Mathematical handbook for scientists and engineers: definitions, theorems, and formulas for reference and review

1152 Pages·2000·80.87 MB·English

by  Granino A. Korn

MATHEMATICAL HANDBOOK
FOR SCIENTISTS
AND ENGINEERS
Definitions, Theorems, and Formulas
for Reference and Review
Granino A. Korn
and
Theresa M. Korn
DOVER PUBLICATIONS, INC.
Mineola, New York
Copyright
Copyright © 1961, 1968 by Granino A. Korn and Theresa M. Korn All rights reserved.
Bibliographical Note
This Dover edition, first published in 2000, is an unabridged republication of the work originally
published in 1968 by McGraw-Hill, Inc., New York. A number of typographical errors have been
corrected.
Library of Congress Cataloging-in-Publication Data Korn, Granino Arthur, 1922-Mathematical
handbook for scientists and engineers : definitions, theorems, and formulas for reference and review /
Granino A. Korn and Theresa M. Korn.
p. cm.
Originally published: 2nd, enl. and rev. ed. New York : McGraw-Hill, cl968.
Includes bibliographical references and index.
ISBN 0-486-41147-8 (pbk.) 1.Mathematics—Handbooks, manuals, etc. I. Korn, Theresa M. II.
Title.
QA40 .K598 2000
510'.2'l—dc21
00-030318
Manufactured in the United States by Courier Corporation
41147806
www.doverpublications.com
This book is dedicated to the memory of Arthur Korn (1870-1945), who excelled
as a mathematician as well as a theoretical physicist and as a communications
engineer.
PREFACE TO THE SECOND EDITION
This new edition of the Mathematical Handbook has been substantially
enlarged, and much of our original material has been carefully, and we hope
usefully, revised and expanded. Completely new sections deal with z transforms,
the matrix notation for systems of differential equations (state equations),
representation of rotations, mathematical programming, optimal-control theory,
random processes, and decision theory. The chapter on numerical computation
was almost entirely rewritten, and the revised appendixes include a discussion of
Polya's counting theorem, several new tables of formulas and numerical data,
and a new, much larger integral table. Numerous illustrations have been added
throughout the remaining text.
The handbook is again designed, first, as a comprehensive reference collection
of mathematical definitions, theorems, and formulas for scientists, engineers,
and students. Subjects of both undergraduate and graduate level are included.
The omission of all proofs and the concise tabular presentation of related
formulas have made it possible to incorporate a relatively large amount of
reference material in one volume.
The handbook is, however, not intended for reference purposes alone; it
attempts to present a connected survey of mathematical methods useful beyond
specialized applications. Each chapter is arranged so as to permit review of an
entire mathematical subject. Such a presentation is made more manageable and
readable through the omission of proofs; numerous references provide access to
textbook material for more detailed studies. Special care has been taken to point
out, by means of suitable introductions, notes, and cross references, the
interrelations of various topics and their importance in scientific and engineering
applications.
The writers have attempted to meet the individual reader's requirements by
arranging the subject matter at three levels:
1. The most important formulas and definitions have been collected in tables
and boxed groups permitting rapid reference and review.
2. The main text presents, in large print, a concise, connected review of each
subject.
3. More detailed discussions and advanced topics are presented in small print.
This arrangement makes it possible to include such material without
cluttering the exposition of the main review.
We believe that this arrangement has proved useful in the first edition.
The arrangement of the introductory chapters was left unchanged, although
additions and changes were made throughout their text. Chapters 1 to 5 review
traditional college material on algebra, analytic geometry, elementary and
advanced calculus, and vector analysis; Chapter 4 also introduces Lebesgue and
Stieltjes integrals, and Fourier analysis. Chapters 6, 7, and 8 deal with
curvilinear coordinate systems, functions of a complex variable, Laplace
transforms, and other functional transforms; new material on finite Fourier
transforms and on z transforms was added.
Chapters 9 and 10 deal with ordinary and partial differential equations and
include Fourier-and Laplace-transform methods, the method of characteristics,
and potential theory; eigenvalue problems as such are treated in Chapters 14 and
15. Chapter 11 is essentially new; in addition to ordinary maxima and minima
and the classical calculus of variations, this chapter now contains material on
linear and nonlinear programming and on optimal-control theory, outlining both
the maximum-principle and dynamic-programming approaches.
Chapter 12, considerably expanded in this edition, introduces the elements of
modern abstract language and outlines the construction of mathematical models
such as groups, fields, vector spaces, Boolean algebras, and metric spaces. The
treatment of function spaces continues through Chapter 14 to permit a modest
functional-analysis approach to boundary-value problems and eigenvalue
problems in Chapter 15, with enough essential definitions to enable the reader to
use modern advanced texts and periodical literature.
Chapter 13 treats matrices; we have added several new sections reviewing
matrix techniques for systems of ordinary differential equations (state equations
of dynamical systems), including an outline of Lyapunov stability theory.
Chapter 14 deals with the important topics of linear vector spaces, linear
transformations (linear operators), introduces eigenvalue problems, and
describes the use of matrices for the representation of mathematical models. The
material on representation of rotations was greatly enlarged because of its
importance for space-vehicle design as well as for atomic and molecular physics.
