Series and Products in the Development of Mathematics. Volume 1

Contents
Contents of Volume 2
Preface
1 Power Series in Fifteenth-Century Kerala
	1.1 Preliminary Remarks
	1.2 Transformation of Series
	1.3 Jyesthadeva on Sums of Powers
	1.4 Arctangent Series in the Yuktibhasa
	1.5 Derivation of the Sine Series in the Yuktibhasa
	1.6 Continued Fractions
	1.7 Exercises
	1.8 Notes on the Literature
2 Sums of Powers of Integers
	2.1 Preliminary Remarks
	2.2 Johann Faulhaber
	2.3 Fermat
	2.4 Pascal
	2.5 Seki and Jakob Bernoulli on Bernoulli Numbers
	2.6 Jakob Bernoulli’s Polynomials
	2.7 Euler
	2.8 Lacroix’s Proof of Bernoulli’s Formula
	2.9 Jacobi on Faulhaber
	2.10 Jacobi and Raabe on Bernoulli Polynomials
	2.11 Ramanujan’s Recurrence Relations for Bernoulli Numbers
	2.12 Notes on the Literature
3 Infinite Product of Wallis
	3.1 Preliminary Remarks
	3.2 Wallis’s Infinite Product for π
	3.3 Brouncker and Infinite Continued Fractions
	3.4 Méray and Stieltjes: The Probability Integral
	3.5 Euler: Series and Continued Fractions
	3.6 Euler: Riccati’s Equation and Continued Fractions
	3.7 Exercises
	3.8 Notes on the Literature
4 The Binomial Theorem
	4.1 Preliminary Remarks
	4.2 Landen’s Derivation of the Binomial Theorem
	4.3 Euler: Binomial Theorem for Rational Exponents
	4.4 Cauchy: Proof of the Binomial Theorem for Real Exponents
	4.5 Abel’s Theorem on Continuity
	4.6 Harkness and Morley’s Proof of the Binomial Theorem
	4.7 Exercises
	4.8 Notes on the Literature
5 The Rectification of Curves
	5.1 Preliminary Remarks
	5.2 Descartes’s Method of Finding the Normal
	5.3 Hudde’s Rule for a Double Root
	5.4 Van Heuraet’s Letter on Rectification
	5.5 Newton’s Rectification of a Curve
	5.6 Leibniz’s Derivation of the Arc Length
	5.7 Exercises
	5.8 Notes on the Literature
6 Inequalities
	6.1 Preliminary Remarks
	6.2 Harriot’s Proof of the Arithmetic and Geometric Means Inequality
	6.3 Maclaurin’s Inequalities
	6.4 Comments on Newton’s and Maclaurin’s Inequalities
	6.6 Hölder
	6.7 Jensen’s Inequality
	6.8 Riesz’s Proof of Minkowski’s Inequality
	6.9 Exercises
	6.10 Notes on the Literature
7 The Calculus of Newton and Leibniz
	7.1 Preliminary Remarks
	7.2 Newton’s 1671 Calculus Text
	7.3 Leibniz: Differential Calculus
	7.4 Leibniz on the Catenary
	7.5 Johann Bernoulli on the Catenary
	7.6 Johann Bernoulli: The Brachistochrone
	7.7 Newton’s Solution to the Brachistochrone
	7.8 Newton on the Radius of Curvature
	7.9 Johann Bernoulli on the Radius of Curvature
	7.10 Exercises
	7.11 Notes on the Literature
8 De Analysi per Aequationes Infinitas
	8.1 Preliminary Remarks
	8.2 Algebra of Infinite Series
	8.3 Newton’s Polygon
	8.4 Newton on Differential Equations
	8.5 Newton’s EarliestWork on Series
	8.6 de Moivre on Newton’s Formula for sin nθ
	8.7 Stirling’s Proof of Newton’s Formula
	8.8 Zolotarev: Lagrange Inversion with Remainder
	8.9 Exercises
	8.10 Notes on the Literature
9 Finite Differences: Interpolation and Quadrature
	9.1 Preliminary Remarks
	9.2 Newton: Divided Difference Interpolation
	9.3 Gregory–Newton Interpolation Formula
	9.4 Waring, Lagrange: Interpolation Formula
	9.5 Euler on Interpolation
	9.6 Cauchy, Jacobi: Waring–Lagrange Interpolation Formula
	9.7 Newton on Approximate Quadrature
	9.8 Hermite: Approximate Integration
	9.9 Chebyshev on Numerical Integration
	9.10 Exercises
	9.11 Notes on the Literature
10 Series Transformation by Finite Differences
	10.1 Preliminary Remarks
	10.2 Newton’s Transformation
	10.3 Montmort’s Transformation
	10.4 Euler’s Transformation Formula
