Handbook of Physics [1 ed.]

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HANDBOOK

OF P H Y S I C S

M cG R A W -H IL L H A N D B O O K S

A b bo tt A ljia n

a n d S m i t h • N ational Electrioal C ode H andbook, Dth ed. * Purchasing H andbook I n s t i t u t e o f P h y s i c s • Am erican Institute o f Physics H andbook

Am erican

A m erican S o cie ty o f

ASM E ASM E ASM I4] ASM E

M e c h a n ic a l E n gin eers

H an db ook : H an db ook : H an db ook : H an d b ook :

Engineering Tables M etals Engineering— Design M etals Engineering— Processes M etals Properties A m e r i c a n S o c i e t y o f T o o l E n g i n e e r s • Die Design H andbook A m e r i c a n Soc i e t y o f T o o l E n g i n e e r s • T ool Engineers H andbook B e e m a n • Industrial Power Systems H andbook B e r r y , H o l l a y , a n d B e e r s • H andbook of M eteorology B r a d y • Materials H an dbook, Sth ed. C o c k r e l l • Industrial Electronics H andbook C o m p r e s s e d A i r a n d G a s I n s t i t c t e • Com pressed Air H andbook, 2d ed. C o n d o n a n d O d i s h a w • H an dbook of Physics C o n s i d i n e • Process Instrum ents and Controls H andbook C r o c k e r • Piping H an dbook, 1th ed. C r o f t • Am erican E lectrician s’ H andbook, 7th ed. D a v i s ■ H andbook o f Applied H ydraulics, 2d ed. E t i i e r i n g t o n • Nuclear Engineering H andbook F a c t o r y M u t u a l E n g i n e e r i n g D i v i s i o n • H andbook of Industrial Loss Prevention F i n k • T elevision Engineering H andbook H a r r i s • H andbook of Noise Control H e n n e y • R adio Engineering H an dbook, 5th ed. 11e n t e r ■ H andbook o f S em iconductor Electronics J o h n s o n a n d A c t i i • Fuels and Com bustion H andbook Jt r a n • (>uality-control H andbook K e t c i i c m • Structural Engineers’ H andbook, dd ed. K i n g • H andbook of H ydraulics, 1th cd. K n o w l t o n • Standard H andbook for Electrical Engineers, Dth ed. K urtz • The Linem an's H an dbook, lid ed. L a h r k r t o n a n d M a r k s • M arine Engineers’ H andbook L a n d e e , D a v i s , a n d A l b r e c h t • E lectronic Designers’ H andbook L a u g i i n e r a n d H a r g a n • H an dbook of Fastening and Joining of M etal Parts L e G r a n d • The New Am erican M a ch in ist’s H andbook L i d d e l l • H an dbook of N onferrous M etallurgy, 2 vols., 2 d e d . M a g i l l , H o l d e n , a n d A c k l e y • Air Pollution H andbook M a n a s • National Plum bing C ode H andbook M a n t e l l • Engineering M aterials H andbook M a r k s a n d B a c m e i s t e r • M echanical Engineers’ H andbook, Oth ed. M a r k i s a n d Z e l u f f • H andbook of industrial Electronic Circuits M a k k i s a n d Z e l c f f • H andbook of Industrial Electronic Control Circuits M a y n a r d • Industrial Engineering H andbook M e r r i t t ■ Building ( ’oustruction H andbook M o r r o w • M aintenance Engineering H andbook O ’ R o u r k e • General Engineering H andbook, 2d ed. P a c i f i c : ( ’o a s t G a s A s s o c i a t i o n • Gas Engineers’ H andbook P e r r y • Chemical Business H andbook P e r r y • Chemical Engineers’ H andbook, ¡Id ed. S i i a n d • Glass Engineering H andbook, 2d ed. S t a n i a r • Plant Engineering H andbook, 2d ed. T e r m a n • R adio Engineers’ H andbook T r u x a l ■ Control Engineers’ H andbook C r q c i i a r t • Civil Engineering H andbook, 1th ed. Y o d e r , I I e n e m a n , T u r n b u l l , a n d S t o n e • H andbook of Personnel M anagem ent and Labor Relations

H A N D B O O K OF PHYSICS Prepared by a staff of specialists E d ite d b y

E. U. C O N D O N , Ph.D. W aym an Crow Professor o f Physics W ashington University, St. Louis Former D irector, National Bureau o f Standards Washinyton, D. C.

H U G H O D I S H A W , D.Sc. Executive Director, U. S. National Connnittee fo r the International Geophysical Year, National Academy o f Sciences, W ashington, D. (\ Former Assistant to the Director, National Bureau o f Standards, Washinyton, D. C.

M c G R A W- I I I LL BOOK C O M P A N Y , I NC. New York T oron to London

1958

HANDBOOK

OF

PHYSICS

C opyrigh t © 1958 b y the M cG raw -H ill B ook C om pany, Inc. Printed in the United States o f Am erica. All rights reserved. This book, or parts thereof, m ay not be reproduced in any form w ithout permission o f the publishers. Library of Congress C atalog Card N u m ber: 57-6387

THE

MAPLE

PRESS

COMPANY,

YORK,

PA.

Contributors

M ilto n A b m m o w itz

C ly d e R . B u rn ett

C h ief, C om putation L aboratory

A ssistan t P rofessor

N a tion a l B u reau o f Standards (deceased)

The P en n sylva n ia Elate U niversity

L a w ren ce H . A ller

H erb ert B. C a llcn

P ro fesso r o f A stron om y

P rofessor o f P h ysics

U n iversity o f M ich iga n

U n iversity o f P en n sylva n ia

F ran z L. A lt

G . M . C lem en cc

A ssista n t C h ief

Scien tific D irector

A p p lie d M athem atics D ivision

U .S. N aval O bservatory

N a tion a l B u reau o f Standards

E. U. C ondon

R . D . A rn o ld

W aym a n Crow P rofessor o f P h ysics

O perations A n a ly st

W ash in gton U niversity

O perations Evaluation G roup

E . R ich a rd C oh en

M assachusetts In stitu te o f T echnology

R esearch A dvisor

K . R . A tk in s

A tom ics In tern a tion a l

P ro fesso r o f P h y sics

A D ivision o f N orth A merican A viation , In c.

U n iversity o f P en n sylva n ia

L . E . C op ela n d

J oh n B ardeen P ro fesso r o f P h y sics and P ro fesso r o f Electrical E n gin eerin g U niversity o f Illin o is R ich a rd A . B eth P h y sicis t, B rookhavcn N a tion al L aboratory R . B y ro n B ird P rofessor o f Chem ical E n gin eerin g U niversity o f W iscon sin

S en ior R esearch Chem ist P ortland C em ent A ssocia tion Research and D evelopm ent Laboratories C . F. C u rtiss A ssociate P rofessor o f C hem istry U n iversity o f W iscoiisin Jesse W . M . D u M o n d P rofessor o f P h ysics C aliforn ia In stitu te o f T echnology L eon a rd E isen bu d

J oh n P . B le w e tt S en ior P h ysicist Brookhavcn N ation al Laboratory

P h ysicist B artol R esearch F ou nd ation C h u rch ill E isen hart

R o b e r t R . B row n

C h ief

A ssista?it P ro fesso r

Statistical E n gin eerin g L aboratory

U niversity o f C a lifo rn ia , B erk eley

N ation al B u reau o f Standards

S a n born C . B ro w n

W illia m E . F o r sy th e

A ssocia te P rofessor o f P h ysics

L a m p D ivision

M assachusetts In stitu te o f T echnology

G eneral E lectric C om pan y

W illiam F u ller B ro w n , Jr.

M . M . E r o d it

P rofessor o f E lectrical E n gin eerin g

R esearch P rofessor o f M ech an ics

U niversity o f M in n esota

and D irector o f E xperim en tal Stress A n a ly sis

S teph en B ru nau er

Illin o is In stitu te o f T echnology

P rin cip a l Research Chem ist

L aw ren ce S. G erm ain

Portland Cem ent A ssocia tion

U n iversity o f C aliforn ia R ad iation Laboratory

R esearch and D evelopm ent L aboratories

Liverm ore

CONTRIBUTORS W a lter G ord y

W a lter H . J oh n son , Jr.

P rofessor o f P h y sics, D uke tfn v e r s ity

Research A ssociate

R . W . G u rn e y Research A ssociate U niversity o f B ristol (deceased)

School o f P h ysics U n iversity o f M in n esota D ea n e B. Ju d d P h ysicist

A n d rew G u th rie

N a tion a l Bureau o f Standards

H ea d , N u cleon ics D ivision U .S. N aval R adiological D efense Laboratory

C . L a n czos S en ior P rofessor

D a v id Ila llid a y P rofessor o f P h y sics U n iversity o f Pittsburgh

D ublin In stitu te fo r A dvanced Studies G erald L. L a n d sm a n A ssistan t D irector fo r Research and D evelopm ent

W a lte r J. H am er

M ilita ry E lectron ics D ivision

C h ief, E lectrochem istry Section

M otorola, In c.

N ation al Bureau o f Standards

H o w a rd R . Lillie

R . W . H a y w a rd

S taff R esearch M a n ag er

R ad ioactivity Section

C orn in g Glass W ork s

N ation a l B u reau o f Standards M a x J. Ile rz b e rg e r S enior R esearch A ssociate K od a k R esearch Laboratories E. L. H ill P rofessor o f P h y sics U niversity o f M in n esota

Julian E llis M a c k P rofessor o f P h ysics U n iversity o f W iscon sin R o lle rt M a u rer P rofessor o f P h ysics U n iversity o f Illin o is J. R a n d M c N a lly , Jr. P h ysicist

John A . H ip p ie

Oak R id ge N ation al L aboratory

D irector o f R esea rch , P h ilip s Laboratories W a lter C . M ich els J. G . H irsch b e rg

M a rio n R eilley P rofessor o f P h ysics

U niversity o f W iscon sin

B ryn M a w r College

Joseph 0 . H irsch felder

E llio tt W . M o n tr o ll

P ro fesso r o f C hem istry

R esearch P rofessor

and D irector o f The U niversity o f W iscon sin N aval Research Laboratory U niversity o f W iscon sin

In stitu te f o r F lu id D yn a m ics and A p p lied M athem atics U n iversity o f M a ryla n d

A lan J. HofTmaii

Philip M . M o rs e

Consultant

P rofessor o f P h ysics

M anagem en t ( 'onsultalion Services

M assach usetts In stitu te o f T echnology

(¡(ta r a i E lectric ('on tp o n y

(\ O . M u eh lh a u se

T h e o d o r Ilu rliniaini

C h ie f o f the R eactor G rou p

R caktor Ltd.

N ation al B u reau o f Standards

W in rcnlitup n Sw itzerland

Ha raid II. N ielsen P rofessor o f P h ysics

U no Ingard

The Ohio State U n iversity

Associait P rofessor o f P h ysics

A lfred O. N ier

M assachusetts Institute o f Technology

P rofessor

F ritz John

t fiiv e r s ity o f M in n esota

Professtw o f M athem atics

R ich a rd M . N oy es

Institute o f M athem atical Sciences

P rofessor o f C hem istry

N ew York U niversity

U n iversity o f Oregon

CONTRIBUTORS H u gh O dish aw

L lo y d P. S m ith

E xecu tive D irector

P resident

U .S , N ation al C om m ittee f o r the In tern ational G eoph ysical Year N a tion a l A ca d em y o f Sciences

R esearch N Advanced D evelopm ent D ivision A vco M a n u fa ctu rin g Corporation R o m a n S in olu ch ow sk i

C h ester II. Page

P rofessor o f P h ysics

Consultant to the D irector

C arnegie In stitute o f Technology

N a tio n a l B ureau o f Standards

E d w a rd S. S teeb

R a y P e p in sk y

P h ysicist

R esearch P ro fesso r o f P h ysics

M ia m i U niversity

and D irector, The Groth In stitute T he P en n sylvan ia State U niversity B.

Peters

W illia m E . Steph ens P rofessor o f P h ysics U n iversity o f P en n sylvan ia

P ro fesso r o f P h y sics

A . H . T a u l)

T ata In stitute o f Fundam ental Research

Research P rofessor o f A p p lied M athem atics

B om ba y

U n iversity o f Illin ois

L ou is A . P ip es

O lga T a u s sk y

P rofessor o f E n gin eerin g

Research A ssocia te (M ath em atics)

U niversity o f C aliforn ia

C aliforn ia Institute o f T echnology

L os A n geles K a r l S. Q u isen b erry A ssista n t P rofessor U niversity o f Pittsburgh

vii

A lan M . T h o rn d ik e P h ysicist Brookhaven N ation al Laboratory John T o d d P rofessor o f M athem atics

N o rm a n F . R a m s e y P ro fesso r o f P h y sics H arvard U n iversity M a rk u s R ein er P rofessor o f A p p lie d M ech a n ics Isra el In stitu te o f T echnology H a ifa M . E . R ose C h ief P h y sicist Oak R idge N a tion a l L aboratory F re d e rick D . R ossin i Sillim an P ro fesso r and H ead, D epartm ent o f C hem istry

C aliforn ia In stitute o f T echnology C.

T o m p k in s

P rofessor o f M athem atics and D irector, N u m erica l A n a ly sis R esearch U n iversity o f C aliforn ia V . Vand A ssociate P rofessor o f P h ysics The P en n sylvan ia State U niversity A rth u r R . v o n H ippel D irector o f the L aboratory fo r Insulation R esearch and P rofessor o f E lectrophysics M assachusetts Institute o f T echnology Joh n A rch ib a ld W h eeler

and D irector, Chem ical and Petroleum

P rofessor o f P h ysics

Research Laboratory

P rin ceton U niversity

C arnegie In stitu te o f T echnology

E u gen e P. W igner

R a y m o n d J. Seeger

Palm er P h ysica l Laboratory

N a tion al S cien ce Fou nd ation

P rin ceton U niversity

E . J. Scld in

R . E . W ilson

P h ysicist

A ssistan t to Laboratory M an ager

N ation al Carbon R esearch Laboratories A D ivision o f U nion Carbide C orporation H a ro ld K . S k ra m sta d

and H ead, A dm in istration T ucson E n gin eerin g Laboratory H ughes A ircra ft C om pan y

A ssistan t C h ief fo r S ystem s

John (lil)so n W in ans

Data J^rocessing System s D ivision

A ssociate P rofessor o f P h ysics

N ation al Bureau o f Standards

U niversity o f 11 isconsin

viii

CONTRIBUTORS H u gh C . W o lfe

W . J. Y o u d e n

H ea d , P h y sics D epartm ent

A p p lied M athem atics D ivision

C ooper U nion School o f E n gin eerin g

N a tion al B u reau o f Standards

R ueben E. W ood

M a rv in Zelen

P ro fesso r o f C hem istry

M athem atician

T he George W ash in gton U niversity

N a tion a l B u reau o f Standards

Preface

T h is b o o k was first planned nearly' ten years ago when we were closely associated at the N ation al Bureau o f Standards. W e set our­ selves the p rob lem o f m aking a ju d icio u s selection from the vast literature of p h ysics o f m aterials w hich m ight reason abl}7 be called “ W h a t every ph ysicist should k n o w .” A s the planning w ent forw ard we becam e increasingly aware o f w hat a difficult task we had undertaken. T h e literature o f physics has be­ com e so great, and is grow in g at such a rate, th at it is v ery difficult for a p h ysicist to be really w ell-in form ed 011 m ore than a relatively narrow sp ecialty within the su b ject. N evertheless the unity o f the science is such that m uch research progress depends con sid era b ly 011 utilization o f ad van ces in one part to p rov id e the means for solvin g problem s in an oth er. T h erefore it is necessary for physicists to m ake stron g efforts to resist tendencies tow a rd over-sp ecialization . O ne w ay in w hich the rapidity of progress has com p lica ted our task is the ten d en cy for parts of the b o o k to b ecom e o u t-o f-d a te while being set up in typ e. W e have m ade efforts to a v oid this b y m aking m ore than the usual num ber o f add ition s and correction s while the b o o k was goin g through galley p ro o f. Our thanks are due the con trib u tin g specialists for their willingness to go to the extra trou ble o f m aking their ch apters as u p -to -d a te as possible in spite o f this difficu lty. B y the v e iy nature of the p reparation and p u b lica tion process, a h a n d b ook ca n n ot be com p letely current with jou rn a l literature, and there is variation even am on g the chapters, as revealed b y their refer­ ences. W ithin this restriction, we believe th at the H a n d b o o k fulfills its fu n ction as a o n e-v olu m e com p en d iu m . I t is our sincere hope that p h ysicists the w orld over will find this selection o f m aterials to be a useful one. W e think that there is co n ­ siderable e con o m y o f effort to be gained in a on e-v olu m e synthesis of the principal parts of the science in that so m an y techniques find use again and again in different parts o f the su b ject and on ly need to be explained on ce in a w ork o f this kind. W e will ap p reciate receiving su ggestions from readers as to how the b o o k 's usefulness m ay be im p roved in future editions. e.

r. c o x n o x

1IUG1I O D I S H A W

Contents

Contributors Preface ix

PART

1

C h a p ter 1

v

MATHEMATICS A r i t h m e t i c b y Franz L. A lt

7 -/

1 . N um bers and Arithm etic Operations.

