Approximate calculation of integrals

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Cover Title page Preface Translator's Preface PART ONE. PRELIMINARY INFORMATION Chapter 1. Bernoulli Numbers and Bernoulli Polynomials 1.1. Bernoulli numbers 1.2. Bernoulli polynomials 1.3. Periodic functions related to Bernoulli polynomials 1.4. Expansion of an arbitrary function in Bernoulli polynomials Chapter 2. Orthogonal Polynomials 2.1. General theorems about orthogonal polynomials 2.2. Jacobi and Legendre polynomials 2.3. Chebyshev polynomials 2.4. Chebyshev-Hermite polynomials 2.5. Chebyshev-Laguerre polynomials Chapter 3. Interpolation of Functions 3.1. Finite differences and divided differences 3.2. The interpolating polynomial and its remainder 3.3. Interpolation with multiple nodes Chapter 4. Linear Normed Spaces. Linear Operators 4.1. Linear normed spaces 4.2. Linear operators 4.3. Convergence of a sequence of linear operators PART TWO. APPROXIMATE CALCULATION OF DEFINITE INTEGRALS Chapter 5. Quadrature Sums and Problems Related to Them. The Remainder in Approximate Quadrature 5.1. Quadrature sums 5.2. Remarks on the approximate integration of periodic functions 5.3. The remainder in approximate quadrature and its representation Chapter 6. Interpolatory Quadratures 6.1. Interpolatory quadrature formulas and their remainder terms 6.2. Newton-Cotes formulas 6.3. Certain of the simplest Newton-Cotes formulas Chapter 7. Quadratures of the Highest Algebraic Degree of Precision 7.1. General theorems 7.2. Constant weight function 7.3. Integrals of the form f b (b - Z) (x - a)p f(x) dx and their application to the calculation of multiple integrals 7.4. The integral , f oa 7x' f(,) dx 7.5. Integrals of the form j ? xa e : f (x) dx Chapter 8. Quadrature Formulas with Least Estimate of the Remainder 8.1. Minimization of the remainder of quadrature formulas 8.2. Minimization of the remainder in the class L, ,(r) 8.3. Minimization of the remainder in the class C, 8.4. The problem of minimizing the estimate of the remainder for quadrature with fixed nodes Chapter 9. Quadrature Formulas Containing Preassigned Nodes 9.1. General theorems 9.2. Formulas of special form 9.3. Remarks on integrals with weight functions that change sign Chapter 10. Quadrature Formulas with Equal Coefficients 10.1. Determining the nodes 10.2. Uniqueness of the quadrature formulas of the highest algebraic degree of precision with equal coefficients 10.3. Integrals with a constant weight function Chapter 11. Increasing the Precision of Quadrature Formulas 11.1. Two approaches to the problem 11.2. Weakening the singularity of the integrand 11.3. Euler's method for expanding the remainder 11.4. Increasing the precision when the integral representation of the remainder contains a short principle subinterval Chapter 12. Convergence of the Quadrature Process 12.1. Introduction 12.2. Convergence of interpolatory quadrature formulas for analytic functions 12.3. Convergence of the general quadrature process PART THREE. APPROXIMATE CALCULATION OF INDEFINITE INTEGRALS Chapter 13. Introduction 13.1. Preliminary remarks 13.2. The error of the computation 13.3. Convergence and stability of the computational process Chapter 14. Integration of Functions Given in Tabular Form 14.1. One method for solving the problem 14.2. The remainder Chapter 15. Calculation of Indefinite Integrals Using a Small Number of Values of the Integrand 15.1. General aspects of the problem 15.2. Formulas of special form Chapter 16. Methods Which Use Several Previous Values of the Integral 16.1. Introduction 16.2. Conditions under which the highest degree of precision is achieved 16.3. The number of interpolating polynomials of the highest degree of precision 16.4. The remainder of the interpolation and minimization of its estimate 16.5. Conditions for which the coefficients a, are positive 16.6. Connection with the existence of a polynomial solution to a certain differential equation 16.7. Some particular formulas Appendix A. Gaussian Quadrature Formulas for Constant Weight Function Appendix B. Gaussian-Hermite Quadrature Formulas Appendix C. Gaussian-Laguerre Quadrature Formulas Index