Sources In Development Of Mathematics, 1st Edition

  • Ranjan Roy
  • Published By:
  • ISBN-10: 1139136224
  • ISBN-13: 9781139136228
  • DDC: 510.72
  • Grade Level Range: College Freshman - College Senior
  • 974 Pages | eBook
  • Original Copyright 2011 | Published/Released December 2011
  • This publication's content originally published in print form: 2011

  • Price:  Sign in for price

About

Overview

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

Front Cover.
Sources in the Development of Mathematics.
Title Page.
Copyright Page.
Contents.
Preface.
1: Power Series in Fifteenth-Century Kerala.
2: Preliminary Remarks.
3: Transformation of Series.
4: Jyesthadeva on Sums of Powers.
5: Arctangent Series in the Yuktibhasa.
6: Derivation of the Sine Series in the Yuktibhasa.
7: Continued Fractions.
8: Exercises.
9: Notes on the Literature.
10: Sums of Powers of Integers.
11: Preliminary Remarks.
12: Johann Faulhaber and Sums of Powers.
13: Jakob Bernoulli's Polynomials.
14: Proof of Bernoulli's Formula.
15: Exercises.
16: Notes on the Literature.
17: Infinite Product of Wallis.
18: Preliminary Remarks.
19: Wallis's Infinite Product for π.
20: Brouncker and Infinite Continued Fractions.
21: Stieltjes: Probability Integral.
22: Euler: Series and Continued Fractions.
23: Euler: Products and Continued Fractions.
24: Euler: Continued Fractions and Integrals.
25: Sylvester: A Difference Equation and Euler's Continued Fraction.
26: Euler: Riccati's Equation and Continued Fractions.
27: Exercises.
28: Notes on the Literature.
29: The Binomial Theorem.
30: Preliminary Remarks.
31: Landen's Derivation of the Binomial Theorem.
32: Euler's Proof for Rational Indices.
33: Cauchy: Proof of the Binomial Theorem for Real Exponents.
34: Abel's Theorem on Continuity.
35: Harkness and Morley's Proof of the Binomial Theorem.
36: Exercises.
37: Notes on the Literature.
38: The Rectification of Curves.
39: Preliminary Remarks.
40: Descartes's Method of Finding the Normal.
41: Hudde's Rule for a Double Root.
42: Van Heuraet's Letter on Rectification.
43: Newton's Rectification of a Curve.
44: Leibniz's Derivation of the Arc Length.
45: Exercises.
46: Notes on the Literature.
47: Inequalities.
48: Preliminary Remarks.
49: Harriot's Proof of the Arithmetic and Geometric Means Inequality.
50: Maclaurin's Inequalities.
51: Jensen's Inequality.
52: Reisz's Proof of Minkowski's Inequality.
53: Exercises.
54: Notes on the Literature.
55: Geometric Calculus.
56: Preliminary Remarks.
57: Pascal's Evaluation of ∫sin x dx.
58: Gregory's Evaluation of a Beta Integral.
59: Gregory's Evaluation of ∫sec θ dθ.
60: Barrow's Evaluation of ∫sec θ dθ.
61: Barrow and the Integral ∫ √(x2 + a2) dx.
62: Barrow's Proof of d/dθ tanθ = sec2θ.
63: Barrow's Product Rule for Derivatives.
64: Barrow's Fundamental Theorem of Calculus.
65: Exercises.
66: Notes on the Literature.
67: The Calculus of Newton and Leibniz.
68: Preliminary Remarks.
69: Newton's 1671 Calculus Text.
70: Leibniz: Differential Calculus.
71: Leibniz on the Catenary.
72: Johann Bernoulli on the Catenary.
73: Johann Bernoulli: The Brachistochrone.
74: Newton's Solution to the Brachistochrone.
75: Newton on the Radius of Curvature.
76: Johann Bernoulli on the Radius of Curvature.
77: Exercises.
78: Notes on the Literature.
79: De Analysi per Aequationes Infinitas.
80: Preliminary Remarks.
81: Algebra of Infinite Series.
82: Newton's Polygon.
83: Newton on Differential Equations.
84: Newton's Earliest Work on Series.
85: De Moivre on Newton's Formula for sin nθ.
86: Stirling's Proof of Newton's Formula.
87: Zolotarev: Lagrange Inversion with Remainder.
88: Exercises.
89: Notes on the Literature.
90: Finite Differences: Interpolation and Quadrature.
91: Preliminary Remarks.
92: Newton: Divided Difference Interpolation.
93: Gregory–Newton Interpolation Formula.