Arithm etic.

C h a pter 2

2 . Logical Foundation of 3. Digital C om puting M achines.

Algebra b y Olga T a u s sk y

7-70

7. P o ly n o m ia ls . 2 . Algebraic Equations in One U nknow n, C om ­ plex N um bers. 3, E quations of Degree 2 (Quadratic Equa­ tions. Jh E quations o f D egree ‘3 (C ubic Equations), 3. Equa­ tions o f D egree 4 (B iquadratic E quations). 6 . Equations o f D egree n. 7. D iscrim inants and General Sym m etric Functions. 8 . C om putational M ethods for O btaining H oots o f Algebraic E quations. y. M atrices. 1 0 , D eterm inants. 1 1 . Systems of Linear Equations. 12 . Num erical M ethods for Finding the In­ verse o f a M atrix and for Solving System s o f Linear Equations. 13. Characteristic R oots o f M atrices and Q uadratic Form s. l/f. C om putation o f Characteristic R oots o f M atrices. 13. F unc­ tions o f M atrices and Infinite Sequences. 1 (1. H ypcrcom plex System s or Algebras. 17. T h eory o f G roups.

C h a pter 3

Analysis b y J oh n Tocld

1-22

1 . Real Numbers, Limits. 2. Real Functions. 3. Finite D iffer­ ences. 4 . Integration. 3. Integral Transform s. 0 . Functions of Several Real Variables. 7. C om plex Num bers. 8 . Series o f F unc­ tions. 3. Functions o f a Com plex Variable. 1 0 . C onform al M apping. 11. O rthogonality. 1 2 . Special Functions.

C h a p ter 4

O rdinary Differential E q u a tio n s b y Olga T au ssk y

7-59

1 . In troduction.

2. Simple Cases. 3. Existence Theorem s. 4M ethods for Solution. 3. Exam ples o f W ell-known E quations. 0 . Some General Theorem s. 7. N on hom ogcncous Equations, G reen’s Function. S. Num erical Integration o f Differential Equations. 0. System s o f Sim ultaneous Differential Equations.

Cha pter 5

Partial Differential E q u a tio n s b y Fritz John 1 . General

1-66

Properties. 2. First-order Equations. 3. E llip­ tic E quations. 4 • Parabolic; Equations o f Second Order. 3. H yperbolic Equations in T w o Independent Variable's. 0 . H yper­ bolic; E quations with M ore than T w o Independent Variables. 7. Num erical Solution o f Partial Differential Equations.

xii

CONTEXTS

Ch a pter 6

In tegra l E q u atio n s b y M . A b ra m o w itz

1-90

1 . Integral Equations o f the Second Kind. 2 . Sym m etric Kernels. 3. N onsym m etric Kernels. 4- Integral Equations o f the First K ind. 5. V olterra’s Equation. 0 . Nonlinear Integral E quation.

Chapter 7

Operators b y Olga T a n ss k y

1-95

1 . V ector

Spaces, A bstract Hilbert Spaces, Hilbert Space. Definition of O perator or T ransform ation. 3. Spectrum Bounded Operators, Eigenvalues, and Eigenfunctions.

C h a pter 8

G e o m e tr y b y A . J. H o ffm a n

2. of

1-97

1. Definition and Assum ptions. 2. P rojective Plane. 3. P rojec­ tive C rou p. 4. Correlations, Polarities, and Conies. 5. Projec­ tive Jane. 6 . Subgroups o f the P rojective G roup. 7. Affine G roup and Plane. S. Euclidean G rou p and Plane. «9. Conies. 10 . Angles. 1 1 . Triangles. 12. Polygons. 13. H yperbolic G roup and Plane. lJh Elliptic G rou p and Plane.

C h a pter 9

V ector Analysis b y E. U. C o n d o n

1-103

1 . A ddition o f V ectors. 2. Scalar and V ector Products. S. V ec­ tors and Tensors in O blique C oordinates. 4. G radient o f Scalar and V ector Fields. 5. D ivergence o f a V ector Field. 0 . Curl o f a V ector Field. 7. Expansion Form ulas. 8 . O rthogonal C u rvi­ linear C oordinates. ,9. Transform ation o f Curvilinear C oord i­ nates.

C h a pter 10

T en s o r Ca lcu lu s b y C. L anzcos

1-111

1 . Scalars, Vectors, Tensors. 2 . A n alytic Operations with V ec­ tors. 3. Unit V ectors; C om ponents. 4* A d join t Set o f Axes. 5. C ovariant and Contravariant C om ponen ts o f a V ector. 6 . T ransform ation o f the Basic Vectors Vi. 7. Transform ation o f V ector Com ponents. S. Radius V ector R. 0. A bstract D efini­ tion o f a V ector. 1 0 . invariants and C ovariants. 11 . A bstract D efinition of a Tensor. 12 . Tensors o f Second Order. 13. E in­ stein Sum C on vention. l/h Tensor Algebra. 15. D eterm inant Tensor. 1 0 . Dual Tensor. 17. T en sor Fields. 18. D ifferentia­ tion o f a Tensor. 1.9. Co variant D erivative o f the M etrical T en ­ sor. 2 0 . Principles o f Special and General R ela tivity . 2 1 . C u rvi­ linear Transform ations. 2 2 . Co variant D erivative o f a Tensor. 23. C ovariant D erivative o f the M etrical Tensor. 2 /t. F u nda­ mental Differential Invariants and C o variants of M athem atical Physics. 25. M axwell E lectrom agnetic Equations. 2 0 . C u rva­ ture Tensor o f Uiemann. 27. Properties o f Riemnnn Tensor. 28. C ontract(‘d Curvature Tensor. 2 0 . The M atter Tensor of Einstein. 30. Einstein’ s T h eory o f G ravity.

C h a pter II

C alculu s o f Variations />y C. 8 . T o m p k i n s

¡-¡2 3

1 . M axim a and M inim a of a Function o f a Single Variable.

2. M inim a o f a Function o f Several Variables. 3. M inim a o f a Definite' Integral— the Euler Equations. 4 . Exam ples. 5. Other First Variations: Weierstruss C on dition, Corner C onditions, Oneside Variations. 0 . Param etric Problem s. 7. Problem s with Variable' End Points. 8 . Isoperim etric Problem s— the Problem of Bolza. .9. Second Variations. 10 . M ultiple-integral Problem s. 11 . M ethods of C om putation . 1 2 . Conclusion.

xiii

CONTENTS

C h a pter 12

E le m e n ts o f Probability b y C h u rehill Eisenhart and M arvin Zelen 1-131 1. Probability. 2 . R andom Variables and D istribution Func­ tions. 3. D istributions in n Dim ensions. /f. E xpected Values, M om ents, Correlation, C ovariance, and Inequalities on D istribu­ tions. 3. M easures of Location, Dispersion, Skewness, and Kurtosis. a. Characteristic Functions and Generating Functions. 7. Limit Theorem s. 8 . T h e N orm al Distribution. 0 . Discrete D istributions. 10 . Sam pling Distributions.

Cha pter 13

PART

Statistical Design o f Exp erim en ts b y W . J. Y oin le n

1-165

2 • M E C H A N I C S OF P A R T I C L E S AND RIGID BODIES

C h a p ter 1

K in e m a tic s b y E. U. C o n d on

2 -3

1 . V elocity and Acceleration. 2 . K inem atics of a Rigid B ody . 3. E uler’s Angles. 4- Relativistic K inem atics. 5. V ector Alge­ bra o f Space-T im e.

Ch a p ter 2

D y n am ie al Principles b y E. U. C o n d on

2-11

1 . Mass. 2 . M om entum . 3. Force. 4 • Impulse, o. W ork and E n ergy: Power. 6 . Potential Energy. 7. Central F orce: Colli­ sion Problem s. S. System o f Particles. 0 . Lagrange’s Equations. 10 . Ignoruble Coordinates. 11 . H am ilton ’s Equations. 12. R el­ ativistic Particle M echanics. 13. Variation Principles.

C h a p ter 3

T heo ry o f Vibra tio n s b y E. U. C o n d on

2-21

1 . Sim ple H arm onic M otion. 2. D am ped H arm onic M otion. 3. Forced H arm onic M otion. 4. M echanical Im pedance, 3. T w o C oupled Oscillators. 0 . Small O scillations about E qu ilib­ rium. 7. Oscillations with Dissipation. 8 . Forced O scillations o f Coupled Systems. 0 . General D riving Force. 10 . Physical Pendulum . 1 1 . N onharm onic Vibrations.

C h a p ter 4

Orb ita l M o tio n b y E. U. C o n d o n

2-2 8

1 . M otion under Constant G ravity. 2 . E ffect of E arth ’s R ota ­ tion. 3. General Integrals o f C entral-foree Problem . 4- Differ­ ential Equation for Orbit. 3. M otion under Inverse-square-law A ttraction. 0 . M otion in Elliptic Orbit.

C h a p ter 5

D y n am ics o f Rigid Bodies b y E. U. C o n d on

2-33

1. Angular M om entum . 2 . Kinetic Energy. 3. Equations of M otion. 4- R otation about a Fixed Axis. 3. R otation about a Fixed Point with No Fxternal Forces. 0 . Asym m etrical T op .

Ch a p ter 6

Q u a n t u m D y n a m ics b y E. U. C o n d on 1 . Particle

2-38

W aves. 2 . The Schroedingcr W ave Equation. 3. M atrix Representations. L The H arm onic Oscillator. 3. A ngu­ lar M om entum . t>. Central-foree Problem s. 7. The Dynam ical E quation. S. Perturbation Th eory for Discrete States. 0 . Variation M ethod. 1 0 . Identical Particles. 11. Collision P rob ­ lems.

xiv

CONTENTS

Cha pter 7

G ravitai ion b y H ugh C. W o lfe

2 -5 5

/ . In versosq u a rc Law. 2. G ravitational Constant, G. celeration of G ravity (j and G eophysical Prospecting.

Cha pter 8

8 . A c­

D y n a m ics o f the Solar S y ste m b y G . M . C lé m e n c e 2 -60 1 . Introduction. 2 . Equations o f M otion. 3. M ethod o f Solu­ tion. /t. Form o f Solution. 5. Precession and N utation. 6 . Frames of Reference. 7. Determ ination o f the Precession. S. Perturbations of Planets and Satellites. 0 . Determ ination o f Tim e. 10 . R elativity. 11 . National Ephem erides. 12. Celestial N avigation. 13. Astronom ical Constants.

Cha pter 9

Control M e c h a n is m s b y Harold K. S k ra m sta d and Gerald L. L a n d sm a n 2-69 1 . In troduction. 2 . Differential E quation Analysis. 3. Frequency-rosponsc Analysis. System Im provem ent by C om pen­ sation. 5. Steady-state Error. 6*. Other M ethods o f Analysis.

PA R T

3

Cha pter 1

• M E C H A N I C S OF DEFORMABLE BODIES K in e m a tic s and D y n a m ic s b y E. U. C o n d o n

3 -3

1 . Kinem atics o f C ontinuous M edia. 2 . Stress. 3. Equations o f M otion. 4 . M olecular Standpoint. 3. Energy Relations for Fluid. 0 . Strain. 7. H ook e’s Law. S. V iscosity.

Ch a pter 2

Fluid M e c h a n ic s b y R. J. Seeger

3 -1 1

1. Statics o f Fluids. 2. In viscid-fluid D ynam ics. 3. Irrotational, Continuous Flows o f In viscid Fluids. 4- D iscontinuous Flows o f Inviscid Fluids. 3. Vortex Flows o f Inviseid Fluids. 6*. Flows o f Com pressible, Inviseid Fluids. 7. Flows o f Viscous Fluids. S. Turbulence. 0 . Fluids with Heat. 10 . Flow s in Electric and M agnetic Fields.

Ch a pter 3

Rhcology b y M . Reiner

3-4 0

1 . Introduction. 2 . Second-order Effects in Elasticity and Vis­ cosity. 3. Rheologieal Properties. 4- Com plex Bodies. 3. V ol­ ume Changes. 0 . Strength. 7. M ierorheologieal Aspects. S. R h com etry.

Ch a j>ter 1

Wave Propagation in Fluids b y A . II. T a u h

3 -3 0

1 . Conservation Laws. 2 . Small Disturbances. 3. Interactions of W aves o f Small A m plitude. 4* Small D isturbances in Shallow Water. 3. Plane W aves of Finite A m plitude. (>. Form ation and Decay of Shocks in One Dimension. 7. Spherical W aves o f Finite Am plitude. ter 7

Dielectrics b y A. von llip pcl

4-103

1 . Introduction. 2 . C om plex P erm ittivity and Perm eability. S. Polarization and M agnetization. 4 . M a croscop ic D escription of Dielectrics b y Various Sots of Param eters. 5. M olecular M echanism s of Polarization. G. R esonance Polarization. 7. Relaxation Polarization. S. Piezoelectricity and Ferroeleetrieity. 9. Polarization by M igrating Charge Carriers. 10 . Electric Breakdown.

('.¡k n itt e r ft

M a g n e tic M ateria ls b y W illiam Fuller Brown, Jr.

4-126

1 . Basic C oncepts. 2 . M acroscopic T h eory. S. Classical M icro­ scopic T h eory. Jt. Q uantum -m echanical C oncepts. 5. D iam ag­ netism. G. Param agnetism. 7. Saturation in Param agnetios and Spontaneous M agnetization in Ferrom agnetics. S. Ferrom agnetic Domains ami the M agnetization Curve. 9. M agnetom echanical Phenom ena in Ferrom agnetics. 10 . D ynam ic Phenom ena.

CONTENTS

C h a pter 9

xvii

Electrolytic C o n d u ctiv ity and Electrode Processes b y W alter J. H a m e r a nd Reuben E. W o od 1-138 1 . E lectrolytic and Electronic C on du ction . 2 . E lectrolytic C on ­ d u cto rs. 3. Io n iz a tio n , /f. D egree o f Io n iz a tio n . 5. Ion ic Charge and the Faraday. 0 . E lectrolytic C on du ctivity. 7. E quivalent and M olar C onductance. S. M easurem ents of Elec­ trolytic C on du ctivity. 0 . Significance of Equivalent C on d u ct­ ance. 10 . Ionic Conductances and Transference Numbers. 11 . Ionic M obilities. 12 . Intcrionic A ttraction and E lectrolytic C on du ctivity. 13. H igh-field Effects in C onductance. 14. C on ­ ductance at High Frequencies. 15. Electrochem ical T h erm ody­ nam ics. 10 . Galvanic Cells at E quilibrium . 17. G alvanic Cells N ot at Equilibrium . IS. Batteries.

