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Sources in the Development of Mathematics

Book Description

The discovery of infinite products by Wallis and infinite series by Newton marked the beginning of the modern mathematical era. It allowed Newton to solve the problem of finding areas under curves defined by algebraic equations, an achievement beyond the scope of the earlier methods of Torricelli, Fermat and Pascal. While Newton and his contemporaries, including Leibniz and the Bernoullis, concentrated on mathematical analysis and physics, Euler's prodigious accomplishments demonstrated that series and products could also address problems in algebra, combinatorics and number theory. In this book, Ranjan Roy describes many facets of the discovery and use of infinite series and products as worked out by their originators, including mathematicians from Asia, Europe and America. The text provides context and motivation for these discoveries, with many detailed proofs, offering a valuable perspective on modern mathematics. Mathematicians, mathematics students, physicists and engineers will all read this book with benefit and enjoyment.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright Page
  4. Contents
  5. Preface
  6. 1 Power Series in Fifteenth-Century Kerala
    1. 1.1 Preliminary Remarks
    2. 1.2 Transformation of Series
    3. 1.3 Jyesthadeva on Sums of Powers
    4. 1.4 Arctangent Series in the Yuktibhasa
    5. 1.5 Derivation of the Sine Series in the Yuktibhasa
    6. 1.6 Continued Fractions
    7. 1.7 Exercises
    8. 1.8 Notes on the Literature
  7. 2 Sums of Powers of Integers
    1. 2.1 Preliminary Remarks
    2. 2.2 Johann Faulhaber and Sums of Powers
    3. 2.3 Jakob Bernoulli’s Polynomials
    4. 2.4 Proof of Bernoulli’s Formula
    5. 2.5 Exercises
    6. 2.6 Notes on the Literature
  8. 3 Infinite Product of Wallis
    1. 3.1 Preliminary Remarks
    2. 3.2 Wallis’s Infinite Product for π
    3. 3.3 Brouncker and Infinite Continued Fractions
    4. 3.4 Stieltjes: Probability Integral
    5. 3.5 Euler: Series and Continued Fractions
    6. 3.6 Euler: Products and Continued Fractions
    7. 3.7 Euler: Continued Fractions and Integrals
    8. 3.8 Sylvester: A Difference Equation and Euler’s Continued Fraction
    9. 3.9 Euler: Riccati’s Equation and Continued Fractions
    10. 3.10 Exercises
    11. 3.11 Notes on the Literature
  9. 4 The Binomial Theorem
    1. 4.1 Preliminary Remarks
    2. 4.2 Landen’s Derivation of the Binomial Theorem
    3. 4.3 Euler’s Proof for Rational Indices
    4. 4.4 Cauchy: Proof of the Binomial Theorem for Real Exponents
    5. 4.5 Abel’s Theorem on Continuity
    6. 4.6 Harkness and Morley’s Proof of the Binomial Theorem
    7. 4.7 Exercises
    8. 4.8 Notes on the Literature
  10. 5 The Rectification of Curves
    1. 5.1 Preliminary Remarks
    2. 5.2 Descartes’s Method of Finding the Normal
    3. 5.3 Hudde’s Rule for a Double Root
    4. 5.4 Van Heuraet’s Letter on Rectification
    5. 5.5 Newton’s Rectification of a Curve
    6. 5.6 Leibniz’s Derivation of the Arc Length
    7. 5.7 Exercises
    8. 5.8 Notes on the Literature
  11. 6 Inequalities
    1. 6.1 Preliminary Remarks
    2. 6.2 Harriot’s Proof of the Arithmetic and Geometric Means Inequality
    3. 6.3 Maclaurin’s Inequalities
    4. 6.4 Jensen’s Inequality
    5. 6.5 Reisz’s Proof of Minkowski’s Inequality
    6. 6.6 Exercises