	10.5 Stirling’s Transformation Formulas
	10.6 Nicole’s Examples of Sums
	10.7 Stirling Numbers
	10.8 Lagrange’s Proof of Wilson’s Theorem
	10.9 Taylor’s Summation by Parts
	10.10 Exercises
	10.11 Notes on the Literature
11 The Taylor Series
	11.1 Preliminary Remarks
	11.2 Gregory’s Discovery of the Taylor Series
	11.3 Newton: An Iterated Integral as a Single Integral
	11.4 Bernoulli and Leibniz: A Form of the Taylor Series
	11.5 Taylor and Euler on the Taylor Series
	11.6 Lacroix on D’Alembert’s Derivation of the Remainder
	11.7 Lagrange’s Derivation of the Remainder Term
	11.8 Laplace’s Derivation of the Remainder Term
	11.9 Cauchy on Taylor’s Formula and l’Hˆopital’s rule
	11.10 Cauchy: The Intermediate Value Theorem
	11.11 Exercises
	11.12 Notes on the Literature
12 Integration of Rational Functions
	12.1 Preliminary Remarks
	12.2 Newton’s 1666 Basic Integrals
	12.3 Newton’s Factorization of xn ± 1
	12.4 Cotes and de Moivre’s Factorizations
	12.5 Euler: Integration of Rational Functions
	12.6 Euler’s “Investigatio Valoris Integralis”
	12.7 Hermite’s Rational Part Algorithm
	12.8 Johann Bernoulli: Integration of √ax^2+bx+c
	12.9 Exercises
	12.10 Notes on the Literature
13 Difference Equations
	13.1 Preliminary Remarks
	13.2 de Moivre on Recurrent Series
	13.3 Simpson andWaring on Partitioning Series
	13.4 Stirling’s Method of Ultimate Relations
	13.5 Daniel Bernoulli on Difference Equations
	13.6 Lagrange: Nonhomogeneous Equations
	13.7 Laplace: Nonhomogeneous Equations
	13.8 Exercises
	13.9 Notes on the Literature
14 Differential Equations
	14.1 Preliminary Remarks
	14.2 Leibniz: Equations and Series
	14.3 Newton on Separation of Variables
	14.4 Johann Bernoulli’s Solution of a First-Order Equation
	14.5 Euler on General Linear Equations with Constant Coefficients
	14.6 Euler: Nonhomogeneous Equations
	14.7 Lagrange’s Use of the Adjoint
	14.8 Jakob Bernoulli and Riccati’s Equation
	14.9 Riccati’s Equation
	14.10 Singular Solutions
	14.11 Mukhopadhyay on Monge’s Equation
	14.12 Exercises
	14.13 Notes on the Literature
15 Series and Products for Elementary Functions
	15.1 Preliminary Remarks
	15.2 Euler: Series for Elementary Functions
	15.3 Euler: Products for Trigonometric Functions
	15.4 Euler’s Finite Product for sin nx
	15.5 Cauchy’s Derivation of the Product Formulas
	15.6 Euler and Niklaus I Bernoulli: Partial Fraction Expansions
	15.7 Euler: Logarithm
	15.8 Euler: Dilogarithm
	15.9 Spence: Two-Variable Dilogarithm Formula
	15.10 Schellbach: Products to Series
	15.11 Exercises
	15.12 Notes on the Literature
16 Zeta Values
	16.1 Preliminary Remarks
	16.2 Euler’s First Evaluation of Σ 1/n^2k
	16.3 Euler: Bernoulli Numbers and Σ (1/n)^2k
	16.4 Euler’s Evaluation of Some L-Series Values by Partial Fractions
	16.5 Euler’s Evaluation of Σ 1/n^2 by Integration
	16.6 N. Bernoulli’s Evaluation of Σ1/(2n+1)^2
	16.7 Euler and Goldbach: Double Zeta Values
	16.8 Secant and Tangent Numbers and ζ(2m)
	16.9 Landen and Spence: Evaluation of ζ(2k)
	16.10 Exercises
17 The Gamma Function
	17.1 Preliminary Remarks
	17.2 Stirling: Γ(1/2) by Newton–Bessel Interpolation
	17.3 Euler’s Integral for the Gamma Function
	17.4 Euler’s Evaluation of the Beta Integral
	17.5 Newman and the Product for Γ(x)
	17.6 Gauss’s Theory of the Gamma Function
	17.7 Euler: Series to Product
	17.8 Euler: Products to Continued Fractions
	17.9 Sylvester: A Difference Equation and Euler’s Continued Fraction
	17.10 Poisson, Jacobi, and Dirichlet: Beta Integrals
	17.11 The Volume of an n-Dimensional Ball
	17.12 The Selberg Integral