Cltapter 10

C o n d u c tio n o f Electricity in Gases b y Sanborn C. Brow n

4-159

1 . P robability o f Collision.

2 . Diffusion. 3. Electron M ob ility. M ob ility. 5. The R atio D / m for Electrons. 0. A m bipolar D iffusion. 7. Electron A ttach m en t. S. Ion R ecom bina­ tion. f). E lectron-Ion R ecom bination. 1 0 . Neutral A tom s and M o le cu le s . 1 1 . Io n iz a tio n b y C o llis io n . 1 2 . H ig h -fre q u e n cy B re a k d o w n . 13. L o w -p re s s u re D -C B rea k d ow n . 14. A tm o s pherie-pressure Spark. 15. Low-pressure G low Discharge. 16. Arc Discharges. 17. Plasma Oscillations. 4. Ionic

PART

5 • IIEAT

C h a pter 1

AND

THERMODYNAMICS

Principles o f T h e r m o d y n a m i c s b y E. U. C o n d on

5 -3

1 . T h e Nature o f H eat. 2 . First Law o f Therm odyn am ics. 3. Second Law of T h erm odyn am ics. 4. Absolute Tem perature Scale. 5. Third Law o f T h erm odyn am ics. 0 . Equilibrium C on­ ditions. 7. R elations betw een T h erm odyn am ic Functions. S. Phase Equilibria o f Sin gle-com p onen t Systems. 0 . Systems of Several Com ponents. 1 0 . Chem ical Equilibrium .

Ch a p ter 2

Principles o f Statistical M e c h a n ic s and K inetic T h e o ry o f Gases b y E. W . M o n tro li 5-11 1 . Scope o f Statistical M echanics. 2 . Identification o f T em pera­ ture with M olecular M otion and the M axw ell V elocity Distribu­ tion. 3. Mean Free Path and Elem entary T h eory o f Transport Processes. 4» T h e Boltzm ann Equation and the System atic K inetic T h eory o f Gases. 5. The Boltzm ann H T heorem . 6 . Averages in Equilibrium Statistical M echanics and the Liouville E quation. 7. The M icrocanonical and Canonical Ensembles. 5. The Partition Function and the Statistical Basis o f T h erm od y­ nam ics. 0 . Som e Sim ple Exam ples. 10. M olecular D istribution Functions. 11 . Calculation o f T h erm odyn am ic Quantities from M olecular D istribution Functions. 12 . The Integrodiffcrcntial Equations for the D istribution Functions. 13. T h eory o f Fluctua­ tions and the Grand Canonical Ensem ble.

C h a pter 3

T h e r m o m e t r y and Py ro m etry b y R. E. W ilson and R. I). Arnold 1.

Th erm odyn am ic

Tem perature

5-30 Scale.

2. The

International

xviii

CONTENTS Tem perature Scale. 3. Calibration o f Tem perature Measuring Instrum ents. 4 . Tem perature Scales below the Oxygen Point.

Cha pter 4

T h e Equ ation o f S tate and T r an sp ort Properties o f Gases and Liquids b y R. 1L Bird, J. O. Hirschfelder, and C. F. Curtiss 5-41 1 . T h e Potential Energy of Interaction between T w o M olecules. 2 . The Equation o f State of D ilute and M oderately Dense Gases.

3. The Equation of State o f Dense Gases and Liquids. / . T he Transport Coefficients o f D ilute Gases. 5. T he Transport Coeffi­ cients of Dense Gases and Liquids. 6*. Som e Applications o f the Principle of Corresponding States.

Cha pter 5

Heat T ran sfer b y E. U. C o n d on

5-66

1 . Ileat C on du ctivity. 2 . Equation o f H eat C on du ction . 3. Simple Boundary Value Problem s. 4- C ooling o f Simple Bodies. 5. Point Source* Solutions. 6*. Periodic Tem perature ("hange. 7. Natural Heat C on vection. S. Forced Heat C on vection . 0 . Condensation and E vaporation. 10 . R adiative Heat Transfer.

C ha pter 6

V a c u u m T e c h n iq u e b y A n d re w G u th r ie

5-78

1 . T he Vacuum C ircu it— C onductance. 2 . Flow o f Gases through Tubes. 3. Pum ping Speed and E vacuation Rate. 4- Vacuum Pumps, o. Vacuum Gauges. 0 . C om ponents and Materials. 7. L eak-detection Instrum ents and Techniques.

Ch a pter 7

Surface T e n s io n , A dsorption b y S teph en B runauer a nd L. E. Copeland 5-91 1 . The T h erm odyn am ic T h eory o f Capillarity. 2 . T he Surface Tension and T otal Surface E nergy o f Liquids and Solids. 3. Adsorption on Liquid Surfaces. Jt. Adsorption on Solid Surfaces. Physical A dsorption o f Gases and Vapors. 5. Chem ical A dsorp­ tion o f Gases on Solids. 0 . A dsorption on Solids from Solutions.

Chapter 8

C h e m ic a l T h e r m o d y n a m ic s b y Frederick I). Rossini

5-119

1 . Introduction.

2 . Useful E n ergy; Free E n ergy; Criteria o f Equilibrium . 3. Equilibrium C onstant and Change in Free Energy for Reactions o f Ideal Gases. / . F u gacity; Standard States. 5. Solutions: Apparent and Partial M olal Properties. 0 . The Ideal Solution. 7. The Dilute Real Solution. S. E quilib­ rium Constant and the Standard Change in Free Energy. 8 . T h erm odyn am ic Calculations.

Chapter 9

Ghem ieal Kinetics b y Richard M . Noyes

5-110

R esu lts ok K in etic O h seiivation s. 1. In trod uction. 2 . Ex­ perimental T echniques. 3. Orders o f Chem ical R eactions. 4C on secutive Reactions. 5. R eversible Reactions. 0 . E ffect o f Tem perature. T h e o r e t i c a l In t e k c k e t a t i o n o f C h em ica l K in etics. 7. Introduction. 3. Collision T h eory o f Bim olecular Gas Reactions. U. Collision T h eory o f U nim olecular Gas R eac­ tions. Hi. S tatistical-T herm odyn am ic T h eory of Reaction Kinet­ ics. 11 . Theoretical Estim ation o f Energies of A ctivation . 12 .

CONTENTS C onsecutive

xix

13. R eactions in Solution. E l u c i d a ­ 14- Criteria for a Satisfactory M echanism . 13. Reactions Involving; N onrepetitive Steps. 10 . Chain Reactions. 17. Branching Chains. IS. Photochem istry. 10 . Heterogeneous Reactions.

tion

Cha pter 10

ok

Reactions.

C h e m ica l

M e ch an ism .

Vibrations o f Crystal Lattices and T h e r m o d y n a m i c Properties o f Solids b y E. \V. M o n troll 1 . In troduction. 2 . D ebye T h eory T h eory of Born and von K arim m . Crystals.

Cha pter 11

Ilea t

Capacities. 3. of State of

4 . Equation

Superflnids b y K. IL Atkin s 1.

PART

of

5-150

Liquid Helium.

5-159

2 . S u percon du ctivity.

6 • OPTICS

Cha pter 1

E le c tro m a g n e tic Waves b y E. U. C o n d on

6-3

I . N ature o f Light. 2 . States o f Polarization. 3. M axwell Field E quations. 4 . P oynting Theorem , 3. Plane W aves in Isotropic M edia. 6 . R eflection and R efraction at a Plane B oundary. 7. Plane W aves in Anisotropic M edia. S. O ptical A ctivity. 9. W aveguides and Transmission Lines. 10 . B laek-body R adiation. I I . R adiation from Oscillating Charge D istribution. 12. Quan­ tization o f the R adiation Field.

Ch a p ter 2

G e o m e tric a l O ptics b y M ax llcrzbcrger

6-20

1 . In troduction. I. G E N E R A L T H E O R Y . 2. Optical Form o f the General Variation Problem . 3. General Problem o f G eo­ metrical Optics. 4 • Characteristic Function o f H am ilton. Laws o f Fermat and o f M alu s-D upin. D escartes’ Law o f R efraction. Lagrange Bracket. II. A N A T O M Y . Ray T racin g. 9. The Refraction Law. 6*. Tracing a R ay through a Surface of R otation. 7. Special Surfaces. S. Transfer Form ulas. 9. General Form u­ las. Diapoint C om putation . B a s i c T o o l s o k O p t i c s . 10 . The Characteristic Functions. 11. T he Direct- M ethod. L a w s o k Im a c e F o r m a tio n . 12. Im age o f a Point. Caustic. 13. Im age of the Points of a Plane. 14- The Im age o f the Points of Space. 19. T he Characteristic Function W for a Singh» Surface. 10 . T he Direct M ethod and the A ddition o f Systems. III. D IA G N O ­ SIS. G a u s s i a n O p t i c s . 17. In troduction. IS. General Laws. 19. Focal Points and N odal Points. 2 0 . Viewing through an Instrum ent. 2 1 . D istance o f C onjugated Points from the Origins and Their M agnification. 2 2 . Gaussian Brackets. 23. Expres­ sion of Basic Data o f Gaussian O ptics with the Help o f Gaussian Brackets. 24 • Vignetting. A n a l y s i s ok a G iven O p tica l S ystem . 23. Introduction. 2 0 . Seidel Aberrations. 27. Exten­ sion o f Seidel T h eory to Finite Aperture and Field. 2 S. The Spot-diagram Analysis and the D iapoint Plot. IV. T H E R A P Y . 29. C orrection o f an Optical System . V. P R O P H Y L A X IS . 30. Introduction. 31. Dispersion o f Glass. 32. C olor-corrected System o f Thin Lenses. A p p e n d i x . 33. Intensity Considera­ tions. 34. Some H istorical Rem arks.

CONTEXTS

XX

Ch a pter 3

P h o to m e try and I llu m in a tio n b y E. S. Steel», Jr., and W . E. Forsythe 6-47 1 . Visual

Photom etry. 2. Physical P h otom etry : The Spherical Integrator. 3. P h otom etry Spectral Response vs. Lum inosity Curve. 4- Production o f Light, 6 .R adiant Energy. 6*. Light Sources.

C h apter 4

Color Vision and Colorim etry b y Deane H. Judd

6-64

1 . Definition o f Color. 2. T ypes o f C olor Vision. 3. Tristim ulus Values. Theories of C olor Vision, o. C h rom aticity Diagram s. 0 . Photoelectric Colorim eters. 7. C olorim etry b y Difference.

Cha pter 5

Diffraction and In te rference b y C. II. lhiriictt, J. G . llirsehberg, and J. E. M a c k

6-77

1 . G eom etrical O ptics as an A pproxim ation. 2. General Aspects of Diffraction and Interference. 3. D iffraction. 4- Resolution and Fringe Shape, o. T w o-b ea m Interference. 0 . Equal-am plitudc M ultibeam Interference. 7. G eom etrically D egraded A m pli­ tude M ultibcam Interference.

C h a pter 6

M o le c u la r Optics b y E. U. C o n d o n

6-109

1 . M olecular R efra ctiv ity. 2 . D ispersion. 3. A bsorption and Selective Reflection. 4 • Crystalline D ouble R efraction, n. Fara­ day E ffect; C otton -M ou ton Effect. 0 . Kerr Effect. 7. Optical R o t a t o r y P ow er. S. P h o to e la s t ic ity . 0, F low b ir e fr in g e n c e : M axwell E ffect. 1 0 . Pleoehroism . 1 1 . Light Scattering.

Ch a pter 7

Fluorescence and Plios ph oresconce b y J. G . W in a n s and E. J. Scldin

6-12H

1 . Introduction. 2 . Fluorescence o f Gases and Vapors. 3. G en­ eral T h eory o f Quenching o f Fluorescence. 4. Polarization of Resonance Radiation. 3. Stepwise Excitation o f Fluorescence in Gases. 6*. O ptical Orientation o f N uclei. 7. Sensitized Fluores­ cence4. S. Selective R eflection. //. Reem ission. 10 . Fluorescence in Liquids. 11 . Therm olum inescence. 12 . Phosphorescence.

C ha pter 3

Optics and Relativity T h e o ry b y E. L. Hill

6-150

I . Introduction. 2 . T he Special T h eory o f R elativity. 3. The Transform ation Formulas o f Special R ela tivity. 4 . 'The Trans­ form ation Equations for Plain; W aves. 3. 'Pin; D ynam ical Proper­ ties o f Photons. (I. Aberration o f Light. 7. D oppler Effect. (S’. The Experiment, of Ives and St dwell. 0 . 'Pin4 M iehelsonM orlev Experim ent. 10 . T he K enn edy-T horn dike Experim ent. I I . Generalizations o f the Lorentz Transform ation G roup. 12. Electrom agnetic Phenom ena in M ovin g M edia. 13. 'Pin* Special T h eory of R elativity and Quantum M echanics. / /. T he General T h eory of R elativity. 13. C osm ological Problems. 10 . Recent I )evelopm enls.

CONTENTS

PART

7 • ATOMIC

C h a pter 1

xxi

PHYSICS

A t o m i c S tru cture b y E. U. C o n d o n

7-5

/. Nuclear Atom M odel. 2 . A tom ic W eights. 3. Periodic T able. it. 'Theory of A tom ic Energy Levels. (>. Series. Isoeleetronic Sequences. 7. M agnetic Spin-orbit Interaction. 5. T w o-electron Spectra. 9. Ionization Potentials. 10 . Zeeman Effect. 4 . A tom ic Units,

Ch a p ter 2

A t o m i c Spectra, In c lu d in g Z e e m a n ami Stark Effects b y J. Rand M c N a lly , Jr.

7-25

1 . In troduction.

2 . Spectroscopic N om enclature. 3. Space Q uantization. 4 • Classical T h eory o f Spectra. 5 . W ave M echan­ ics. (>. Interaction Energy and Fine Structure. 7. Zeeman Effect. S. Intensity of Zeeman C om ponents. 9. T he Stark E ffect. 10 . Intensity of Stark Lincs.

C h a p ter 3

A t o m i c Line S tre n g th s b y Lawrence Allcr

7-48

1 . A tom ic R adiation Processes. 2 . Form ulas and Tables for Line Strengths. 3. C ontinuous A tom ic A bsorption Coefficients. 4• Forbidden Lines. if. Tin; A tom ic Line A bsorption Coefficient. 0 . Experim ental D eterm ination of / Values. 7. Tests and A ppli­ cations o f the T h eory.

C h a p ter 4

Hypcrfinc Stru cture and A t o m i c Ream M e th o d s b y N o r m a n F. R a m s e y

7-58

1. In troduction. 2 . M u ltipole Interactions. 3. M agnetic D ipole Interactions. 4. Electric Q uadrupole Interaction. 5. M agnetic O etupole Interaction. 6*. Optical Studies o f H yperfine Structure. 7. A tom ic Beam -dettcction Experim ents. S. A tom ic Beam M ag­ netic Resonance Experim ents. 9. H ydrogen Fine Structure. The Lam b Shift.

C h a p ter 5

T h e Infrared Spectra o f M o le cu le s b y llarald II. Nielsen

7-64

1. In troduction. 2. The Energies o f a M olecule. 3. The V ibra­ tion o f a M olecule. 4- T he R otational Energies o f M olecules. i>. The Energy of Interaction, Ei. 6 . T he Selection ILdes for the R otator. 7. The Interpretation of Band Spectra. S. The Ram an S pectroscopy o f M olecules. 9. Resonance Interactions of Levels.