    7. 6.7 Notes on the Literature
  12. 7 Geometric Calculus
    1. 7.1 Preliminary Remarks
    2. 7.2 Pascal’s Evaluation of ∫ sin x dx
    3. 7.3 Gregory’s Evaluation of a Beta Integral
    4. 7.4 Gregory’s Evaluation of ∫ secθ dθ
    5. 7.5 Barrow’s Evaluation of ∫ secθ dθ
    6. 7.6 Barrow and the Integral ∫ √ x2 + a2dx
    7. 7.7 Barrow’s Proof of d/dθ tanθ = sec2 θ
    8. 7.8 Barrow’s Product Rule for Derivatives
    9. 7.9 Barrow’s Fundamental Theorem of Calculus
    10. 7.10 Exercises
    11. 7.11 Notes on the Literature
  13. 8 The Calculus of Newton and Leibniz
    1. 8.1 Preliminary Remarks
    2. 8.2 Newton’s 1671 Calculus Text
    3. 8.3 Leibniz: Differential Calculus
    4. 8.4 Leibniz on the Catenary
    5. 8.5 Johann Bernoulli on the Catenary
    6. 8.6 Johann Bernoulli: The Brachistochrone
    7. 8.7 Newton’s Solution to the Brachistochrone
    8. 8.8 Newton on the Radius of Curvature
    9. 8.9 Johann Bernoulli on the Radius of Curvature
    10. 8.10 Exercises
    11. 8.11 Notes on the Literature
  14. 9 De Analysi per Aequationes Infinitas
    1. 9.1 Preliminary Remarks
    2. 9.2 Algebra of Infinite Series
    3. 9.3 Newton’s Polygon
    4. 9.4 Newton on Differential Equations
    5. 9.5 Newton’s Earliest Work on Series
    6. 9.6 De Moivre on Newton’s Formula for sin nθ
    7. 9.7 Stirling’s Proof of Newton’s Formula
    8. 9.8 Zolotarev: Lagrange Inversion with Remainder
    9. 9.9 Exercises
    10. 9.10 Notes on the Literature
  15. 10 Finite Differences: Interpolation and Quadrature
    1. 10.1 Preliminary Remarks
    2. 10.2 Newton: Divided Difference Interpolation
    3. 10.3 Gregory–Newton Interpolation Formula
    4. 10.4 Waring, Lagrange: Interpolation Formula
    5. 10.5 Cauchy, Jacobi: Lagrange Interpolation Formula
    6. 10.6 Newton on Approximate Quadrature
    7. 10.7 Hermite: Approximate Integration
    8. 10.8 Chebyshev on Numerical Integration
    9. 10.9 Exercises
    10. 10.10 Notes on the Literature
  16. 11 Series Transformation by Finite Differences
    1. 11.1 Preliminary Remarks
    2. 11.2 Newton’s Transformation
    3. 11.3 Montmort’s Transformation
    4. 11.4 Euler’s Transformation Formula
    5. 11.5 Stirling’s Transformation Formulas
    6. 11.6 Nicole’s Examples of Sums
    7. 11.7 Stirling Numbers
    8. 11.8 Lagrange’s Proof of Wilson’s Theorem
    9. 11.9 Taylor’s Summation by Parts
    10. 11.10 Exercises
    11. 11.11 Notes on the Literature
  17. 12 The Taylor Series
    1. 12.1 Preliminary Remarks
    2. 12.2 Gregory’s Discovery of the Taylor Series
    3. 12.3 Newton: An Iterated Integral as a Single Integral
    4. 12.4 Bernoulli and Leibniz: A Form of the Taylor Series
    5. 12.5 Taylor and Euler on the Taylor Series
    6. 12.6 Lacroix on d’Alembert’s Derivation of the Remainder
    7. 12.7 Lagrange’s Derivation of the Remainder Term
    8. 12.8 Laplace’s Derivation of the Remainder Term
    9. 12.9 Cauchy on Taylor’s Formula and l’Hôpital’s Rule
    10. 12.10 Cauchy: The Intermediate Value Theorem
    11. 12.11 Exercises
    12. 12.12 Notes on the Literature
  18. 13 Integration of Rational Functions
    1. 13.1 Preliminary Remarks
    2. 13.2 Newton’s 1666 Basic Integrals
    3. 13.3 Newton’s Factorization of xn ± 1
    4. 13.4 Cotes and de Moivre’s Factorizations
    5. 13.5 Euler: Integration of Rational Functions
    6. 13.6 Euler’s Generalization of His Earlier Work
