Series and Products in the Development of Mathematics. Volume 2

Contents
Contents of Volume 1
Preface
25. q-Series
	25.1 Preliminary Remarks
	25.2 Jakob Bernoulli’s Theta Series
	25.3 Euler’s q-Series Identities
	25.4 Euler’s Pentagonal Number Theorem
	25.5 Gauss: Triangular and Square Numbers Theorem
	25.6 Gauss Polynomials and Gauss Sums
	25.7 Gauss’s q-Binomial Theorem and the Triple Product Identity
	25.8 Jacobi: Triple Product Identity
	25.9 Eisenstein: q-Binomial Theorem
	25.10 Jacobi’s q-Series Identity
	25.11 Cauchy and Ramanujan: The Extension of the Triple Product
	25.12 Rodrigues and MacMahon: Combinatorics
	25.13 Exercises
26. Partitions
	26.1 Preliminary Remarks
	26.2 Sylvester on Partitions
	26.3 Cayley: Sylvester’s Formula
	26.4 Ramanujan: Rogers–Ramanujan Identities
	26.5 Ramanujan’s Congruence Properties of Partitions
	26.6 Exercises
	26.7 Notes on the Literature
27. q-Series and q-Orthogonal Polynomials
	27.1 Preliminary Remarks
	27.2 Heine’s Transformation
	27.3 Rogers: Threefold Symmetry
	27.4 Rogers: Rogers–Ramanujan Identities
	27.5 Rogers: “Third Memoir”
	27.6 Rogers–Szegö Polynomials
	27.7 Feldheim and Lanzewizky: Orthogonality of q-Ultraspherical Polynomials
	27.8 Exercises
28. Dirichlet L-Series
	28.1 Preliminary Remarks
	28.2 Dirichlet’s Summation of L(1,χ)
	28.3 Eisenstein’s Proof of the Functional Equation
	28.4 Riemann’s Derivations of the Functional Equation
	28.5 Euler’s Product for Σ 1/n^2
	28.6 Dirichlet Characters
29. Primes in Arithmetic Progressions
	29.1 Preliminary Remarks
	29.2 Euler: Sum of Prime Reciprocals
	29.3 Dirichlet: Infinitude of Primes in an Arithmetic Progression
	29.4 Class Number and L_χ(1)
	29.5 Vallée-Poussin’s Complex Analytic Proof of L_χ (1) ≠ 0
	29.6 Gelfond and Linnik: Proof of L_χ (1) ≠ 0
	29.7 Monsky’s Proof That L_χ (1) ≠ 0
	29.8 Exercises
	29.9 Notes on the Literature
30. Distribution of Primes: Early Results
	30.1 Preliminary Remarks
	30.2 Chebyshev on Legendre’s Formula
	30.3 Chebyshev’s Proof of Bertrand’s Conjecture
	30.4 De Polignac’s Evaluation of Σ_{p≤x} ln p/p
	30.5 Mertens’s Evaluation of Π_{P≤x} (1-1/p)^{-1}
	30.6 Riemann’s Formula for π(x)
	30.7 Exercises
	30.8 Notes on the Literature
31. Invariant Theory: Cayley and Sylvester
	31.1 Preliminary Remarks
	31.2 Boole’s Derivation of an Invariant
	31.3 Differential Operators of Cayley and Sylvester
	31.4 Cayley’s Generating Function for the Number of Invariants
	31.5 Sylvester’s Fundamental Theorem of Invariant Theory
	31.6 Hilbert’s Finite Basis Theorem
	31.7 Hilbert’s Nullstellensatz
	31.8 Exercises
	31.9 Notes on the Literature
32. Summability
	32.1 Preliminary Remarks
	32.2 Fejér: Summability of Fourier Series
	32.3 Karamata’s Proof of the Hardy–Littlewood Theorem
	32.4 Wiener’s Proof of Littlewood’s Theorem
	32.5 Hardy and Littlewood: The Prime Number Theorem
	32.6 Wiener’s Proof of the PNT
	32.7 Kac’s Proof of Wiener’s Theorem
	32.8 Gelfand: Normed Rings
	32.9 Exercises
	32.10 Notes on the Literature
33. Elliptic Functions: Eighteenth Century
	33.1 Preliminary Remarks
	33.2 Fagnano Divides the Lemniscate
	33.3 Euler: Addition Formula
	33.4 Cayley on Landen’s Transformation
	33.5 Lagrange, Gauss, Ivory on the agM
	33.6 Remarks on Gauss and Elliptic Functions
	33.7 Exercises
	33.8 Notes on the Literature
34. Elliptic Functions: Nineteenth Century
	34.1 Preliminary Remarks
	34.2 Abel: Elliptic Functions
	34.3 Abel: Infinite Products
	34.4 Abel: Division of Elliptic Functions and Algebraic Equations
	34.5 Abel: Division of the Lemniscate
	34.6 Jacobi’s Elliptic Functions
	34.7 Jacobi: Cubic and Quintic Transformations
	34.8 Jacobi’s Transcendental Theory of Transformations
	34.9 Jacobi: Infinite Products for Elliptic Functions
	34.10 Jacobi: Sums of Squares
	34.11 Cauchy: Theta Transformations and Gauss Sums
	34.12 Eisenstein: Reciprocity Laws
	34.13 Liouville’s Theory of Elliptic Functions
	34.14 Hermite’s Theory of Elliptic Functions
	34.15 Exercises
	34.16 Notes on the Literature
35. Irrational and Transcendental Numbers
	35.1 Preliminary Remarks
	35.2 Liouville Numbers
	35.3 Hermite’s Proof of the Transcendence of e
	35.4 Hilbert’s Proof of the Transcendence of e
	35.5 Exercises
	35.6 Notes on the Literature
36. Value Distribution Theory
	36.1 Preliminary Remarks
	36.2 Jacobi on Jensen’s Formula
	36.3 Jensen’s Proof
	36.4 B¨acklund Proof of Jensen’s Formula
	36.5 R. Nevanlinna’s Proof of the Poisson–Jensen Formula
	36.6 Nevanlinna’s First Fundamental Theorem
	36.7 Nevanlinna’s Factorization of a Meromorphic Function
	36.8 Picard’s Theorem
	36.9 Borel’s Theorem
	36.10 Nevanlinna’s Second Fundamental Theorem
	36.11 Exercises
	36.12 Notes on the Literature
37. Univalent Functions
	37.1 Preliminary Remarks
	37.2 Gronwall: Area Inequalities
	37.3 Bieberbach’s Conjecture
	37.4 Littlewood: |a_n| ≤ en
	37.5 Littlewood and Paley on Odd Univalent Functions
	37.6 Karl Löwner and the Parametric Method
	37.7 De Branges: Proof of Bieberbach
	37.8 Exercises
	37.9 Notes on the Literature
38. Finite Fields
	38.1 Preliminary Remarks
	38.2 Euler’s Proof of Fermat’s Little Theorem
	38.3 Gauss’s Proof That Z^×_p Is Cyclic
	38.4 Gauss on Irreducible Polynomials Modulo a Prime
	38.5 Galois on Finite Fields
	38.6 Dedekind’s Formula
	38.7 Finite Field Analogs of the Gamma and Beta Integrals
	38.8 Weil: Solutions of Equations in Finite Fields
	38.9 Exercises
	38.10 Notes on the Literature
Bibliography
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Index
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