C h a pter 6

Microwave Spectroscopy l>y W a lte r G ordy

7-82

1 . In troduction. 2. T he M icrow ave Spectroscope. S. M icro­ wave Spectra of Free A tom s. 4. Pure R otational Spectra. S. Inversion Spectra. 6 . Electronic Effects in M olecular Spectra. 7. Nuclear Efforts in M olecular Spectra. S. Stark and Zeeman Effects in R otational Spectra. 9. Shapes and Intensities o f M icro­ w ave A bsorption Lincs. 1 0 . Electronic M agnetic Resonance in Solids.

xxii

CONTENTS

Cha pter 7

Electronic S tru cture o f M olecules b y E. U. C o n d on 7-100 1 . Energy Levels of D iatom ic M olecules. 2. E lectronic Band Spectra, o f D iatom ic M olecules. S. F ranck-C ondon Principle. 4» D issociation Energy, 3. Continuous and Diffuse Spectra. Predissociation. (k H ydrogen M olecule. 7. Sketch of Chem ical Bond T h eory. S. Bond Energies, Lengths, and Force Constants. 9. Ionic Bonds and Dipole M om ents.

C h a pter 8

X Rays b y E. U. C o n d on

7-118

1 . Main Phenom ena. 2 . E m ission: Continuous Spectrum . 3. Em ission: Characteristic Line Spectrum . 4- A bsorption. 3. Angular Distribution o f Photoelectrons. (>. Intensity M easure­ m ent. 7 . Internal C on version: Auger E ffect. S. Pair P rodu c­ tion. 9. Coherent Scattering. 10. Incoherent Scattering: C om p ­ ton Effect..

Ch a pter 9

.Mass Spectroscopy and Io nization Processes b y John A. Hippie

7-131

1 . Introduction.

2. Study of Ionization Processes. 3. Ionization o f Atom s b y E lectron Im pact. Jh D iatom ic M olecules. 5. P oly­ atom ic M olecules, (i. Analysis.

C h a pter 10

F u n d a m e n t a l C o n sta n ts o f A t o m i c Physics b y Jesse W . M . D u M o n d and E. Richard Cohen 7-113 1 . T he G roup Known as the A tom ic Constants.

2 . T he Pioneer W ork and M ethods of It. T . Birge and Others Prior to 1941). 3. D ata of G reatly Increased A ccuracy Subsequent to 1949. 4- C on­ sistency Diagrams and G raphic M eth od s: T h e Ellipsoid o f Error. 3. The M ethod o f Least Squares. 0. Calculation o f Standard Errors and Correlation Coefficients. 7. R ejection o f Certain Input D ata in the Present Least-squarcs A djustm en t. nv. It can be proved, again by purely logical reasoning and without recourse to intuition, that the natural num bers possess the elem entary properties usually associated with them . A m ong these arc: Theorem 1. T h e natural num bers arc ordered. This means that there exists a relation betw een n um ­ bers, a > b ( “ a is greater than b ” ), such th at: a. If a > b and b > c, then a > c (the relation is transitive). b. If a > b is true, then b > a is not true (the rela­ tion is asym m etric). c. For any tw o num bers, a,b, at least one o f the statem ents a > b, a = b, b > a is true (it follow s that exactly one is true). W e write a > b instead of “ a > b or a = b ” a ^ b instead of “ a > b or a < b,” a < b instead of b > a. d. 1 > 0. Theorem 2. A ddition and m ultiplication are defined; the sum , a + b, and product, ab, of tw o natural num bers arc again natural num bers. T h ey satisfy these relations: a. a + b = b -b a (com m u tative law of addition ) b. (a + b) - f c = a -b (/; + e) (associative law of a ddition ) c. a + 0 = a d. ab = ba (com m u ta tive law of m ultiplication) e. (ab)c = a(bc) (associative law o f m u ltiplication) / . a. • 1 = a (/. a(b -f- c) = ab + ae (distribu tive law of m u ltiplica­ tion) h. If a > b} then a + c > b - f c i. If a > b, then ae > be Oilier Typ es o f N u m b e r s. It is possible in principle, b u t cum bersom e in practice, to express statem ents a b ou t physical measurem ents in terms of natural num bers alone. Such statem ents arc m ade briefer by the introduction of other types o f n um bers: negative num bers, fractions, and irrational and com ­ plex num bers. In order to introduce fractions (rational numbers), we consider ordered pairs of natural num bers (a,b) with b 0, calling the first num ber o f such a pair numerator and the second denom inator; we call tw o pairs (a,I)) and (a',b') equivalent if ab' — a 'b ; we define the class of all pairs which are equivalent with a given pair (a,b) as the rational num ber a/b. W e then define the rela­ tion a/b > a'/b' as m eaning that ab' > a 'b ; the sum a/b + a'/b' as (ab' - f a'b) / (bbf) ; the prod u ct (a/b) (a'/b') as (aa')/(bb'). Again, as in the case of natural num bers, we can prove the usual elem entary p rop ­ erties o f rational num bers. A m ong these are m ost, but not all, of the theorem s about natural num bers. By the a b ove definitions, fractions with den om in a­ tor 1 have properties exactly analogous to those o f the natural num bers. Thus, (a/\) + (/>/1) = (a + b )/ 1; (a/\)(b/\) = (ab/ 1); and a/\ > b/i if and on ly if a > b. For b revity, we shall write a instead of a /1 , etc., although there is a logical difference betw een the natural num ber a and the fraction a/1

A R ITH M ETIC Custom arily, the introduction of rational num bers, ju st given, is preceded b y a similarly sim ple in troduc­ tion of relative (positive or negative) integers. A fter this is done, rational num bers are defined as classes of pairs o f integers, with slight m odifications in the pro­ cedure shown a b ove. T h e introduction of irrational num bers is som ew hat m ore com plicated and will not be explained here (sec Chap. 3 ). Finally, complex num bers are defined as ordered pairs of real (i.e., rational or irrational) num bers (ft,6), where a is called the real part and b the imaginary part, with the con ­ ventional definitions (a, 6) -j" (ft', 6') (ft, 6) (ft', 6')

= (ft -J- a', b 6') — (aa' — bb'f ah' + a'b)

For pairs w hose im aginary part is 0, these operations give the same result as those with real num bers, that is, (ft, 0) + (a', 0) = (ft + o ', 0 ); (ft, ())(« ', 0) = (ft«', 0 ); for abbreviation we write « for (ft,0), etc. If we fu r­ ther set i = (0,1), we obtain (ft,b) = (ft,0) + (0,1)(6,0) = ft - f ib E very time we widen the dom ain of num bers, som e theorem s lose their v a lid ity. For instance, when we go from natural to (positive and negative) integer n um bers, T h eorem 2 i is no longer true; in its place is Theorem 2i': Theorem 2 i'. If ft > 6 and c > 0, then ae > be. On the other hand, new theorem s can be proved for the enlarged dom ain which do not hold for the nar­ rower one. For exam ple, for the dom ain of all integers (positive, negative, and ze ro ): Theorem 3ft. For any two integers x ,?/, there is an integer z such that x = y + z. Ry w ay of definition we set 2 = x — y. For the dom ain o f all rational num bers: Theorem 36. For any tw o rational num bers p,q, where q ^ 0, there exists a rational num ber r such that p = rq. T h e follow ing fundam ental property o f rational num bers can be derived from the simple theorem s listed so far: If x ,y are tw o rational num bers such that y < x, then there exists a rational num ber 2 such that y < z < x. T h is property is expressed briefly by saying that the rational num bers arc dense. T h e real num bers have the im portan t fundam ental property o f being continuous. In order to form ulate this property, we introduce tw o definitions: a set A/ of real num bers is said to precede another set N if any num ber in il/ is sm aller than every num ber in N\ and a num ber z is said to separate the sets ftl and N if z > x for every num ber x in M and z < y for every n u m ber y in N . T h e con tin uity theorem states: Theorem 3c. W henever tw o sets M and iV o f real num bers h ave the property that M precedes N f there exists a real num ber z w hich separates M and N . It should be noted that this theorem is false as long as only rational num bers are considered. Once it is established, a num ber o f frequently applied state­ m ents a b ou t real num bers can be p roved. One of these states that each infinite sequence ri,r2, . . • r„, . . . o f real num bers, all of which lie between two fixed real num bers, has at least one “ point o f accum u­ la tion ,” that is, a real num ber r such that each n eighborhood o f r, no m atter how small, contains infi­

1 -7

nitely m any of the num bers ?•„. If, m oreover, the sequence r* is m on oton ic (for exam ple, m on oton ically increasing, that is, each num ber of the sequence is greater than all preceding num bers), then it has exactly one point o f accum ulation (which is then called the limit of the sequen ce). A special case of the last statem ent is the fact that each decim al num ber, w hether term inating or not, represents a real num ber. While T heorem 3c and Theorem s 1ft to Id, 2a to 2i', and 3 « to 3c must be proved from the definition o f real num bers, all other statem ents a bout such num bers can instead be derived from these few num ­ bered theorem s, w ithout further reference to the definition of real num bers. T h at is to say, these theorem s form a com plete set of axiom s for the sys­ tem of real num bers. 3.

D ig it a l C o m p u t i n g M a c h i n e s

N o n an tom atic M achines. For facilitating largescale com pu tation , com pu tin g machines are used. T h e sim plest of these is the adding machine. It con ­ tains a numerical keyboard, a counter, and a printing mechanism. T he person operating t he m achine enters a num ber on the keyboard and depresses an operating key ( ‘ 'a d d k e y ” ), whereupon the num ber is a u tom ati­ cally set up in the counter. In m ost machines the key­ board is autom atically cleared, in som e the num ber is printed on a paper tape, and in m any these functions are optional. T h e operator enters another num ber and again depresses the add k ey; the m achine now adds the num ber from the keyboard to the one in the counter, so that the counter now contains the sum of the tw o num bers entered. In the same w ay further terms m ay be added. At any time the operator m ay, b y depressing the proper key, cause the m achine to print the contents of the counter, i.e., the sum or “ to t a l” of terms entered; in doing so he m ay cither retain the num ber in the counter for adding further terms (in which case the total is considered a “ sub­ t o t a l” ) or clear the counter (the total is a “ grand total ” ). M any adding machines can be made to sub­ tract as well as add. All can be used for m ultiplying, by repeated addition, but this is practicable on ly for small m ultipliers (usually containing not more than one nonzero digit). If the transmission betw een keyboard and counter is m ade so flexible that the num ber in the keyboard m ay be added in any decim al position of the counter, then the m achine can be used for m ultiplyin g and dividing. Such machines arc com m on ly called calculators. T h ey usually dispense with the printer and, instead, m ake the contents of the counter visible on a set of dials (product dials). In m ultiplication the m ultipli­ cand is entered in the keyboard and is repeatedly added into the counter. The num ber of additions perform ed in each decim al position is determ ined by the m ultiplier, and is exhibited on a separate set of dials (m ultiplier dials). In division, the operator enters the dividend in the counter, the divisor in the k eyboard, and causes the m achine to subtract repeatedly so as to m ake the contents of the counter as small as possible. T h e num ber of subtractions per­ form ed in each decim al position appears in the m ulti­ plier dials and indicates the quotient. D ifferent m odels differ in the details of autom atic shifting to

1 -8

MATHEMATICS

the proper decim al position, of clearing of the dials, of how the operator sets up the m ultiplier, etc. P u n c h e d -ca rd M a c h in e s . The machines d e­ scribed so far arc often called ‘ ‘ desk m ach in es’ ’ because of their m oderate size. Their ca pacity and speed are also relatively m oderate. The need for faster and m ore flexible machines has caused the developm ent, beginning around 1S70, of a series of m achines called “ pu nched-card m ach in es,” and in recent years of sev­ eral types of very large autom atic digital com puters. In all types of punched-card m achines num bers are entered in the form of holes punched into paper cards. T h e cards are fed into the m achine, and each hole establishes an electrical con tact. T h e location of the holes indicates what num bers arc represented. Inside the m achine, arithm etic and allied operations arc car­ ried out in essent ially the same w ay as in desk machines except that some of the newer types of punched-card m achines use m ore advanced techniques, such as electronic circuits. T h e sim pler types of machines perform one operation, or a small group of allied operations, for each card fed, for exam ple, adding and printing or m ultiplying, adding the product to a previously accum ulated total, and punching the result into a card. M ore advanced types perform fairly long sequences o f operations. Details concerning the operations to be perform ed are selected b y the opera­ tor of the m achine b y plugging a num ber of wires. A few types of machines use cards for introducing instructions as to the operations to be perform ed, and can thereby perform arbitrarily long sequences of operations; these types also make provision for storage of interm ediate results of com pu tation , and thereby com e close to the large machines described below . H igh-speed A utom atic, M ach in e s. T he devel­ opm ent of high-speed autom atic com pu tin g machines began around H) 10. Several different t ypes have been built. Except- for the earliest, few, these machines agree in the essential features of their organization. T h ey deal with numbers on which com pu tation s are to be perform ed or which result from com putations, and with instructions as to the sequence of com p u ­ tations to be carried out. in most machines these instructions are represented by (rode num bers, so that the machines chad with num bers only. This has the double advan tage of sim plifying their design and of enabling them to perform arithm etic oper­ ations on their own instructions so as to m od ify them in tin1 courser of a problem . All machines an* capable of adding, subtracting, m ultiplying, and m any of them can divide and extract square roots (if the* latter two operations are not explicitly provided for, they can be replaced b y iterated sequences of the form er three). All can choose one of two alternative instructions, the ch oice depending on som e previously obtained result. M any provide for various other operations such as isolating digits of a num ber, shifting a num ber or a portion of a num ber from one part of the machine to another, and the like. M any of the large com puting m achines represent num bers in the binary system , but some1 use* tin* d eci­ mal system . The latter requires a somewhat m ore com plicated machine design but facilitates use of the machine'. Even when the decim al system is used, each decim al digit is represented by a com bination of binary signals, since tin* physical means available for

num ber storage— such as electric or acoustic pulses or one of the tw o stable states of a vacu um -tube circuit or of a small m agnet— are essentially binary. A num ­ ber of different system s o f num ber representation arc in use, each of which has certain specific advantages. T h e m achine usually provides for a fixed num ber of (decim al or binary) positions for each num ber with w hich it deals. A m on g these positions, the location of the units position is cither fixed or variable; we speak o f a “ fix ed ” or “ floa tin g ” decim al (binary) poin t. Som e fixed-point m achines enable the operator to vary the location of the units position m anually. F loating­ point machines must store, with each num ber, som e form of inform ation about the location of the units position and m ust make provision for this variable location when carrying ou t arithm etic operations. T h e floatin g-point feature entails a certain co n ­ venience for the planner o f a com pu ta tion program , b u t it is ob ta in ed at the expense of m uch additional equipm ent. These m achines have several m ajor organs or co m ­ ponents. (1) T h e arithmetic organ carries ou t the arithm etic and allied operations. (2) A control organ establishes the sequence in which operations are carried out. (3) T h e memory organ stores the num ­ bers put into the m achine at the start of a problem and those occu rrin g as interm ediate or final results. In practically all m achines, the same m em ory organ also stores all instructions. Then there arc (4) the input organ (both of num bers and of instructions) and (5) the output organ (norm ally o f num bers o n ly ). A rith m etic Organ. T h e arithm etic and con trol organs consist principally of electronic circuits. Num bers and instructions, while passing through these organs, arc represented b y sequences o f electric pulses. Circuits can be designed for very high pulse repetition rates, from about 100,000 to several million pulses per second. As a result these m achines are capable o f perform ing several thousands o f arithm etic operations per second. A ddition is carried out either in parallel or serially. In the parallel mode all digits o f the addend arc added to tlie corresponding digits of the augend sim ul­ taneously and all carries are taken care o f subse­ quen tly. In the serial mode the digits arc added one at a time, starting with the least significant, and as each digit is added the carry coining from the next lower digit is taken into accou nt. T h e distinction between the serial m ode and the parallel one pervades most com pon en ts o f the m achines, especially the m em ory. T h e parallel m ode tends to be faster b ut to require more equipm ent. M ultiplication is usually done b y repeated addition. Short cuts arc som etim es used, especially in machines which represent num bers in the decim al system . For exam ple, some' m achines use subtraction instead of addition when a m ultiplier digit is greater than 5 while others have built in “ m ultiplication ta b les” for all com binations of m ultiplier and m ultiplicand digits. If provision is made for division and extraction of square roots, it usually consists in some process of repeated subtruction. C o n t r o l O r g a n . T h e principal functions o f the con trol organ are to cause the arithm etic unit to carry out the desired type of operation, to feed the proper operands into the arithm etic unit, to dispatch the