94: Waring, Lagrange: Interpolation Formula.
95: Cauchy, Jacobi: Lagrange Interpolation Formula.
96: Newton on Approximate Quadrature.
97: Hermite: Approximate Integration.
98: Chebyshev on Numerical Integration.
99: Exercises.
100: Notes on the Literature.
101: Series Transformation by Finite Differences.
102: Preliminary Remarks.
103: Newton's Transformation.
104: Montmort's Transformation.
105: Euler's Transformation Formula.
106: Stirling's Transformation Formulas.
107: Nicole's Examples of Sums.
108: Stirling Numbers.
109: Lagrange's Proof of Wilson's Theorem.
110: Taylor's Summation by Parts.
111: Exercises.
112: Notes on the Literature.
113: The Taylor Series.
114: Preliminary Remarks.
115: Gregory's Discovery of the Taylor Series.
116: Newton: An Iterated Integral as a Single Integral.
117: Bernoulli and Leibniz: A Form of the Taylor Series.
118: Taylor and Euler on the Taylor Series.
119: Lacroix on d'Alembert's Derivation of the Remainder.
120: Lagrange's Derivation of the Remainder Term.
121: Laplace's Derivation of the Remainder Term.
122: Cauchy on Taylor's Formula and l'Hôpital's Rule.
123: Cauchy: The Intermediate Value Theorem.
124: Exercises.
125: Notes on the Literature.
126: Integration of Rational Functions.
127: Preliminary Remarks.
128: Newton's 1666 Basic Integrals.
129: Newton's Factorization of xn ±1.
130: Cotes and de Moivre's Factorizations.
131: Euler: Integration of Rational Functions.
132: Euler's Generalization of His Earlier Work.
133: Hermite's Rational Part Algorithm.
134: Johann Bernoulli: Integration of √(ax2 + bx + c).
135: Exercises.
136: Notes on the Literature.
137: Difference Equations.
138: Preliminary Remarks.
139: De Moivre on Recurrent Series.
140: Stirling's Method of Ultimate Relations.
141: Daniel Bernoulli on Difference Equations.
142: Lagrange: Nonhomogeneous Equations.
143: Laplace: Nonhomogeneous Equations.
144: Exercises.
145: Notes on the Literature.
146: Differential Equations.
147: Preliminary Remarks.
148: Leibniz: Equations and Series.
149: Newton on Separation of Variables.
150: Johann Bernoulli's Solution of a First-Order Equation.
151: Euler on General Linear Equations with Constant Coefficients.
152: Euler: Nonhomogeneous Equations.
153: Lagrange's Use of the Adjoint.
154: Jakob Bernoulli and Riccati's Equation.
155: Riccati's Equation.
156: Singular Solutions.
157: Mukhopadhyay on Monge's Equation.
158: Exercises.
159: Notes on the Literature.
160: Series and Products for Elementary Functions.
161: Preliminary Remarks.
162: Euler: Series for Elementary Functions.
163: Euler: Products for Trigonometric Functions.
164: Euler's Finite Product for sin nx.
165: Cauchy's Derivation of the Product Formulas.
166: Euler and Niklaus I Bernoulli: Partial Fractions Expansions of Trigonometric Functions.
167: Euler: Dilogarithm.
168: Landen's Evaluation of ζ (2).
169: Spence: Two-Variable Dilogarithm Formula.
170: Exercises.
171: Notes on the Literature.
172: Solution of Equations by Radicals.
173: Preliminary Remarks.
174: Viète's Trigonometric Solution of the Cubic.
175: Descartes's Solution of the Quartic.
176: Euler's Solution of a Quartic.
177: Gauss: Cyclotomy, Lagrange Resolvents, and Gauss Sums.
178: Kronecker: Irreducibility of the Cyclotomic Polynomial.
179: Exercises.
180: Notes on the Literature.
181: Symmetric Functions.
182: Preliminary Remarks.
183: Euler's Proofs of Newton's Rule.
184: Maclaurin's Proof of Newton's Rule.
185: Waring's Power Sum Formula.
186: Gauss's Fundamental Theorem of Symmetric Functions.
187: Cauchy: Fundamental Theorem of Symmetric Functions.
188: Cauchy: Elementary Symmetric Functions as Rational Functions of Odd Power Sums.
189: Laguerre and Pólya on Symmetric Functions.
190: MacMahon's Generalization of Waring's Formula.
191: Exercises.
192: Notes on the Literature.
193: Calculus of Several Variables.
194: Preliminary Remarks.