Handbook of Integration

✍ Scribed by Daniel Zwillinger

• Publisher: Jones and Bartlett Publishers
• Year: 1992
• Tongue: English
• Leaves: 384
• Edition: 1
• Category: Library

✦ Synopsis

In performing mathematical analysis, analytic evaluation of integrals
is often required. Other times, an approximate integration may be more
informative than a representation of the exact answer. (The exact representation
could, for example, be in the form of an infinite series.) Lastly, a
numerical approximation to an integral may be all that is required in some
applications.
This book is therefore divided into five sections:
• Applications of Integration which shows how integration is used in
differential equations, geometry, probability and performing summations;
• Concepts and Definitions which defines several different types of integrals
and operations on them;
• Exact Techniques which indicates several ways in which integrals may
be evaluated exactly;
• Approximate Techniques which indicates several ways in which integrals
may be evaluated approximately; and
• Numerical Techniques which indicates several ways in which integrals
may be evaluated numerically.
This handbook has been designed as a reference book. Many of the
techniques in this book are standard in an advanced course in mathematical
methods. Each technique is accompanied by several current references.

✦ Table of Contents

Title Page
Table of Contents
Preface
Introduction
How to Use This Book
I Applications of Integration
1 - Differential Equations: Integral Representations
2 - Differential Equations: Integral Transforms
3 - Extremal Problems
4 - Function Representation
5 - Geometric Applications
6 - MIT Integration Bee
7 - Probability
8 - Summations: Combinatorial
9 - Summations: Other
10 - Zeros of Functions
11 - Miscellaneous Applications
II Concepts and Definitions
12 - Definitions
13 - Integral Definitions
14 - Caveats
15 - Changing Order of Integration
16 - Convergence of Integrals
17 - Exterior Calculus
18 - Feynman Diagrams
19 - Finite Part of Integrals
20 - Fractional Integration
21 - Liouville Theory
22 - Mean Value Theorems
23 - Path Integrals
24 - Principal Value Integrals
25 - Transforms: To a Finite Interval
26 - Transforms: Multidimensional Integrals
27 - Transforms: Miscellaneous
III Exact Analytical Methods
28 - Change of Variable
29 - Computer Aided Solution
30 - Contour Integration
31 - Convolution Techniques
32 - Differentiation and Integration
33 - Dilogarithms
34 - Elliptic Integrals
35 - Frullanian Integrals
36 - Functional Equations
37 - Integration by Parts
38 - Line and Surface Integrals
39 - Look Up Technique
40 - Special Integration Techniques
41 - Stochastic Integration
42 - Tables of Integrals
IV Approximate Analytical Methods
43 - Asymptotic Expansions
44 - Asymptotic Expansions: Multiple Integrals
45 - Continued Fractions
46 - Integral Inequalities
47 - Integration by Parts
48 - Interval Analysis
49 - Laplace's Method
50 - Stationary Phase
51 - Steepest Descent
52 - Approximations: Miscellaneous
V Numerical Methods: Concepts
53 - Introduction to Numerical Methods
54 - Numerical Definitions
55 - Error Analysis
56 - Romberg Integration / Richardson Extrapolation
57 - Software Libraries: Introduction
58 - Software Libraries: Taxonomy
59 - Software Libraries: Excerpts from GAMS
60 - Testing Quadrature Rules
61 - Truncating an Infinite Interval
VI Numerical Methods: Techniques
62 - Adaptive Quadrature
63 - Clenshaw-Curtis Rules
64 - Compound Rules
65 - Cubic Splines
66 - Using Derivative Information
67 - Gaussian Quadrature
68 - Gaussian Quadrature: Generalized
69 - Gaussian Quadrature: Kronrod's Extension
70 - Lattice Rules
71 - Monte Carlo Method
72 - Number Theoretic Methods
73 - Parallel Computer Methods
74 - Polyhedral Symmetry Rules
75 - Polynomial Interpolation
76 - Product Rules
77 - Recurrence Relations
78 - Symbolic Methods
79 - Tschebyscheff Rules
80 - Wozniakowski's Method
81 - Tables: Numerical Methods
82 - Tables: Formulas for Integrals
83 - Tables: Numerically Evaluated Integrals
Mathematical Nomenclature
Index