دانلود کتاب Series and Products in the Development of Mathematics. Volume 2

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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 B C D E F G H I JK L M NO P R S T VW YZ Index AB CD EF GH IJKL MNOP QRS TUVW YZ