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

1151 Pages·2000·76.75 MB·English

by  Granino Arthur Korn

\
MATHEMATICAL^
HANDBOOK FOR x
SCIENTISTS AND
ENGINEERS
Definitions,Theorems, and
Formulas for Reference and
Review
Granino A. Korn and
Theresa M. Korn
DOVER BOOKS ON MATHEMATICS
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(continuedonbacknap)
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, NewYork
Copyright
Copyright©1961,1968byGraninoA.KornandTheresaM.Korn
All rights reserved under Pan American and International
CopyrightConventions.
Published inCanada byGeneral Publishing Company, Ltd.,30
LesmillRoad, Don Mills,Toronto, Ontario.
BibliographicalNote
This Dover edition, first published in 2000, is anunabridged
republicationoftheworkoriginallypublishedin1968byMcGraw-
Hill, Inc.,NewYork. Anumberoftypographical errors havebeen
corrected.
LibraryofCongress Cataloging-in-Publication Data
Korn, Granino Arthur, 1922-
Mathematical handbook for scientists andengineers :defini
tions,theorems, andformulas forreferenceandreview/Granino
A.KornandTheresaM. Korn.
p. cm.
Originallypublished:2nd,enl.andrev.ed.NewYork: McGraw-
Hill, cl968.
Includesbibliographicalreferencesandindex.
ISBN0-486-41147-8(pbk.)
1. Mathematics—Handbooks, manuals, etc. I. Korn, Theresa
M. II.Title.
QA40 .K5982000
510'.2'1—dc21
00-030318
Manufacturedin theUnitedStatesofAmerica
DoverPublications, Inc.,31East2nd Street, Mineola, N.Y. 11501
This bookisdedicatedto the memoryofArthur Korn
(1870-1945), who excelledasamathematicianas wellasa
theoreticalphysicistand as a communications engineer.

PREFACE TO THE
SECOND EDITION
Thisnew edition ofthe MathematicalHandbook hasbeensubstantially
enlarged, and muchofour original materialhas beencarefully, and we
hopeusefully, revisedandexpanded. Completelynewsectionsdealwith
z transforms, the matrix notation for systemsof differential equations
(state equations), representation of rotations, mathematical program
ming, optimal-control theory, random processes, and decision theory.
The chapter on numerical computation wasalmost entirely rewritten,
and the revised appendixes include a discussion of Polya's counting
theorem, severalnewtablesofformulas andnumerical data, and a new,
much larger integral table. Numerous illustrations have been added
throughout the remainingtext.
The handbook is again designed, first, as a comprehensive reference
collection of mathematical definitions, theorems, and formulas for
scientists, engineers,and students. Subjects ofboth undergraduate and
graduate level are included. The omissionof all proofsand the concise
tabularpresentationofrelatedformulashavemadeit possibleto incorpo
rate a relatively large amount of reference material in one volume.
The handbook is, however,not intended for referencepurposes alone;
it attemptsto presentaconnectedsurveyofmathematicalmethodsuseful
beyond specialized applications. Each chapter is arranged so as to
permit review of an entire mathematical subject. Such a presentation
is made more manageableand readable through the omissionof proofs;
numerousreferencesprovideaccessto textbookmaterialfor moredetailed
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 scientificand engineering applications.
The writers have attempted to meet the individual reader's require
ments 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.
PREFACE TO THE SECOND EDITION viii
2. The maintext presents, inlargeprint, a concise, connectedreview
of each subject.
3. More detailed discussions and advanced topics are presented in
small print. Thisarrangement makes it possible to include such
material without cluttering the exposition ofthe main review.
Webelievethatthis arrangement has provedusefulinthe firstedition.
The arrangement of the introductory chapters was left unchanged,
althoughadditionsandchangesweremadethroughouttheirtext. Chap
ters 1 to 5 review traditional college material on algebra, analytic geom
etry, elementary and advanced calculus, andvector analysis; Chapter4also
introducesLebesgueandStieltjesintegrals, andFourieranalysis. Chapters
6,7,and8dealwithcurvilinearcoordinate systems,functions ofacomplex
variable, Laplace transforms, and other functionaltransforms; newmate
rial on finite Fourier transforms and on ztransforms was added.
Chapters 9 and 10dealwithordinary and partial differential equations
and include Fourier- andLaplace-transformmethods, the methodofcharac
teristics, andpotential theory; eigenvalue problems assucharetreated in
Chapters14and 15. Chapter11 isessentially new; inadditionto ordi
nary maxima and minima and the classical calculus ofvariations, this
chapternow contains material onlinear and nonlinearprogrammingand
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
mathematicalmodelssuchasgroups,fields, vectorspaces, Booleanalgebras,
and metric spaces. Thetreatmentoffunction spaces continues through
Chapter14topermitamodestfunctional-analysisapproachtoboundary-
value problems and eigenvalue problems in Chapter 15, with enough
essential definitions to enable the reader to use modern advanced texts
and periodical literature.
Chapter13treats matrices;wehaveaddedseveralnewsectionsreview
ing matrix techniques for systems ofordinary differential equations (state
equations of dynamical systems), includingan outline of Lyapunov sta
bility theory. Chapter 14 deals with the important topics of linear
vector spaces, linear transformations (linear operators), introduces eigen
value problems, and describes the use ofmatricesforthe representation of
mathematical models. The material on representation of rotations was
greatly enlarged because of its importance for space-vehicle design as
wellas for atomic and molecularphysics. Chapter 15reviewsa variety
of topics related to boundary-value problems and eigenvalue problems,
including Sturm-Liouville problems, boundary-value problems in two and
three dimensions, and linear integral equations, considering functions as
vectors in a normed vector space.
ix PREFACE TO THE SECOND EDITION
Chapters 16and 17respectively outline tensor analysis and differential
geometry, including the description of plane and space curves, surfaces,
and curvedspaces.
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
mathematicalstatisticsand includes extensive tables of formulas involving
special sampling distributions. A subchapter on Bayes tests and esti
mation was added.
The new Chapter 20 introducesfinite-difference methods and difference
equations and reviews a number of basic methods of numericalcompu
tation. Chapter 21 is essentially a collection of formulas outlining the
properties of higher transcendentalfunctions; 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, largerintegraltable,anda new setoftablesofsumsandseries.
The treatment of combinatorial analysis was enlarged to outline the
use of generatingfunctions and a statementof Polya's countingtheorem.
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
widenhisbackgroundorto correlatehis specializedknowledgewith 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