1-9

A RITHM ETIC result o f the operation to the proper location in the machine (usually in the m em ory organ ), and to select the code num ber representing the next instruction to be carried out b y the con trol organ itself. M ost of these functions consist in sw itching (or “ g a tin g ” ) the channels along which pulses travel in the m achine so as to establish con nection to the proper locations in the m em ory organ. As a rule an arithm etic operation has tw o operands and one result to dispatch, and therefore the instruction calling for the operation m ust contain three designations o f m em ory locations or addresses. In som e machines instructions contain four addresses, the fourth indicating the location of the next instruction. O ther machines dispense with the fourth address and take their instructions from a predeterm ined sequence of locations unless a different location is called for b y a special type of instruction. Finally, there arc m achines using a single-address co d e; in these a single operation like m ultiplication requires several instructions. M em o ry Organ. T h e m em ory organ m ust have sufficient ca p a city to store all the instructions and all the initial data and interm ediate results needed at any one tim e in the course o f a relatively com plex com p u ta ­ tion. M ost m achines store from a few hundred to a few thousand w ords (a word designates either the cod e num ber for an instruction or a true n um ber). T o accom plish this storage w ith ou t excessive space or pow er requirem ents is a m ajor engineering problem . Four types of system s are currently in use: m agnetic surfaces, acoustic pulses, electrostatic mem ories, and m agnetic cores. T h e first stores binary digits in the form o f m agnetized spots on the surface of a wire, tape, disk, drum , or the like. In the second a binary digit (1 or 0) is represented b y the presence or absence, at a given instant, o f an ultrasonic pulse that travels the length o f a colum n (usually o f m ercury) and is picked up b y a piezoelectric crystal, con verted to an electric pulse, carried back to the beginning of the colum n, and reconverted to a sound pulse. Provision must be m ade for proper tim ing and shaping of the pulses. Electrostatic m em ories indicate binary digits b y electric charges on the face o f a cath ode ray tube. M agnetic cores arc small rings with several wires passing through; the core is m agnetized b y a current pulse in one of the wires; its state is “ re a d ” b y observ­ ing whether another pulse causes further m agnetiza­ tion. M ore often than not, the first tw o kinds of mem ories store words in serial fashion and arc used in m achines in w hich the arithm etic unit, too, works serially. T h e latter tw o are usually con nected with parallel operation. T h e average access time, i.e., the time which the m achine has to wait before a desired w ord can be read ou t o f the m em ory, is relatively long with m agnetic m em ories, shorter with acoustic ones, and alm ost nil w ith electrostatic tubes and m agnetic cores. On the other hand, m agnetic surface m em o­ ries can be built w ith greater capa city than the other types. Inpu t and O u t p u t Organs. M edia suitable for input and ou tpu t are punched cards, punched paper tape, and the m agnetic wire or tape m entioned in connection with m em ory organs. O ccasionally input is accom plished b y ph otoelectric means. O utput is som etim es m ade directly to a typew riter. If input and ou tpu t use the same m edium and the same code,

the ou tpu t o f a com pu tation m ay be fed back into the m achine for use in subsequent com pu tation . If this is done autom atically under the control of the m achine, the ou tpu t-in pu t m edium is referred to as the external memory; the m em ory organ proper, by contrast, is called internal memory. Usually the external m em ory has greater ca p a city (10,000 to 100,000 words) but longer access time. Analogue C o m p u ter s. T h e com puting machines described so far are called digital or discrete-variable com puters. T h is means that num bers are repre­ sented b y the cou nt of certain discrete events in time (electric pulses) or discrete ob jects in space (teeth o f a counter w heel). B y contrast, there are com pu tin g instrum ents in which num bers are repre­ sented b y physical m easurem ent o f continuously vari­ able m agnitudes: voltages, displacem ents, and time. Such instrum ents are called analogue or continuousvariable com puters. T h e sim plest o f these is the slide rule, which measures the displacem ent of one linear scale against another and the sum or difference o f two linear m agnitudes. If logarithm ic scales are marked off on tw o edges so that the num ber a is m arked at distance log a from som e origin, the slide rule, b y locating the sum or difference o f tw o distances, gives the product or quotient o f tw o num bers, because of the relationships log a + log b = log ab log a -

log b = log j b

M ore elaborate analogue com puters use system s of interconnected rotating shafts, or m ore recently electronic circuits, to perform not on ly the arithm etic operations but also integration o f continuous fun c­ tions. All such devices are severely lim ited in their accu racy. M an y of them are m ade for the solution of specific types o f problem s, w hile others are of fairly general a pplicability. Because of the ease w ith which integration can be perform ed, machines of this kind arc particu larly useful for the solution of (ordinary) differential equations. Analogue machines built for this purpose are called differential analyzers. Other m achines have been built for solving system s of linear equations, for the analysis of electric circuits, etc. Re fe re n c e s 1. Ilartrcc, D . R .: “ Calculating Instrum ents and M a ­ chines,” U niversity of Illinois Press, Urbana, 111., 1949. 2. M eyer zur Capellen, W .: “ M athcniatischc Instrum en te,” Becker and Erler, Leipzig, 1944, and Edwards, Ann Arbor, M ich ., 1947. 3. M urray, F. J .: “ T he T h eory of M athem atical M ach in es,” K in g ’s Crown Press, N ew Y ork , 1947. 4. Russell, B .: “ Introduction to M athem atical Phi­ lo sop h y,” 2d ed., London, 1921. 5. Schubert, L . I I .: Grundlagen der A rithm etik, “ E n cyklopadic der M athem atischcn W issenschaftcn,” IA 1 , Teubner, Leipzig, 1 S 9S -1904. 6. Tarski, A .: “ Introduction to Logic and the M e th ­ odology of the D eductive Sciences,” 2d ed., N ew Y ork , 1940. 7. T om pkins, C . B ., and J. II. W akelin (Staff of Engineer­ ing Research Associates, In c .): “ Iligh-speed C o m p u t­ ing D evices,” M cG ra w -H ill, N ew Y ork , 1950. S. Richards, R. K ., “ Arithm etic Operations in Digital C o m pu ters,” Van N ostrand, Princeton, N .J ., 1955.

Chapter 2 Algebra By OLG A T A U S S K Y , California Institute of T ech n ology

1.

P olynom ials

Hy definition

T h e main task of algebra is the solution o f algebraic equations in one or m ore unknowns and of system s of such equations. An algebraic equation in n unknowns Xi, • • • , x n is an equation w hich can be brought into the form 2c«!

These num bers (‘an be arranged in the so-called Pascal triangle: n n n n n

= 0

tnXieix«e*

=

where ct. are num bers, called coefficients; the c» are integers. T h e sum on the left-han d side o f this equation is called a polynomial in the n unknowns. Th e single elem ent ctl . . . x nen is called a term. T he sum 2 c; is its degree; the largest actually occurring degree in a polynom ial is the degree o f the polynomial. If all terms have the same degree, the polynom ial is called homogeneous or a form . T h e best known form s are the quadratic form s

= = — — =

0 1 2 3 4

(ai - f 02

+

a t) *

l

ri+ ♦•• +/•< = «

i,k = l

Q

Even if n is not an integer, the num ber

r

2« A lg e b r a i c K q n a t i o n s in O n e U n k n o w n , Com plex N u m b ers

— T +

1)

a 0x n -f- a ix n~l +

(") = ( " ) ---------\ rj \n - r ) r\(n = n{n -

C

• • • + an- i x + an = 0 (2.1)

1 . 2 • • • /•

)

If the a-i are rational num bers, then the solutions or roots o f such equations are called algebraic numbers. All other num bers are called transcendental, for exam ple, e and v. M any algebraic equations would not have any roots unless the idea of num ber is extended to include quantities m ore general than rational num bers and num bers com posed o f surds or lim its o f sequences of such num bers (sec Chap. 3, Sec. 1). All these num bers form what is called the sot o f real num bers. T h e sim plest exam ple o f an equation w ith ou t real roots is the equation x 2 + 1 = 0 . It has been found essential to define the im aginary

are called binomial coefficients. T h e num ber 1 •2 ♦ • • r is denoted b y r! (factorial r). (See Stirling form ula and V fu n ction .) The binom ial coefficients have the properties that

(2)

has

These equations are o f the form _ n(n — 1) • • • ( / ?

( 1)

atr
a2rî r I ! r 2!

If / = 3, the coefficients can be arranged in a pyram id and m ultidim ensional analogues exist in the higher cases.

(see under sym m etric m atrices). A polynom ial which has only one term is called a monomial, if tw o a binomial. T he binom ial theorem gives the polynom ial expan ­ sion for the nth power of the sum of two quantities, when n is a positive integer:

= £

1 4 6 4 1

Using relation (2), each row is easily com pu ted from the preceding one. A generalization of the binom ial theorem is the m ultinom ial theorem :

a 4

/ ( * . + h) = / (x „ ) + hf' On) +

/"(* » ) +

Consider

• ••

and so on . A few practical m eth o d s for the d e te r­ m in ation of real roots of an algebraic, e q u ation follow .

If all but the first tw o terms on the right are neg­ lected, then if f i x n + b) — 0, the correction h to be added to x n to get to the zero is given b y

Synthetic Division. A convenient, m eth o d for tin* d ete rm in a tio n o f the q u o tie n t (J w hen a p o ly n o m ia l f i x ) = a,,s n -j- a i.r’1-1 + • • • - { - a n is div id ed b y a b in o m ia l, say x and for the e va lu ation o f the remainder II = /(**.): Suppose

that is

f i x ) = {x —

oc)(J

+

It

fiXn) +

h =

hfixn)

-

= 0

fM / V h)

Discussion o f the validity of the (Conclusion (2.11) and o f the rate of con vergen ce can be given when / is

ALGEBRA su b ject to certain restrictions. These will n ot be given here, nor will various developm ents of the process. T h e m ethod is valid even if / is not a p o ly ­ nom ial. An im portant exam ple occurs when f(x ) = X "1 -

(3) T h e equation whose roots arc ten times those of f ( x ) = 0 is, in the cu bic case, ciox3 +

x3 -

that is, x,t+i = x n{ 2 — A x n) In this case x n —> A ~ l, provided that 0 < x 0 < 2,4 “ h T h e N ew ton process is often m ore con venien t to determ ine the roots of a cubic or quartic than the explicit form ulas already given, especially when a suitable Xo can be obtain ed easily. H orner Process. It is n ot always con venien t to get exact ideas on the accu racy of the N ew ton process. T h e follow ing process, due to H orner, obtains, at each succeeding step the root correct to one more decim al place. It is, how ever, slower than the N ew ­ ton process, w hich, upon each new application, obtains the root correct roughly to double the num ber of decim al places that it does on the previous application . 1. D eterm ine betw een which tw o integers « , a - f 1 the root a lies. 2. Obtain a new equation whose roots p are those of the original one dim inished b y a ; the new equation has a root betw een 0 and 1. 3. D eterm ine the equation whose roots are 10/1. A p p ly now the first process again to determ ine betw een w hich integers p, p + 1 the root lies. T h e root is then a = a.p . . . . R epetition o f this p roc­ ess will give any desired a ccu racy. T h e process can be speeded up, b y special devices, or b y use o f N ew ­ ton ’s process. Steps ( 1), ( 2 ), and (3) are norm ally carried out as follow s: (1) Use the fact th at if f ( x i), f ( x 2) have different signs there is a root (or an odd num ber of roots) betw een x i ,x 2. (2) R earrange the polynom ial, for exam ple, f i x ) = dox 3 + a ex 2 - f a 2x -J- a 3 in powers of ix — a ): f i x ) = A oix -

a ) 2 + A 2(x -

a) + A a

Here A 3 is the rem ainder when f i x ) is divided b y {x — « ) , the quotient being a ) 2 + A xix -

a) + A 2

where A 2 is the rem ainder when this is divided b y {x — a ), etc. T h e algorithm described in ( 1) can be set up in the follow ing condensed w ay. Choose f i x ) = x 3 — 4 x 2 + 2 as an exam ple. This has a root between — 1 and 0 . 1

- 4 -1

1

—5

0

2 5 -5

x3 -

5 —3

has a root betw een 0

-1

6

1

-0 -1

11

1

-7

7()x2 + 1,1UOx -

3,000 = 0

This equation has a root betw een 3 and 4. This means that the equation x 3 — 7x2 -f- l l x — 3 = 0 has a root between 0.3 and 0.4 and therefore the original equation has a root —0.6 . . . . A m ethod due to Gtaeffe depends on the fact that it is easy to obtain the equation whose roots arc the squares of the roots of a given equation aoXn -J- • • • - f a„ = 0 R epetition o f this process gives equations a«(wV* - f • • • + o n(m) = 0 whose roots are the 2 W powers of those of the original equation. If there is one root a of the original equa­ tion whose m odulus exceeds that of all the rest, say, |«| > |/3| ^ |t| = ■ • • » then for com paratively small values o f in

and a can be obtained b y taking logarithm s. 9.

M atrices

An n X ni m atrix A — [«a], with i = 1, • • • ,?i and k = 1, • • • ,w, is an array o f num bers arranged in n rows and m colu m n s: a n

ai2

• •

aim "

a 21

a->2

• •

a 2m

_a«i

a , 12

• ■ •

a nm_

Certain operations arc defined for such arrays: if another m atrix B = [/>^1 has the same num ber of rows and colum ns, then the sum of the tw o m atrices is A + B = [«¿A- -f- bifc)

a ) 3 + A x(x -

A oix -

lOtiiiC2 + lOOuox + l,000(/3 = 0

In the special case this becom es

A

T hen

1

1 -1 3

7 x2 +

lb

- 3 = 0 and

1

T h e product of a num ber r and a matrix .1 = [m*] gives the m atrix rA = [ra,-*]. If [bik] is a m atrix with m rows and p colum ns, then the product A B of [«,•*] and [bik] is the n X p m atrix [ca], where Cik =

C lu b Ik 4 -

• • '

+

(lin b n k

In general the p rodu ct A B differs from the produ ct B A . If the rows and colum ns o f a m atrix are inter­ changed, the resulting m atrix is called the transpose A ' of A . T h e transposed m atrix of a product A B is B 'A '. M atrices are clearly con nected with linear trans­ form ations of variables or linear substitutions, and it is from them that the produ ct definition originates. Of particular usefulness arc the square m atrices for which m = n. There the elem ents an , i = 1,- • • ,?i, form the principal or main diagonal; their sum is called the trace of the m atrix. If a square matrix coincides with its transpose, it is called sym metric; if it

MATHEMATICS

1 -1 4

coincides with the com plex con ju gate of its transpose, it is called H erm itian; if it coincides with its negative transpose, it is culled skew or antisymmetric. T he square matrix o f n rows with ones in the principal diagonal and zeros everywhere else is called the unit matrix or A matrix all of whose elem ents are zeros is called a zero matrix. A square m atrix /1 of n rows for which a matrix B exists such that A B = / „ is called nonsingular; otherwise it is called singular. T he matrix B is unique, is called the inverse m atrix of .1, and is usually denoted by A ~ l. It is also easily seen that A ~ b 1 = In general, A B ^ B A for arbitrary m atrices A and B ; however, for any two m atrices A ,B we have trace A B = trace B A . Both this property and the fact that ( A “ 1) “ 1 — A do not hold in general for infinite matrices. Also the fact that the inverse matrix is unique is in general not true for infinite m atrices. Explicit expressions for the inverse of a nonsingular m atrix will be m entioned later (Sec. 12); these expres­ sions are, how ever, of little use for the com pu tation o f the inverse o f numerical matrices. An orthogonal matrix is a real matrix whose transpose coincides with the inverse; a unitary m atrix is one for which the com plex con ju ga te transpose coincides with the inverse. Using the product definition for general m atrices, a system of linear equations UnXl 4" • • • 4" dlmXm = bl anXi -h ■ • • + a-imXm = b2 a„lX\ +

(2.12)

• • • “h OnmXm = l>n

can be written in the abbreviated form Ax = b

(2.13)

where x stands for the colum n vector (xi, . . . ,x m) and b for (b {, . . . ,bn). If n — m and the m atrix A is nonsingular, the solution of the system can be written sym bolically its x -

A hi

(2.14)

If it is desirable to transform the unknowns x, in (2.13) to a sot of unknowns y = //, which are con ­ nected with the x, b y the transform ation y = Bx, where B is again a nonsingular n X n m atrix, then it appears that the system (2.13) is equivalent with the system BA B ~ ly = Bu Th e m atrix A is said to have been transformed by B and B A B ~ X is also called similar to A. Similar matrices h ave the same traces. Another very im por­ tant con cept, which is invariant under transform a­ tions, is the n characteristic roots or eigenvalues or proper values of the matrix (see Sec. 13). 10.