    7. 13.7 Hermite’s Rational Part Algorithm
    8. 13.8 Johann Bernoulli: Integration of √ax2 + bx + c
    9. 13.9 Exercises
    10. 13.10 Notes on the Literature
  19. 14 Difference Equations
    1. 14.1 Preliminary Remarks
    2. 14.2 De Moivre on Recurrent Series
    3. 14.3 Stirling’s Method of Ultimate Relations
    4. 14.4 Daniel Bernoulli on Difference Equations
    5. 14.5 Lagrange: Nonhomogeneous Equations
    6. 14.6 Laplace: Nonhomogeneous Equations
    7. 14.7 Exercises
    8. 14.8 Notes on the Literature
  20. 15 Differential Equations
    1. 15.1 Preliminary Remarks
    2. 15.2 Leibniz: Equations and Series
    3. 15.3 Newton on Separation of Variables
    4. 15.4 Johann Bernoulli’s Solution of a First-Order Equation
    5. 15.5 Euler on General Linear Equations with Constant Coefficients
    6. 15.6 Euler: Nonhomogeneous Equations
    7. 15.7 Lagrange’s Use of the Adjoint
    8. 15.8 Jakob Bernoulli and Riccati’s Equation
    9. 15.9 Riccati’s Equation
    10. 15.10 Singular Solutions
    11. 15.11 Mukhopadhyay on Monge’s Equation
    12. 15.12 Exercises
    13. 15.13 Notes on the Literature
  21. 16 Series and Products for Elementary Functions
    1. 16.1 Preliminary Remarks
    2. 16.2 Euler: Series for Elementary Functions
    3. 16.3 Euler: Products for Trigonometric Functions
    4. 16.4 Euler’s Finite Product for sin nx
    5. 16.5 Cauchy’s Derivation of the Product Formulas
    6. 16.6 Euler and Niklaus I Bernoulli: Partial Fractions Expansions of Trigonometric Functions
    7. 16.7 Euler: Dilogarithm
    8. 16.8 Landen’s Evaluation of ζ(2)
    9. 16.9 Spence: Two-Variable Dilogarithm Formula
    10. 16.10 Exercises
    11. 16.11 Notes on the Literature
  22. 17 Solution of Equations by Radicals
    1. 17.1 Preliminary Remarks
    2. 17.2 Viète’s Trigonometric Solution of the Cubic
    3. 17.3 Descartes’s Solution of the Quartic
    4. 17.4 Euler’s Solution of a Quartic
    5. 17.5 Gauss: Cyclotomy, Lagrange Resolvents, and Gauss Sums
    6. 17.6 Kronecker: Irreducibility of the Cyclotomic Polynomial
    7. 17.7 Exercises
    8. 17.8 Notes on the Literature
  23. 18 Symmetric Functions
    1. 18.1 Preliminary Remarks
    2. 18.2 Euler’s Proofs of Newton’s Rule
    3. 18.3 Maclaurin’s Proof of Newton’s Rule
    4. 18.4 Waring’s Power Sum Formula
    5. 18.5 Gauss’s Fundamental Theorem of Symmetric Functions
    6. 18.6 Cauchy: Fundamental Theorem of Symmetric Functions
    7. 18.7 Cauchy: Elementary Symmetric Functions as Rational Functions of Odd Power Sums
    8. 18.8 Laguerre and Pólya on Symmetric Functions
    9. 18.9 MacMahon’s Generalization of Waring’s Formula
    10. 18.10 Exercises
    11. 18.11 Notes on the Literature
  24. 19 Calculus of Several Variables
    1. 19.1 Preliminary Remarks
    2. 19.2 Homogeneous Functions
    3. 19.3 Cauchy: Taylor Series in Several Variables
    4. 19.4 Clairaut: Exact Differentials and Line Integrals
    5. 19.5 Euler: Double Integrals
    6. 19.6 Lagrange’s Change of Variables Formula
    7. 19.7 Green’s Integral Identities
    8. 19.8 Riemann’s Proof of Green’s Formula
    9. 19.9 Stokes’s Theorem
    10. 19.10 Exercises
    11. 19.11 Notes on the Literature
  25. 20 Algebraic Analysis: The Calculus of Operations
    1. 20.1 Preliminary Remarks
    2. 20.2 Lagrange’s Extension of the Euler–Maclaurin Formula
    3. 20.3 Français’s Method of Solving Differential Equations