	17.13 Good’s Proof of Dyson’s Conjecture
	17.14 Bohr, Mollerup, and Artin on the Gamma Function
	17.15 Kummer’s Fourier Series for ln Γ(x)
	17.16 Exercises
	17.17 Notes on the Literature
18 The Asymptotic Series for ln Γ(x)
	18.1 Preliminary Remarks
	18.2 De Moivre’s Asymptotic Series
	18.3 Stirling’s Asymptotic Series
	18.4 Binet’s Integrals for ln Γ(x)
	18.5 Cauchy’s Proof of the Asymptotic Character of de Moivre’s Series
	18.6 Exercises
	18.7 Notes on the Literature
19 Fourier Series
	19.1 Preliminary Remarks
	19.2 Euler: Trigonometric Expansion of a Function
	19.3 Lagrange on the Longitudinal Motion of the Loaded Elastic String
	19.4 Euler on Fourier Series
	19.5 Fourier and Linear Equations in Infinitely Many Unknowns
	19.6 Dirichlet’s Proof of Fourier’s Theorem
	19.7 Dirichlet: On the Evaluation of Gauss Sums
	19.8 Schaar: Reciprocity of Gauss Sums
	19.9 Exercises
	19.10 Notes on the Literature
20 The Euler–Maclaurin Summation Formula
	20.1 Preliminary Remarks
	20.2 Euler on the Euler–Maclaurin Formula
	20.3 Maclaurin’s Derivation of the Euler–Maclaurin Formula
	20.4 Poisson’s Remainder Term
	20.5 Jacobi’s Remainder Term
	20.6 Bernoulli Polynomials
	20.7 Number Theoretic Properties of Bernoulli Numbers
	20.8 Exercises
	20.9 Notes on the Literature
21 Operator Calculus and Algebraic Analysis
	21.1 Preliminary Remarks
	21.2 Euler’s Solution of a Difference Equation
	21.3 Lagrange’s Extension of the Euler–Maclaurin Formula
	21.4 Franc¸ais’s Method of Solving Differential Equations
	21.5 Herschel: Calculus of Finite Differences
	21.6 Murphy’s Theory of Analytical Operations
	21.7 Duncan Gregory’s Operational Calculus
	21.8 Boole’s Operational Calculus
	21.9 Jacobi and the Symbolic Method
	21.10 Cartier: Gregory’s Proof of Leibniz’s Rule
	21.11 Hamilton’s Algebra of Complex Numbers and Quaternions
	21.12 Exercises
	21.13 Notes on the Literature
22 Trigonometric Series after 1830
	22.1 Preliminary Remarks
	22.2 The Riemann Integral
	22.3 Smith: Revision of Riemann and Discovery of the Cantor Set
	22.4 Riemann’s Theorems on Trigonometric Series
	22.5 The Riemann–Lebesgue Lemma
	22.6 Schwarz’s Lemma on Generalized Derivatives
	22.7 Cantor’s Uniqueness Theorem
	22.8 Exercises
	22.9 Notes on the Literature
23 The Hypergeometric Series
	23.1 Preliminary Remarks
	23.2 Euler’s Derivation of the Hypergeometric Equation
	23.3 Pfaff’s Derivation of the 3F2 Identity
	23.4 Gauss’s Contiguous Relations and Summation Formula
	23.5 Gauss’s Proof of the Convergence of F(a,b,c,x) for c − a −b > 0
	23.6 Raabe’s Test for Convergence
	23.7 Gauss’s Continued Fraction
	23.8 Gauss: Transformations of Hypergeometric Functions
	23.9 Kummer’s 1836 Paper on Hypergeometric Series
	23.10 Jacobi’s Solution by Definite Integrals
	23.11 Riemann’s Theory of Hypergeometric Functions
	23.12 Exercises
	23.13 Notes on the Literature
24 Orthogonal Polynomials
	24.1 Preliminary Remarks
	24.2 Legendre’s Proof of the Orthogonality of His Polynomials
	24.3 Gauss on Numerical Integration
	24.4 Jacobi’s Commentary on Gauss
	24.5 Murphy and Ivory: The Rodrigues Formula
	24.6 Liouville’s Proof of the Rodrigues Formula
	24.7 The Jacobi Polynomials
	24.8 Stieltjes: Zeros of Jacobi Polynomials
	24.9 Askey: Discriminant of Jacobi Polynomials
	24.10 Chebyshev: Discrete Orthogonal Polynomials
	24.11 Chebyshev and Orthogonal Matrices
	24.12 Chebyshev’s Discrete Legendre and Jacobi Polynomials
	24.13 Exercises
	24.14 Notes on the Literature
Bibliography
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Index
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