D eterm inants

num bers 1, . . . ,n and the sign ± is chosen a ccord ­ ingly as the perm utation is even or odd. T h e value of the determ inant is rarely com pu ted from this expression, but b y using som e o f the properties of determ inants: (1) A m atrix and its transpose have the same determ inants. (2) If all num bers in a fixed row (or colum n) arc m ultiplied b y the same num ber, the determ inant is m ultiplied by that num ber too. (3) If a m ultiple (by the same num ber) of the elem ents of a row (or colu m n ) is added to another row (or colu m n ), the value o f the determ inant is unchanged. (4) D enote b y Aik the value of the subdeterm inant of the m atrix obtain ed from A b y om itting the ith row and the Ath colum n, m ultiplied b y ( — l ) i+*. T h e subdeterm inant alone is called a minor; Aik is called the cofa ctor of the elem ent T h e determ inant can be expressed in the form |aijfc| = an A n 4- a» 2 A ¿2 4~ * * * On the other hand 0 = mu A

4~ rttsAjfeo 4- • • •

if f 9^ k

Using (4), it is clear that the inverse m atrix o f a nonsingular m atrix /1 is

T his is, however, not generally o f m uch use for the com pu ta tion of the inverse o f a numerical matrix (see Sec. 12). (5) T h e determ inant of the p rod u ct of tw o m atrices is equal to the p rodu ct of the deter­ m inants o f the tw o m atrices. (G) From the definition of the determ inant it is evid ent that a m atrix w hich has a row or a colum n o f zeros has a vanishing deter­ m inant. Further, the determ inant vanishes if there is a linear dependence betw een the rows or the colum ns o f the m atrix, that is, if num bers « i , . . . ,a n exist, such that not all « , = 0 and tt)«it 4~ « 'idii 4“ • • • 4~ ctnanx = 0 or

i = 1, . . . ,u

nn«ii 4- « 2«*2 4" * • * 4“ ««a»i. — 0

A square matrix is singular if and on ly if its deter­ m inant vanishes. A lthough much w ork has been done on properties of special determ inants, on ly three will he m entioned. Let Xi, . . . ,x„ be n unknow ns; then the correspond­ ing Vandermonde determinant is \iiik\} where « a = x t*_1. T h e value o f this determ inant is

(_!)»(»-.)/. [ | (I( _ xt) i< k its square is the discrim inant (see Sec. 7). T h e sec­ ond determ inant, the so-called W ronskian, concerns n functions f i( x ). It, is defined as |/*№)|, where

/,. « A n = A , if 1im n—► oo aik{n) = aik for all values o f i and k. From this it is evident that an infinite series 2 A „ of matrices converges if the infinite scries corresponding to each elem ent converges. T h e con vergen ce o f a power series in A is linked up w ith the characteristic roots of A . In particular the geom etric series / + A + A 2 ~b • • • converges if and on ly if the characteristic roots of A all lie inside the unit circle. T h e exp o­ nential series

converges for all m atrices A . N ot all properties o f the exponential fu n ction remain if m atrices arc intro­ duced as exponents, e.g., in general eAe B ^ eA+B If, how ever, A and B com m u te, eAeB = eA+B. 16.

H y p e r c o m p l c x S y s t e m s or A lg e b r a s

A h ypercom plcx system w ith respect to the real (or com p lex) num bers has a finite set n of base elements or units Ci, . . . ,e„ such that every elem ent of the system is of the form aiCi + • • • + «*»««, where a» are real (or com plex) num bers. T h e base elem ents have m ultiplication rules etCk = Saijt/Cy, w here a;*/ are again real (or com plex) num bers w hich are arbitrary as long as (c,ejt)cy = c,-(cjtcy). If this last con dition is n ot fulfilled, the system is called nonassociative. From the m ultiplication o f the base elem ents a m u lti­ plication o f any tw o elem ents a\e\ + • • •+ a nen and b xe\ + * • • -b bne n is given b y putting (uiCi +

• * * + « »c„)(5 ici ~b * • • + bne n) = 'ZaibkCiCk

A dd ition o f tw o elem ents is (ai + />i)ci ~b * • * -b («n *b bn)e n

T h e clem ent 0 •ci -b * * • + 0 •en plays the role o f the 0 am ong ordinary num bers. An elem ent cor­ responding to 1 am on g ordinary num bers does n ot always exist. T h e follow ing arc several exam ples of h ypercom plcx system s, the first four of associative system s and the last tw o o f n onassociative system s: 1. The complete matrix algebra with real or com plex coefficients, i.e., the set o f all n X n m atrices with real (or com plex) coefficients. T h e n2 matrices which have one elem ent 1 and 0 elsewhere m ay be taken as base elem ents. T h is set has a I elem ent, nam ely, /„ . H ow ever, in m any other respects, as already m en­ tioned earlier, it does not behave like ordinary num ­ bers, for exam ple, A B ^ B A in general; further, the product of tw o m atrices m ay be the zero m atrix, w ithout either factor being the zero m atrix. It m ay even happen that A B = 0 and B A ^ 0. T h e nonsingular m atrices have reciprocals; the singular m atrices do not.

ALGEBRA 2. Quaternions. This system has four units, usu­ ally called or l,/i,* 2,f 3> with the m ultiplication rules 1 •ia • 1 = it iaip = iflift ( « 7* ft), ia2 = “ 1 ? 11-2 “ 13 = ¿1 ?3¿1 = ¿2 T h e (piatcrnions have no divisors of zero, i.e., no product (ao T" a\i\ T (z2f 2 T « 3*3) (bo 4~ b\i\ H~ ^2/2 ~h ^2/ 3) — 0 w ithout cither all «» = 0 or all = 0. quaternion 9^ 0 has an inverse, nam ely,

E very

ao — a p i — a^i-i — a,d^ do2 T* a 12 T d 22 T a 32 The (piatcrnions arc a m ost im portan t discovery b y H am ilton, who used them for expressing the rotations of the sphere as ordinary com plex num bers describe the rotations o f the circle. T h ey arc the on ly h yper­ com plex system , apart from the reals and the com plex num bers, which has no divisors of zero. One o f the applications of quaternions is to furnish a proof that every integer is a sum of four squares. 3. Clifford algebras have 2 n base elem ents which are generated b y n elem ents Ci with the relations

c;2 = - 1

CiCj = - CjCi (i 9± j)

Sets of m atrices which are anticommuting have been studied in various connections. T h e Pauli spin matrices form a set of anticom m u tin g 2 X 2 matrices whose square is the unit m atrix:

[

ï

; ] . [ ?

-

a

u

-

D irac obtains four 4 X -1 m atrices properties: ~0 0 0 0 1 0 1 0 0 _1 0 0

a

with the same r 0 0 0_

“0 0 0 _i

0 0 —i 0

0 i 0 0

—i 0 Ü 0_

'0 0 1 _0

0 0 0 -1

1 0 0 0

0 -1 0 0.

'1 0 0 0

0 1 0 0

0 0 -1 0

o' 0 0 -1

E ddington proved that there cannot be more than five anticom m uting matrices Ki in four dim ensions, such that Ei 2 = — 1. If all are real or pure im aginary, then two arc real and three are pure im aginary. 4. Dual numbers have tw o base elem ents 1 and e with e2 = 0 . 5. Cayley numbers have eight base elem ents: l,£i, . . . ,€7 with a 2 — — 1, a e j = —c>ct- and cie 2 = c3,

C \C \

1 -lü =

Cg,

C 3Cfi =

Cfi.

C iC g

*

€ 7,

C2C0

— 67,

C 2C 4 =

— Cf,,

C 3C 4 — € 7 ,

All other products of base elem ents are obtained from the further rule that C i C k — e,• im plies C k C , = d ,

CjCi = Cjfc. G. M atrix algebra with (A B -J- B A ) as composition instead of A B . This is a com m u ta tive (even if A B t* B A ) but not associative com position . It plays a role in quantum -m echanical theory. While com plete matrix algebras are hypercom plex system s, a certain converse is true to o : T o every hypercom plex system S there corresponds an isomor­ phic set of m atrices, i.e., a set which is in a one-to-on e correspondence with S such that the sum (or product) o f tw o elem ents of *S corresponds to the sum (or p rod­ u ct) o f the corresponding matrices. For the com plex num bers this isom orphism is established through the correspondence a + ib 1 the corn \spondo nee

a -+■ hi - f cj + die the set of all points with integral coefficients form a lattice; another lattice is found if also the centers o f the squares o f the previous lattice are adm itted as lattice points (sec Figs. 2.2 and 2.3). •

• •

•

•

t

t

•

•

•

•

•

•

t

t

•

• •

•

•

• • F io. 2.2

Fin. 2.3

In modern crystallography the lattices in / i 3 are used as the main tool for the classifications; there are seven different- types of lattices, called triclinic, m onoclinic, tetragonal, hexagonal, rhombocdric, rhombic, and cubic. A symmetry is a m otion o f the R n which transform s lattice points into lattice points. T h e sym m etries of a lattice form a group. It is easy to see that in R 2 sym m etries which leave ((),()) invariant must be of period 2, 3, 1, or (>. Generally, a m otion in R n is given by the transform ation y = A x + a, where .1 is an //-dim ensional orthogonal matrix and a a colum n of n real num bers. 'Fin' matrix A alone defines a rotation, the colum n a alone a translation. G roups whose elem ents are m otions are called space groups. Special attention is confined to such space groups

where the translations contain n linearly independent elem ents and which leave the lattice that is defined b y these elem ents invariant. T h ey arc called discrete groups. T h ere are 230 such space groups in R 3 and 17 in R 2. A set o f m atrices w hich is isom orphic w ith a group is called a representation o f the group. E very finite grou p has representations; m any infinite groups, although n ot all of them , can be represented b y finite m atrices. T h e dim ension o f the m atrices of a representation is called the degree o f the representation. T h e sets of traces o f the m atrices which occu r in the representa­ tion are called its character. If all the m atrices of a representation arc transform ed b y the same nonsingular m atrix, an equivalent representation is obtain ed. If a representation is equivalent to one in which all the m atrices have the form

[;:■

1

]

where A*, A 2 are square m atrices and P,Q consist of zeros only, then it is called reducible; otherwise it is called irreducible. All irreducible representations o f a finite group are equivalent to representations b y matrices whose elem ents arc algebraic num bers. All representations o f a finite group arc eq u iv a ­ lent to representations b y u nitary m atrices. T h e num ber o f nonequivalcnt irreducible representations is equal to the sum o f classes in the group. E very representation o f a finite group can be reduced com pletely, i.e., an equivalent representation o f the form

M i 0 _0

0 A 2

• 0

• •

*

• •

• 0 • 0 0 • A r_

can be found when» the .1» arc square m atrices and w here 0 means a zero m atrix of appropriate size, and where the representation A i is irreducible. T h e irreducible representations o f a com m u ta tive group are all -one-dim ensional. T h e reduction is always unique. T w o irreducible representations which have the same character are equivalent. T h e degrees m o f the irre­ d u cible representations divide the order o f the group. T h e sum is equal to the order of the group. T h e on ly m atrices which com m u te with all m atrices o f an irreducible representation are the scalar matrices, i.e., the m atrices which have the sam e con stant along the main diagonal and zeros elsewhere. T h e group ring is a h ypercom plex system associated with finite groups which plays a big role in the study o f the representations. T h e base elem ents arc the elem ents of the group, and their prod u ct is defined b y the group com position . B e fo re n ees 1. A lbert, A . A .: “ M od ern Higher A lg e b r a ," U niversity of C hicago, 1937. 2. Birklioff, G ., and S. M a c b a n e : “ A Survey of M odern A lg e b r a ," N ew Y ork , 1911). 3 . Boclier, M .: “ Introduction to Higher A lg e b r a ," New Y ork , 1907.

ALGEBRA 4. Burckardt, J. J .: “ Die Bewegungsgruppen der K ristallographie,” Basel, 1947. 5. Burnside, \V.: “ T heory of Groups of Finite O rd e r," C am bridge, 1911. 6. Burnside, W . S., and A. W . Panton: ‘ ‘ T heory of E q u a tio n s," D ublin, 1924. 7. Dickson, L. E .: “ Algebras and Their A rith m e tic s," Chicago, 1923. 8 . Dirac, P. A . M .: “ T he Principles of Quantum M e ch a n ic s," Oxford, 1947. 9 . E ddington, A . S .: “ R elativity T heory of Protons and E le c tro n s," Cam bridge, 193G. 10. Faddecva, V. N .: “ C om putational M ethods of Linear A lg e b r a ," M oscow and Leningrad, 1950. 11. Frazer, 11. A ., W . J. D uncan, and A . It. C ollar: “ Elem entary M atrices and Som e Applications to D ynam ics and Differential E q u a tio n s," Cam bridge, 1938. 12. G antm aher, F . R .: “ T heory of M a tric e s," M oscow , 1953. 13. H alm os, P. R .: “ Finite Dim ensional Vector S p a c e s," Annals of M athem atics Studies 7, Princeton Univer­ sity Press, Princeton, N . J., 194S. 14. H am burger, II. L ., and M . E. G rim shaw : “ Linear T ran sform a tion s," Cam bridge, 1951. 15. Kuros, A . G .: “ Course of Higher A lg e b r a ," M oscow and Leningrad, 1949. 1G. K uros, A . G .: “ T h corv of G r o u p s," M oscow , 1953. 17. L ittlew ood, D . E .: “ The T heory of Group Characters and M atrix Representations of G r o u p s ," Oxford, 1950. 18. M acD u ffcc, C . C .: “ T he T heory of M a tric e s," Ergebnissc der Mathcmatik and Hirer Grenzgebiete, vol. 2, Berlin, 1933. 19. M acD u ffcc, C . C .: “ Vectors and M a tric e s," C am s M ath. Monograph 7, 1949.