    4. 20.4 Herschel: Calculus of Finite Differences
    5. 20.5 Murphy’s Theory of Analytical Operations
    6. 20.6 Duncan Gregory’s Operational Calculus
    7. 20.7 Boole’s Operational Calculus
    8. 20.8 Jacobi and the Symbolic Method
    9. 20.9 Cartier: Gregory’s Proof of Leibniz’s Rule
    10. 20.10 Hamilton’s Algebra of Complex Numbers and Quaternions
    11. 20.11 Exercises
    12. 20.12 Notes on the Literature
  26. 21 Fourier Series
    1. 21.1 Preliminary Remarks
    2. 21.2 Euler: Trigonometric Expansion of a Function
    3. 21.3 Lagrange on the Longitudinal Motion of the Loaded Elastic String
    4. 21.4 Euler on Fourier Series
    5. 21.5 Fourier: Linear Equations in Infinitely Many Unknowns
    6. 21.6 Dirichlet’s Proof of Fourier’s Theorem
    7. 21.7 Dirichlet: On the Evaluation of Gauss Sums
    8. 21.8 Exercises
    9. 21.9 Notes on the Literature
  27. 22 Trigonometric Series after 1830
    1. 22.1 Preliminary Remarks
    2. 22.2 The Riemann Integral
    3. 22.3 Smith: Revision of Riemann and Discovery of the Cantor Set
    4. 22.4 Riemann’s Theorems on Trigonometric Series
    5. 22.5 The Riemann–Lebesgue Lemma
    6. 22.6 Schwarz’s Lemma on Generalized Derivatives
    7. 22.7 Cantor’s Uniqueness Theorem
    8. 22.8 Exercises
    9. 22.9 Notes on the Literature
  28. 23 The Gamma Function
    1. 23.1 Preliminary Remarks
    2. 23.2 Stirling: Γ(1/2) by Newton–Bessel Interpolation
    3. 23.3 Euler’s Evaluation of the Beta Integral
    4. 23.4 Gauss’s Theory of the Gamma Function
    5. 23.5 Poisson, Jacobi, and Dirichlet: Beta Integrals
    6. 23.6 Bohr, Mollerup, and Artin on the Gamma Function
    7. 23.7 Kummer’s Fourier Series for ln Γ(x)
    8. 23.8 Exercises
    9. 23.9 Notes on the Literature
  29. 24 The Asymptotic Series for ln Γ(x)
    1. 24.1 Preliminary Remarks
    2. 24.2 De Moivre’s Asymptotic Series
    3. 24.3 Stirling’s Asymptotic Series
    4. 24.4 Binet’s Integrals for ln Γ(x)
    5. 24.5 Cauchy’s Proof of the Asymptotic Character of de Moivre’s Series
    6. 24.6 Exercises
    7. 24.7 Notes on the Literature
  30. 25 The Euler–Maclaurin Summation Formula
    1. 25.1 Preliminary Remarks
    2. 25.2 Euler on the Euler–Maclaurin Formula
    3. 25.3 Maclaurin’s Derivation of the Euler–Maclaurin Formula
    4. 25.4 Poisson’s Remainder Term
    5. 25.5 Jacobi’s Remainder Term
    6. 25.6 Euler on the Fourier Expansions of Bernoulli Polynomials
    7. 25.7 Abel’s Derivation of the Plana–Abel Formula
    8. 25.8 Exercises
    9. 25.9 Notes on the Literature
  31. 26 L-Series
    1. 26.1 Preliminary Remarks
    2. 26.2 Euler’s First Evaluation of ∑ 1/n2k
    3. 26.3 Euler: Bernoulli Numbers and ∑ 1/n2k
    4. 26.4 Euler’s Evaluation of Some L-Series Values by Partial Fractions
    5. 26.5 Euler’s Evaluation of ∑ 1/n2 by Integration
    6. 26.6 N. Bernoulli’s Evaluation of ∑ 1/(2n + 1)2
    7. 26.7 Euler and Goldbach: Double Zeta Values
    8. 26.8 Dirichlet’s Summation of L(1, χ)
    9. 26.9 Eisenstein’s Proof of the Functional Equation
    10. 26.10 Riemann’s Derivations of the Functional Equation
    11. 26.11 Euler’s Product for ∑ 1/ns
    12. 26.12 Dirichlet Characters
    13. 26.13 Exercises
    14. 26.14 Notes on the Literature
  32. 27 The Hypergeometric Series
    1. 27.1 Preliminary Remarks
    2. 27.2 Euler’s Derivation of the Hypergeometric Equation
    3. 27.3 Pfaff’s Derivation of the 3F2 Identity
