |Γx+iy|for |y|→∞
4. Gudermann''s series
5. Stirling''s series
6. Delicate estimates for the remainder term
7. Binet''s integral
8. Lindelof''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
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