1 21

20. M a l’cev, A . I .: “ Foundations of Linear A lg e b r a ," M oscow and Leningrad, 1948. 21. M iller, G . A ., H . F. Blichfeldt, and L. E . D ickson : “ T h eory and A pplications of Finite G ro u p s ," New Y ork , 1 9 1G. 22. M uir, T .: “ T he History of D ete rm in a n ts," 5 vols., London and Glasgow , 190G -1930. 23. M urnaghan, F. D .: “ T he T heory of Group Repre­ se n ta tio n s," Baltim ore, 193S. 24. Schoenfliess, A .: “ K ristallsystem e und K ristall­ stru ktur,” Leipzig, 1891. 25. Schwerdtfeger, I I .: “ Linear Algebra and T heory of M a tric e s," Groningen, 1950. 2G. Speiser, A .: “ Theorie der Gruppen von endlicher O rd n u n g ," Springer, Berlin, 1937. 27. T urnbull, H . W ., and A . C . A itk e n : “ T heory of Canonical M a tr ic e s ," G lasgow , 1932. 28. Turnbull, H . W .: “ The T heory of D eterm inants, M atrices and In v a ria n ts," G lasgow , 1945. 29. van der W acrden, B . L .: “ M oderne A lg e b r a ," vols. I. II, Berlin, 1931. 3 0. van der W acrden, B. L .: “ Die gruppentheoretische M ethode in der Q uan tenm echanik," Springer, Berlin, 1932. 31. Von N eum ann, J .: “ M athem atical Foundations of Quantum M e ch a n ic s," Princeton University Press, Princeton, N . J., 1905. 3 2. W cdderburn, J. H . M .: Lectures on M atrices, A m . M ath. Soc. Colloquium Publications, vol. X V I I , 1934. 33. W cy l, I I .: “ Gruppentheorie und Q uantenm echanik,’ ’ Leipzig, 1928. 34. W hittaker, E . T ., and G . R obinson: “ T he Calculus of O b se rva tion s," London and Glasgow , 1944. 35. W igner, E .: “ Gruppentheorie und ihre Anw endung auf die Quantenm echanik der A t o m e ," Berlin, 1931.

Chapter 3 Analysis B y JO H N T O D D ,

1.

California Institute of T ech n olog y

tone is used to cover both these cases. The follow ing result is fundam ental: a m on oton e sequence is con ­ vergent if and on ly if it is bounded. This is proved b y use o f the Dedekind section theorem: if all the real num bers are divided into tw o classes L,R such that any m em ber o f L is less than any m em ber o f R, and if each class L,R contains at least on e m em ber, then there is a real num ber £ such that if x < £, then x belongs to L ; and if x > £, then x belongs to R. Other applications of this theorem establish the existence o f exact boun ds o f a bounded

Beal N u m b e r s , L i m i t s Beal

Num bers,

Convergence

of

Sequences.

From the non-negative integers 0,1,2, . . . we first generate the negative integers, — 1, —2, . . . and then the fractions or rational num bers, ± p / q (p,q integers, q ^ 0) (see Chap. 1, Sec. 2). W e m ay then generate the real num bers as limits of convergent sequences o f rational numbers. W e then prove that the limit of a convergent sequence o f real num bers is a real num ber, so that the real num bers have a com ­ pleteness property which is not possessed b y the rational numbers. For exam ple, the sequence

sequence (xn ), the least upper boun d, bound x n, and the greatest low er boun d, bound x„. If a sequence (x rt| is such that to every A , how ever large, there corresponds an n 0 = no(A ) such that if n > n 0 then x» > A , we say that the sequence tends to plus infinity and write x„ —> + — «>, lim x n — — If x n — n , then x n —> + °°. T he follow ing results, which can not be extended to the infinite ease in general, are valid when the lim its (on the right) are finite:

1, 11/10, 1 11/100, 1,11 1/1,000, . . . converging to tin* real num ber \/'2f does not have a rational lim it. The set o f real num bers (or points) x such that a < x < b is called a closed interval and is denoted b y [n,b]. An open interval is the set o f real num bers (or points') x such that a < x < b ; it is denoted b y (a}b). A neighborhood o f a point a is an open interval containing it: (a — e, a + ij) is a neighborhood o f a for any t > 0, i) > 0. We shall speak sim ply of intervals when tin* question o f their closure or open ­ ness is not relevant. T he sequence |x„j converges to x if any neighborhood o f x, no m atter how small, contains all but a finite* num ber o f terms o f the sequence. A rithm etically this means that corre­ sponding to any € > 0 (no matter how small) there is an integer /o, (depending on e) such that, if n > /¿0 then 'xn — x\ < c. We denote* this situation by the sym bols x n —» x or by lim „_>00 x n = x or sim ply by lim x n = x if there is no doubt about the current variable. Exam ples o f convergent sequences, with their limits, follow , lim an — 0 if |«| < l;lim ? m n = 0, if |«| < 1; lim (1 - a - 1)71 = e = 2.71828 . . . . A sequence ¡x „j is said to he bounded if there is a num ber B such that, for all n, |xM| < B. A bounded sequence always contains a convergent- sub-sequence, i.e., there are num bers i\\ < n-> < n 3 < • • • such that lim Xni

lim (a x n + byn) = a lim x n + b lim y n lim (xnVn) = lim x n lim ?/„ lim x n = (lim x M)_1 if liiii x „ 5^ 0 A sequence need have no limit, finite or infinite. Let us confine our attention to hounded sequences—■ for exam ple, |xtl] where x w = ( — l ) n. Here then* arc sub-sequences, |x««} for instance, convergent to + 1 and sub-sequences (x 2» - i ! con vergen t to — 1. No convergent- sub-sequence has a lim it greater than + 1 or less than — 1. T his holds in general: There is (1) a num ber L, denoted b y lim x„, such that, for all € > 0, there is an infinite num ber o f the terms o f the sequence greater than L — e and on ly a finite num ber greater than L + e and (2) a num ber /, denoted by lim x H, such that, for all e > 0, t here is an infinite num ­ ber o f terms o f the sequence less than I + e and on ly a finite num ber less than / — c. A necessary and sufficient condition for the convergence o f the se­ quence is the (‘quality o f L and /: if L — I, then lim x„ exists and has this value and, conversely, if lim x„

i —» ao

exists. This is the BolzonoAVeicrstrass theorem. For instance, 1Ik? sequence }x„|, where x n = ( — I ) n, is hounded but is not con vergen t; the sub-sequence \x2n\ is convergent- to 1. A sequence is called increasing if X\ < x-> < • • • and decreasing if X\ > x 2 > • * ■ ; the* term mono­

exists, then L = I = lim x H. T h e num bers lim x n and lim x n are called the upper and lower limits of the sequence; they arc to be carefully distinguished from the upper and lower bounds o f the sequence. All rational num bers between 0 and 1 can be ar-

1-22

1 -2 3

ANALYSIS ranged in the form o f a sequence. ordered as follow s

T h ey can be

0 /1 ; 1/1 ; 1 /2 ; 1/:L 1 /4 ,2 /d ; 1 /5 ; 1/(», 2 /o ,:V -l; 1 / 7 i / 5 ; 1 /8 ,2 /7 ,4 /5 ;

• • •

where the groups o f fractions m/n between sem icolons arc all those which are irreducible and have a constant value (m - f n). In the nth group we include num ­ bers p/q (in their lowest terms) such that p -f- q = n and we arrange these in order o f m agnitude. This group is called the Farcy series o f order n. T his sequence has I = 0 , L = 1 and there are sub-sequences in it which converge to any real num ber between 0,1. Cantor has proved that it is not possible to arrange all the real num bers between 0,1 in the form o f a sequence.

T h e convergence properties o f the follow ing se­ quences are o f interest. T h e geom etrical interpreta­ tion is indicated in Fig. 3.1. T he first sequence is Xn+i = x n(2 - Nxn)

(3.1)

If this sequence has a lim it, it is zero or Ar_1. Sup­ pose 0 < Xo < N ~ l, so that N ~ l — x 0 > 0. T h en, since x w+i — x n — N x n(N ~ l — xM), we have j h+i > x n. On the other hand Xn+i that is,

A -1 = - A ja r « -

A “ 1) 2 < 0

is said to be converge at if the sequence partial sums S n =

is convergent.

X\ +

Xj

+

• • •

If lim *S„ =

n—*»

+

o f its

Xn

we write

00

71—1 and call the sum of the series. The geom etric series a + ar ar2 + • ■ * is c o n ­ vergent if |r| < 1. An arithm etic series a - f (a + d) -J- (a + 2d) +

•••

can only be convergent in the trivial case a = 0 = d. Th e terms o f a convergent series them selves form a sequence which converges to zero; such a sequence is called a null sequence. This con dition, while neces­ sary for convergence, is not sufficient, as is shown by the case o f the series 1 -j- 1 /2 + 1 /3 + • • • + 1 /n + • • • for which the partial sums are un­ bounded. A series which docs not converge is called divergent. Several types o f divergence are possible, as indicated by the follow ing exam ples: 1 -f- 1 + 1 + 1 + ••• ; 1 - 1 + 1 - 1 + « •• ; l - 2 + 3 4 + •••. T he theory o f sequences and the theory o f series are coextensive: given a series 2 x n we consider the sequence o f its partial sums while, on the other hand, given any sequence S Mwe can construct a series £*,■, which has (£«1 for its partial sums by putting X{ = Si i ^ 1, Si = x Efforts have been m ade, since the time o f Euler, to assign conventional sums to series divergent in the sense ju st defined. This theory is o f considerable interest m athem atically and has occasional physi­ cal significance. T w o exam ples follow . T he scries 1 — x + x 2 + • • • has sum (1 - f z ) -1 for \x\ < 1; it is reasonable to consider assigning the sum

Xn+i < N ~ ]

Thus {rr„J is a bounded increasing sequence: it must have a lim it and this can on ly be A -1 . Let 8n denote the error x n — A -1 . W e have 5w+i = — Ar5„2. C on ­ vergence of this type is called q uadratic; roughly speaking, this implies that if x n and N ~ ] coincide to a certain num ber o f decim als, x n+i and N ~ l will coincide to about twice that num ber. As an exam ple take Ar = Yi and Xo = 1. Then the sequence is 1, 1.5, 1.875, 1.9021875, . . . . T he second sequence is x,l+1 = (1 -

N )x n + 1

(3.2)

Here we have linear convergence to Ar_l: Vn+i = (x n+l -

A "* ) = (1 -

N )(x n -

AM)

= (I - N)Vn with the same Xo — 1 we obtain the sequence 1,1.5, 1.75.1.875, . . . . I n fin ite S e rie s . An infinite series qo

Xn = Xx + x 2 + n= 1

• • • + Xn +

•••

(1 + I ) " 1 = v> to this series when x = 1. This is the basis o f the Abel or Poisson sum m ation m ethod. An alternative approach is the method o f arithm etic means due to Cesaro. W e associate with the series 2x{ having partial sums S n the value

71—* ao

H

if this exists. In the case o f 1 — 1 + 1 — 1 + •• • the partial sums are 1,0,1,0,1, . . . and we reach the same value Loth these m ethods are consistent, i.e., when applied to convergent series they produce the sum in the ordinary sense. T he behavior of series all (but a finite num ber) of whose terms are of one sign is particularly simple. We begin with this case, then consider alternating series, and conclude with scries o f real terms whose signs are arbitrary. S eries w ith P o s itiv e T e r m s . A series o f positive terms must either converge or diverge to This follows from the remark on m onotone se­ quences. T o prove that such a series converges,

1 -2 4

MATHEMATICS

it is sufficient to show that its partial sums are bounded above. Thus since /¿! > 2“-1 for n = 1, 2, . . . it follow s that the partial sums of 2 ( / i ! ) _1 do not exceed those of 22 n; since those of the lat­ ter series are ob viou sly bounded b y 2, so arc the former. It follow s that 2 ( « ! ) _l is convergent. This criterion is not always convenient. M an y other practically useful tests for the convergence or diver­ gence o f series have been devised. Cauchy's Test. 2 x n is convergent if lim {x,i)l/n < 1 and divergent if lim ( x „ )1/n > 1. 1У AlemberCs Test. 2 x ri is convergent if lim ( x H/xH+i) > 1, and divergent if lim (x„/ xH+i) < 1. W hen the limits are actually unity, m ore refined tests are necessary. Convergence follow s from lim (a*,,)1 n < 1; similar extensions are available for the other cast's. It is often more convenient to operate with integrals than with sums. For this reason the follow ing test is useful. For definitions of infinite integrals, see Sec. 4. Integral Test. If ф(х) > 0, x > 1 then the series 2ф (п) and

Ji

ф(т) dr

converge or diverge together.

For instance, since

is convergent for a > 1 and divergent for a < 1 , it follows that 2n~a is convergent- for a > 1 and diver­ gent for a < 1. Even inthe divergent case the difference between the partial sums o f the series and the corresponding definite integral has a finite lim it, for exam ple, in the case « = 1 using

This is not so even for 2 ( — l ) ”n l, for it can be rear­ ranged in the form

! 2

4

3

85

,

10

-J

12

L _ _ _ L + ...

2« -

1

-Ik -

2

An

which is convergent to the sum Yi log 2. A ny co n ­ vergent series which is not absolutely convergent can be rearranged in such a w ay that the resulting series has any prescribed behavior, for exam ple, converges to any assigned sum or diverges to -}- 00 or to — oo, or oscillates between any assigned lim its (possibly infinite). C onvergent series can be added and subtracted term b y term. In fact for any is convergent to the sum a A + /3/?. T he question o f multiplication o f tw o series is m ore com plicated. T h e product o f tw o series 2 a n, 2 b n can be arranged in the form 2c,,, where c,t = a j ) n + dibn-i +

• • • + a n-\b\ + \ d‘2 + dn- 1+ dn/ Periodic continued fractions such as

! + — — — — — ••• 1 + 2 + 1 + 2 + 1+ represent quadratic surds, in this case \ /3 , and con ­ versely. Elaborate theories have been developed. One im portant application is in the representation o f the characteristic values o f M athieu functions (Sec. 12). A nother nontrivial exam ple is tan x

1

x2 x2 x2

1^ 3 ~ 5-

7^

1 -2 5

o f f ( x ) and /(?/), the function is said to have an addi­ tion theorem, for exam ple, tan {x + y) =

tan x - f tan y 1 — tan x tan y

Functions are often represented graphically. For this purpose various coordinate system s can be used. T h e m ost usual are the rectangular cartesian. T he graph o f the inverse function is then obtain ed b y interchanging the x and y coordinate axes. L i m i t s o f F u n c t i o n s o f a C o n t i n u o u s V a r ia b le .

W e have already considered the limits of functions o f a positive integral variable, n. Similar definitions and theorem s are available for the limits of functions o f a real variable, x. For exam ple, .. sin x Inn ------- = 1 x—► () X

lim (x — I ) “ 2 = + oo

x—*1

lim

2.