    4. 27.4 Gauss’s Contiguous Relations and Summation Formula
    5. 27.5 Gauss’s Proof of the Convergence of F(a, b, c, x) for c – a – b > 0
    6. 27.6 Gauss’s Continued Fraction
    7. 27.7 Gauss: Transformations of Hypergeometric Functions
    8. 27.8 Kummer’s 1836 Paper on Hypergeometric Series
    9. 27.9 Jacobi’s Solution by Definite Integrals
    10. 27.10 Riemann’s Theory of Hypergeometric Functions
    11. 27.11 Exercises
    12. 27.12 Notes on the Literature
  33. 28 Orthogonal Polynomials
    1. 28.1 Preliminary Remarks
    2. 28.2 Legendre’s Proof of the Orthogonality of His Polynomials
    3. 28.3 Gauss on Numerical Integration
    4. 28.4 Jacobi’s Commentary on Gauss
    5. 28.5 Murphy and Ivory: The Rodrigues Formula
    6. 28.6 Liouville’s Proof of the Rodrigues Formula
    7. 28.7 The Jacobi Polynomials
    8. 28.8 Chebyshev: Discrete Orthogonal Polynomials
    9. 28.9 Chebyshev and Orthogonal Matrices
    10. 28.10 Chebyshev’s Discrete Legendre and Jacobi Polynomials
    11. 28.11 Exercises
    12. 28.12 Notes on the Literature
  34. 29 q-Series
    1. 29.1 Preliminary Remarks
    2. 29.2 Jakob Bernoulli’s Theta Series
    3. 29.3 Euler’s q-series Identities
    4. 29.4 Euler’s Pentagonal Number Theorem
    5. 29.5 Gauss: Triangular and Square Numbers Theorem
    6. 29.6 Gauss Polynomials and Gauss Sums
    7. 29.7 Gauss’s q-Binomial Theorem and the Triple Product Identity
    8. 29.8 Jacobi: Triple Product Identity
    9. 29.9 Eisenstein: q-Binomial Theorem
    10. 29.10 Jacobi’s q-Series Identity
    11. 29.11 Cauchy and Ramanujan: The Extension of the Triple Product
    12. 29.12 Rodrigues and MacMahon: Combinatorics
    13. 29.13 Exercises
    14. 29.14 Notes on the Literature
  35. 30 Partitions
    1. 30.1 Preliminary Remarks
    2. 30.2 Sylvester on Partitions
    3. 30.3 Cayley: Sylvester’s Formula
    4. 30.4 Ramanujan: Rogers–Ramanujan Identities
    5. 30.5 Ramanujan’s Congruence Properties of Partitions
    6. 30.6 Exercises
    7. 30.7 Notes on the Literature
  36. 31 q-Series and q-Orthogonal Polynomials
    1. 31.1 Preliminary Remarks
    2. 31.2 Heine’s Transformation
    3. 31.3 Rogers: Threefold Symmetry
    4. 31.4 Rogers: Rogers–Ramanujan Identities
    5. 31.5 Rogers: Third Memoir
    6. 31.6 Rogers–Szegő Polynomials
    7. 31.7 Feldheim and Lanzewizky: Orthogonality of q-Ultraspherical Polynomials
    8. 31.8 Exercises
    9. 31.9 Notes on the Literature
  37. 32 Primes in Arithmetic Progressions
    1. 32.1 Preliminary Remarks
    2. 32.2 Euler: Sum of Prime Reciprocals
    3. 32.3 Dirichlet: Infinitude of Primes in an Arithmetic Progression
    4. 32.4 Class Number and Lχ(1)
    5. 32.5 De la Vallée Poussin’s Complex Analytic Proof of Lχ(1) ≠ 0
    6. 32.6 Gelfond and Linnik: Proof of Lχ(1) ≠ 0
    7. 32.7 Monsky’s Proof That Lχ(1) ≠ 0
    8. 32.8 Exercises
    9. 32.9 Notes on the Literature
  38. 33 Distribution of Primes: Early Results
    1. 33.1 Preliminary Remarks
    2. 33.2 Chebyshev on Legendre’s Formula
    3. 33.3 Chebyshev’s Proof of Bertrand’s Conjecture
    4. 33.4 De Polignac’s Evaluation of ∑p ≤ x inp/p
    5. 33.5 Mertens’s Evaluation of ∏p ≤ x(1 – 1/p)–1
    6. 33.6 Riemann’s Formula for π(x)
    7. 33.7 Exercises
    8. 33.8 Notes on the Literature
  39. 34 Invariant Theory: Cayley and Sylvester
    1. 34.1 Preliminary Remarks
    2. 34.2 Boole’s Derivation of an Invariant
    3. 34.3 Differential Operators of Cayley and Sylvester
