Approximate calculation of integrals

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