Chapter 15 reviews a variety of topics related to boundary-value problems and
eigenvalue problems, including Sturm-Lioiwille problems, boundary-value
problems in two and three dimensions, and linear integral equations, considering
functions as vectors in a normed vector space.
Chapters 16 and 17 respectively outline tensor analysis and differential
geometry, including the description of plane and space curves, surfaces, and
curved spaces.
In view of the ever-growing importance of statistical methods, the completely
revised Chapter 18 presents a rather detailed treatment of probability theory and
includes material on random processes, correlation functions, and spectra.
Chapter 19 outlines the principal methods of mathematical statistics and
includes extensive tables of formulas involving special sampling distributions. A
subchapter on Bayes tests and estimation was added.
The new Chapter 20 introduces finite-difference methods and difference
equations and reviews a number of basic methods of numerical computation.
Chapter 21 is essentially a collection of formulas outlining the properties of
higher transcendental functions; various formulas and many illustrations have
been added.
The appendixes present mensuration formulas, plane and spherical
trigonometry, combinatorial analysis, Fourier-and Laplace-transform tables, a
new, larger integral table, and a new set of tables of sums and series.
The treatment of combinatorial analysis was enlarged to outline the use of
generating functions and a statement of Polya's counting theorem. Several new
tables of formulas and functions were added. As before, there is a glossary of
symbols, and a comprehensive and detailed index permits the use of the
handbook as a mathematical dictionary.
The writers hope and believe that this handbook will give the reader an
opportunity to scan the field of mathematical methods, and thus to widen his
background or to correlate his specialized knowledge with more general
developments. We are very grateful to the many readers who have helped to
improve the handbook by suggesting corrections and additions; once again, we
earnestly solicit comments and suggestions for improvements, to be addressed to
us in care of the publishers.
Granino A. Korn
Theresa M. Korn
CONTENTS
Preface
Chapter 1. Real and Complex Numbers. Elementary Algebra
1.1. Introduction. The Real-number System
1.2. Powers, Roots, Logarithms, and Factorials. Sum and Product Notation
1.3. Complex Numbers
1.4. Miscellaneous Formulas
1.5. Determinants
1.6. Algebraic Equations: General Theorems
1.7. Factoring of Polynomials and Quotients of Polynomials. Partial
Fractions
1.8. Linear, Quadratic, Cubic, and Quartic Equations
1.9. Systems of Simultaneous Equations
1.10. Related Topics, References, and Bibliography
Chapter 2. Plane Analytic Geometry
2.1. Introduction and Basic Concepts
2.2. The Straight Line
2.3. Relations Involving Points and Straight Lines
2.4. Second-order Curves (Conic Sections)
2.5. Properties of Circles, Ellipses, Hyperbolas, and Parabolas
2.6. Higher Plane Curves
2.7. Related Topics, References, and Bibliography
Chapter 3. Solid Analytic Geometry
3.1. Introduction and Basic Concepts
3.2. The Plane
3.3. The Straight Line
3.4. Relations Involving Points, Planes, and Straight Lines
3.5. Quadric Surfaces
3.6. Related Topics, References, and Bibliography
Chapter 4. Functions and Limits. Differential and Integral Calculus
4.1. Introduction
4.2. Functions
4.3. Point Sets, Intervals, and Regions
4.4. Limits, Continuous Functions, and Related Topics
4.5. Differential Calculus
4.6. Integrals and Integration
4.7. Mean-value Theorems. Values of Indeterminate Forms. Weierstrass's
Approximation Theorems
4.8. Infinite Series, Infinite Products, and Continued Fractions
4.9. Tests for the Convergence and Uniform Convergence of Infinite Series
and Improper Integrals
4.10. Representation of Functions by Infinite Series and Integrals. Power
Series and Taylor's Expansion
4.11. Fourier Series and Fourier Integrals
4.12. Related Topics, References, and Bibliography
Chapter 5. Vector Analysis
5.1. Introduction
5.2. Vector Algebra
5.3. Vector Calculus: Functions of a Scalar Parameter
5.4. Scalar and Vector Fields
5.5. Differential Operators
5.6. Integral Theorems
5.7. Specification of a Vector Field in Terms of Its Curl and Divergence
5.8. Related Topics, References, and Bibliography
Chapter 6. Curvilinear Coordinate Systems
6.1. Introduction
6.2. Curvilinear Coordinate Systems
6.3. Representation of Vectors in Terms of Components
6.4. Orthogonal Coordinate Systems. Vector Relations in Terms of
Orthogonal Components
6.5. Formulas Relating to Special Orthogonal Coordinate Systems
6.6. Related Topics, References, and Bibliography
Chapter 7. Functions of a Complex Variable
7.1. Introduction
7.2. Functions of a Complex Variable. Regions of the Complex-number
Plane
7.3. Analytic (Regular, Holomorphic) Functions
7.4. Treatment of Multiple-valued Functions
7.5. Integral Theorems and Series Expansions
7.6. Zeros and Isolated Singularities
7.7. Residues and Contour Integration
7.8. Analytic Continuation
7.9. Conformal Mapping
7.10. Functions Mapping Specified Regions onto the Unit Circle
7.11. Related Topics, References, and Bibliography
Chapter 8. The Laplace Transformation and Other Functional
Transformations
8.1. Introduction
8.2. The Laplace Transformation
8.3. Correspondence between Operations on Object and Result Functions
8.4. Tables of Laplace-transform Pairs and Computation of Inverse Laplace
Transforms
8.5. “Formal” Laplace Transformation of Impulse-function Terms
8.6. Some Other Integral Transformations
8.7. Finite Integral Transforms, Generating Functions, and z Transforms
8.8. Related Topics, References, and Bibliography
Chapter 9. Ordinary Differential Equations
9.1. Introduction
9.2. First-order Equations
9.3. Linear Differential Equations
9.4. Linear Differential Equations with Constant Coefficients
9.5. Nonlinear Second-order Equations
9.6. Pfaffian Differential Equations
9.7. Related Topics, References, and Bibliography
Chapter 10. Partial Differential Equations
10.1. Introduction and Survey