    4. 34.4 Cayley’s Generating Function for the Number of Invariants
    5. 34.5 Sylvester’s Fundamental Theorem of Invariant Theory
    6. 34.6 Hilbert’s Finite Basis Theorem
    7. 34.7 Hilbert’s Nullstellensatz
    8. 34.8 Exercises
    9. 34.9 Notes on the Literature
  40. 35 Summability
    1. 35.1 Preliminary Remarks
    2. 35.2 Fejér: Summability of Fourier Series
    3. 35.3 Karamata’s Proof of the Hardy–Littlewood Theorem
    4. 35.4 Wiener’s Proof of Littlewood’s Theorem
    5. 35.5 Hardy and Littlewood: The Prime Number Theorem
    6. 35.6 Wiener’s Proof of the PNT
    7. 35.7 Kac’s Proof of Wiener’s Theorem
    8. 35.8 Gelfand: Normed Rings
    9. 35.9 Exercises
    10. 35.10 Notes on the Literature
  41. 36 Elliptic Functions: Eighteenth Century
    1. 36.1 Preliminary Remarks
    2. 36.2 Fagnano Divides the Lemniscate
    3. 36.3 Euler: Addition Formula
    4. 36.4 Cayley on Landen’s Transformation
    5. 36.5 Lagrange, Gauss, Ivory on the agM
    6. 36.6 Remarks on Gauss and Elliptic Functions
    7. 36.7 Exercises
    8. 36.8 Notes on the Literature
  42. 37 Elliptic Functions: Nineteenth Century
    1. 37.1 Preliminary Remarks
    2. 37.2 Abel: Elliptic Functions
    3. 37.3 Abel: Infinite Products
    4. 37.4 Abel: Division of Elliptic Functions and Algebraic Equations
    5. 37.5 Abel: Division of the Lemniscate
    6. 37.6 Jacobi’s Elliptic Functions
    7. 37.7 Jacobi: Cubic and Quintic Transformations
    8. 37.8 Jacobi’s Transcendental Theory of Transformations
    9. 37.9 Jacobi: Infinite Products for Elliptic Functions
    10. 37.10 Jacobi: Sums of Squares
    11. 37.11 Cauchy: Theta Transformations and Gauss Sums
    12. 37.12 Eisenstein: Reciprocity Laws
    13. 37.13 Liouville’s Theory of Elliptic Functions
    14. 37.14 Exercises
    15. 37.15 Notes on the Literature
  43. 38 Irrational and Transcendental Numbers
    1. 38.1 Preliminary Remarks
    2. 38.2 Liouville Numbers
    3. 38.3 Hermite’s Proof of the Transcendence of e
    4. 38.4 Hilbert’s Proof of the Transcendence of e
    5. 38.5 Exercises
    6. 38.6 Notes on the Literature
  44. 39 Value Distribution Theory
    1. 39.1 Preliminary Remarks
    2. 39.2 Jacobi on Jensen’s Formula
    3. 39.3 Jensen’s Proof
    4. 39.4 Bäcklund Proof of Jensen’s Formula
    5. 39.5 R. Nevanlinna’s Proof of the Poisson–Jensen Formula
    6. 39.6 Nevanlinna’s First Fundamental Theorem
    7. 39.7 Nevanlinna’s Factorization of a Meromorphic Function
    8. 39.8 Picard’s Theorem
    9. 39.9 Borel’s Theorem
    10. 39.10 Nevanlinna’s Second Fundamental Theorem
    11. 39.11 Exercises
    12. 39.12 Notes on the Literature
  45. 40 Univalent Functions
    1. 40.1 Preliminary Remarks
    2. 40.2 Gronwall: Area Inequalities
    3. 40.3 Bieberbach’s Conjecture
    4. 40.4 Littlewood: |an| ≤ en
    5. 40.5 Littlewood and Paley on Odd Univalent Functions
    6. 40.6 Karl Löwner and the Parametric Method
    7. 40.7 De Branges: Proof of Bieberbach’s Conjecture
    8. 40.8 Exercises
    9. 40.9 Notes on the Literature
  46. 41 Finite Fields
    1. 41.1 Preliminary Remarks
    2. 41.2 Euler’s Proof of Fermat’s Little Theorem
    3. 41.3 Gauss’s Proof that Z×p Is Cyclic
    4. 41.4 Gauss on Irreducible Polynomials Modulo a Prime
    5. 41.5 Galois on Finite Fields
    6. 41.6 Dedekind’s Formula
    7. 41.7 Exercises
    8. 41.8 Notes on the Literature
  47. References
  48. Index