Keal F u n c t i o n s

D e fin it io n s a n d E x a m p l e s . A law which defines a correspondence betw een the individuals o f two given sets o f num bers is called a function. D enote a typical num ber o f one o f the sets b y x , the w hole set b y A" = |x), the num bers that correspond to x in the second set V b y ?/. If to every x t there corresponds exactly one value o f y , then y is called a single-valued fun ction o f x , frequently denoted b y y = } ( x ) . T he function m aps A” on Y. A m any-valued function o f x is one w hich assumes m ore than one value for one value o f x. T he set o f all values o f x for which y is defined is called the domain (of definition) o f the function. Since x stands for any element in the set X = \x], it is called a variable. So is y, which stands for any elem ent in the range (of values) o f / : x is called the independent variable (or argum ent) and y the dependent variable. A nalogously functions fixi,Xu, . . . ,Xn) o f several independent variables are defined. A function m ay be given through a m athem atical form ula or through a table. If the m athem atical relation which defines the function is of the form y = /(:r ), the function is said to be given explicitly. If the relation is defined b y an expression o f the form f (x ,y ) = 0, then the function is said to be given im plicitly. If y is a function o f x , then x is some function o f y — som etim es denoted b y x = f ~ l(y), the so-called in­ verse function with respect to the original one. If x = F (y ) is inverse to y = {x), then y = is inverse to x = F (y ). An inverse function need not be unique; for exam ple, the inverse o f y = tan x, which is denoted b y x = arctan ?/, is infinitely m an y­ valued. G iven any particular determ ination o f arctan y, say the so-called principal one denoted b y Arctan y which satisfies — A l ir < A rcta n ?/ < A n , then iiir + Arctan y is also an inverse function o f tan x. Som e functions satisfy functional equations, for exam ple, the function f i x ) = ax, where a is a constant, satisfies the functional equation: f ( x + iy) = f (x ) - f /07) T he function f ( x ) — ax satisfies f ( x ) f ( y ) = f ( x + ?/). If f ( x + y) can be expressed as a rational function

x—>4*30

x"c~x — 0

for any n

In certain cases we must specify in what w ay the argum ent approaches its lim it: for example*, x ~ l has no limit as x —> 0 unrestrictedly, but if it approaches x through positive values it has a limit + < » , while if x approaches through negative values it has a limit — oo. T he follow ing notation is used to indicate these situations: lim

lim x ~ l = - f 00 r->0 +

c—*0 —

x ~ l = — oo

An im portant case is that o f infinite integrals dis­ cussed in Sec. 1. Th e follow ing notations are convenient. W e shall say that f ( x ) is o f smaller order than (x), for exam ­ ple, as x —> oo f if lim

fix )

= 0

In these circum stances we write / = oi)

X

> oo

For instance, x 2 = o(cz), x oo and x~ 2 = o ( l ) , x —* oo. W e also write \x/(x — 1)1 = 1 + o ( I ), x —> oo . W e shall say that /( x ) is of the same order as (x), for exam ple, as x —> oo ; we denote this by / = 0(4>), x —» oo. T h us we write [x/(x — 1)] = 0 ( 1) as x —> oc , sin x = 0 ( x ) as x —> 0. T h e standard function with which we com pare / i s usually a m on oton e function tending to oo, or to 0. T h e notation is also applicable to functions o f a positive integral variable: + - = 0 (log n) n

n-

or to functions o f a com plex variable. ( 'o n t i i i i i o u s F u n c t i o n s . Let. f i x ) be defined in an interval. It is continuous at a point c in this inter­ val if w henever x n * c th e n /(.r„) —> /( c ) , or, expressed arithm etically, when corresponding to any e > 0, no matter how small, there is a 5 = 5(e) > 0 such that when \x — c\ < 5 then |f i x ) — /(c)| < e. If a

1 -2 6

MATHEMATICS

function is continuous at all points o f a closed inter­ val, it is boun ded in that interval and it is also uni­ form ly continuous there, that is, we can choose one 6, independent o f x , such that if |* — x\ < 5 then l/(£) ~ f ( * ) \ < « for all x, ¿j in the interval. It is easy to see that these properties do not necessarily hold in an interval which is not closed, for exam ple, x _1 in the interval 0 < x < 1. As exam ples o f points where a function is not con ­ tinuous we mention the origin in the case of the fun c­ tions f i x ) and ( j { x) defined by fix ) = and

C +

1*1

x *

g(x) = sin x i

j (x — b )(x — e )(x — d)

+

x ^ 0, j/(0) = 0

ft (* ~ (l^ x ~~

(a — b)(a — e)(a — d)

~~

(b — a)(b — c)(b — d)

(x — a) (x -

b) (x — d)

(c — a)(e - b)(c — d) + I)

(x + l) (x -

l)(i -

2)

/(0 )

0, /(0) = 0

This concept o f con tin uity is far from being as strong as the intuitive idea o f a function whose graph can be draw n; there are, indeed, continuous functions which are nowhere differentiable. The theorems o f Sec. 1 a b ove about limits of sums, products, and quotients o f sequences give rise to similar results about continuous functions. Using them , and the obviou s fact that / ( x ) = x is a con ­ tinuous function o f x, we can deduce that x r is continuous for any integral value o f n} that any p oly ­ nomial 0 {)Xr + a ixr_1 + • • • + flr-iX + (ir is con ­ tinuous, «and that any rational function (i.e., quotient of two polynom ials) is continuous except at points where the denom inator vanishes. The follow ing theorem asserts the existence o f an inverse function in certain circum stances: If F (y ) is a function o f //, continuous and strictly increasing in the neighborhood o f y = b, and F(b) = «, then there is a unique continuous function y = 4>{x), such that («) = f>, which satisfies F (y ) = x identically in the neighborhood of x = a. It is im portant to note this merely asserts the existence o f an inverse function locally. See also Sec. 3, the special case of a power series. If a function is not defined for all points in an interval (e.g., if its values are only given b y a table), it is o f interest to find a continuous function, prefer­ ably a polynom ial, which coincides with the function wherever it is known. T h e construction o f such a function is called (polyn om ial) interpolation. T he problem o f polynom ial interpolation is solved b y the Lagrange formula. The polynom ial

+ C

coincides with / ( x ) at the n + 1 distinct points Xo,xi, . . . ,x„. In the cases where the x, form an arithm etic progression with com m on difference 1, the coefficients o f / ( x t) in L n(x) have been tabulated for varyin g x so as to make available a practical m ethod o f interpolation. As an exam ple consider the case a — — 1 ,6 = 0, c = 1, d = 2; then

(x

a )(x

b )(x — c)

id — a)(d — b)(d — c)

assumes the values .1, I f (\ I), for «arguments a, b, r, d, respectively. "Flic* general form ula is the follow ing. Let Wii(x) = (x - x f)(x — X i ) • • • (x — X,i)

— (x + 1 )x(x

In particular, for x = A , =

1 tf>[ - / ( -

1)

+ 9 /( 0 )

+

9 /(1 )

D iffe r e n t ia t io n . Suppose / ( x ) defined neighborhood o f x = x 0. If the limit

L Jx) =

(x

-

_

Xi)œ'n(Xi)

the

exists, it is called the derivative or differential co­ efficient o f /( x ) at x 0; the lim it is variously denoted b y / ' ( x 0), (df/dx)xo, D f(xo). Unless otherwise stated, it is understood that the variable h is unrestricted (except b y h ^ 0). In special cases one can dis­ tinguish betw een right and left derivatives, for exam ple, f ( x ) = lxl, where / ;+(0) = l , / '_ ( 0 ) = - 1 . A necessary con dition for the existence o f a deriva­ tive at a poin t is the con tin u ity o f the function at that poin t, but this is not sufficient. F or instance, f i x ) = lx[ is con tin uou s b u t n ot differentiable at x = 0. If f i x ) is differentiable at all points in a neighbor­ h ood o f xo, we can ask whether / '( x ) is differentiable at Xo. If it is, the derivative o f / '( x ) at Xo is called the second derivative o f /( x ) at x 0 and den oted b y / " ( x ) , id2f/ d x 2)xo, D 2f i x 0). In the same w ay higher derivatives can be defined. T h e nth derivative will be denoted b y f n or D nf. When n — 0, these are interpreted as / itself. It is not difficult to construct .functions (of a real variable) which «are differentiable n times at a point x 0 b ut which can not be differen­ tiated n + 1 times. Th e follow ing general results are available, where / and g are differentinble functions of x and a , (3 are con stants: ( « / + fly)' = otf' + (i(*0 + o,h)

where 0h 02 are num bers between 0 and 1. A special case o f T a y lo r’s theorem is Maclauriri’s, where x 0 — 0, h = x: fix)

= / (0 ) +

+ / '(( )) +

^ / "(0 ) +

+

in

■ . •

- ~ r

^ / ”-1(0) + Rn

where R n = x nf niOx)/n !, 0 < 0 < 1. If f ix ) is indefinitely differentiable and if it can b< proved that 7?„ —* 0, then (1) the series 2 x n/ (w)( 0 ) / « is convergent and (2) it converges to the correct, sun

1-2S

MATHEMATICS

f { x ) . Statement (1) does not im ply statement (2). This is shown b y f i x ) = exp ( —x " 1), x ^ 0 ,/( 0 ) = 0, for which / (M)((J) = 0 for all n = 0,1,2, . . . so that the series 2 x " /n(())///! is identically zero, whereas /(()) ^ 0 except at x = 0.

where the B n are the Bernoulli numbers ju st intro­ duced. /m • , 1 x3 1 •3 x 5 , (9) arcsm x = x + - — + - — - — +

23

1*1 < 1

2-45

P ow er Scries fo r E lem en tary F unctions X

all x (2) ln (1 + x) = x -

2n + 1

+

•)

+e>'

all x

+

(7)

+ (-1)*

(2/ 0 !

all x + tan x = x + A x 3 + H r)x 5 + 174 i s * 7 + ••• 1*1 < 2*-

T he coefficients in this series are expressible in terms o f the Bernoulli numbers Iin. T he coefficient o f x 2n_1 is ( _ l ) n 122«(22» - l ) / i 2n

(2/0!

H0 = 1

£

( - D—

Ihn +1 = 0

1*1 < A tt

W e have (1 + x 2) -1 = 1 — x 2 + x 4 — • • • . In te­ grating this over the interval [0,x], we obtain (see See. 4) arctan x — x — A x 3 + 3\->*5 — • • • (5) y = sin (in arcsin x) Since y' = cos (tn arcsin x) X m (l — x 2)_1/2, we have (1 — x 2) 0 /(1)) 2 = m2(l — y 2). If we differentiate this n + 1 times b y L eibn iz’ theorem , (1 - x 2)// = 0

Putting x = 0,

(ill2 - n2)?/(0)

in > 0) . , . . m (in2 — l 2) sin (m arcsm x) = mx — --------x3

- : (2” )! y ± (2tt)2'‘ u k2“ ¿=1

( « > 1) ~

. m (m 2 — l 2)(m 2 — 32) + --------------- 51--------------*

. A m(in2 - l 2) . sm mO = m sin 0 ------------ —-------sin3 0

!»)(,»» - 8»)

_

5!

9 15 ’

In another fo rm : X 1U. r > x ) 2n

( - ' )m '(2,0! ‘

, --------------

If we put x = sin 0 in this, we obtain

m(m* — 1x - —1 • 3 Tr)

Z

all x

By successive use o f this, beginning with ?/(0) = 0, ?/(1)(0) = ///, we obtain

(8) It is not possible to differentiate cot x at the origin and thus it is not possible to express cot x as a power series in x. H owever, cot x — x _1 can be expressed in this w av: cot x — 1 = x

•••

b ^ lo * 7

y = arctan x

n*2)(()) =

The Bernoulli numbers an* defined as follow s:

1

>ä*3 + K 5 *5 -

T h e coefficients o f this last series are the same as those for tan x except for the signs. T he evaluation o f / u ) (0) is generally not the simplestw ay to obtain the form al pow er series representing a given function. T w o other m ethods are the follow ing: in)

.j«2n+1

— +

(2»)!

+ ••*

• ♦ • + ( - 1 ) " -—----------- (2/i + 1)!

all x

x:

+

+ (13) tanh x = x

1*1 < 1

'j*3

(G) cos x = 1 -

X“

1*1 < 1

For a definition o f the coefficients, see Chap. 2, Sec. 1. +

w< 1

b •••

(2/, + 1)! + •••

•••

*r +

3!

X

/

+ 7.r

(12) cosh x = 1

(4) (1 + * ) » ■ - 1 + ( * ) x + ( * ) *» +

/y

—

5

1*1 < 1

+

® 1,1 ( i ^ f ) - - ( * + ! + f +

sin x = - 1!

X

a rccot x = arctan x~] = A n — arctan x

+

(5)

1 3

^

+

X

(10) arctan x = - — — H

W
0. This is known as a principal value integral: PY

[ b b —c / (x — c) 1 dx = l o g Ja c —a

G e n e ra liza tio n s

of

th e C o n c e p t

o f Integral.

In addition to the im m ediate extensions o f the con ­ cept of integral already introduced there are m any o f interest. T h e Stieltjes integral o f a fun ction f ( x ) w ith respect to a weight function ?r(:r) is defined as J b f ( x ) dw(x) = lim

-

w’ (o\_i)]

which exists as an im proper Riem ann integral with the value J^tt, b u t this docs n ot exist in the Lebesgue sense, for it is a fundam ental property of the L cbcsguc integral that if / / ( x ) dx exists so does /|/(x)| dx} w hich is easily seen to be im possible in the present ease. N um erical Q u a dratu re. T h e numerical evalu­ ation o f definite integrals, where the integrand is tabulated, or where the indefinite integral is unknown (or very com plica ted) can be accom plished in various w ays. One o f the sim plest is Simpsojds rule:

J f " '‘/ (x )d x = H h

( /( < > ) + / ( 2 / i /»)

+ 2 | /(2/t) + / ( 4 / 0 +

• • • + / [(2 ,1 -

+ • »{/(*) + f & h ) + where the lim it is taken as the m axim um length o f the subintervals (x£_i,ari) tends to zero, and where, as before, $,-_i is a point in this interval (a*r_i,^»)- T his integral exists w h e n e v e r / is continuous and w(x) is a m on oton e function. T his type of integral is often used in statistics. It reduces to the Riem ann integral when iv(x) = x and when w(x) has a continuous derivative, it can be evaluated as the Riem ann in­ tegral f b I f ( x ) w ' ( x ) dx Another im portant t3rpe of integral is the Lcbcsguc integral. W ithout this a satisfactory account o f the theory of Fourier series, for instance, is im possible. T o introduce this we essentially subdivide the range o f values assumed b y f ( x ) for a < x < b, instead of the interval [a,b]. W e then consider sums o f the form

s =

'Z iji- im

S

-

Sy/igi

where is the measure o f the set Ei o f points x for which thef i x ) assumes values between ?/*_iand ?/t. In the ease when f i x ) is continuous, Ei consists o f a se*t of intervals on the x axis and then the correspond­ ing measure is the sum o f the lengths o f the intervals. T he definition o f (Lebosgue) measure applies to much more general cases, but not every set is measurable, i.e., has a measure in this sense. I f / i s bounded and is such that all the sets Ei are measurable, then the upper bound o f the sums s and the lower bound o f the sums S coincide. T h is com ­ mon bound is called the Lebesgue integral o f / . In considering the integration of a function / in the present, sense, we can therefore alw ays disregard the values id f i x ) on a set o f measure zero, since t hese make no contribution to the sums s,S. 'Phis im plies that we can integrate such functions as the one described at the end o f Sec. 2 a b ov e; for if we disre­ gard the values at the rational points, which form a set of measure zero, the function becom es (essentially) constant and is certainly integrable in the Lebesgue sense.

2)/i) |

• • • + /1 (2 » -

D M ))

T his is derived b y approxim atin g f ( x ) in each interval (0 , 2 / i) , ( 2 /i, 4 / i) , . . . , ( ( 2 i i — 2 j/i, 2 nh) b y a parab­ ola y = a x 2 + bx + c. T h e error com m itted is estim ated as F = - n h sf " ( t ) 90 where £ is som e point 0 < £ < 2nh. If / is given num erically, or if / iv is com plicated, we can estim ate h4f ,v(£) as the fourth difference. In oth er w ords, the error is a b ou t — - X (length o f range o f in tegration) X (a mean 180 o f the fourth difference) It is som etim es more convenient to use a m ore pow er­ ful form ula and fewer points o f subdivision. One form ula is Weddle’s rule, which is appropriate when Iho num ber o f points o f subdivision is a m ultiple o f G: j ^ ' / U) dx =

[ / « » + 5 /(1 ) + / ( 2 ) + 0 /(3 )

+ / ( t ) + 5 /(5 ) +/((> )! T he error is a bout ^ (length o f range o f integration) X /lve*s on ly elilTe*re*ne*e*s e>f the* function which ran be com pu ted fe>r values within the range o f integration. (T h e central-

1 -3 5

ANALYSIS difference form ula requires the estim ation of differ­ ences not directly available.)

f ' f f (x ) dx

=

+

/1C 2 /0 + / , + a

—

-

i 2., a =

/ 1/ K

2Y

-f

•••

+ /r -1

+ « » 720 a 3 12 4 V 2 -f

+

H fr )

• • •)/.»

1 ;* 7 2 o Y 3 +

can be interpreted as a volu m e: the volum e bounded b y the surface; 2 = f ( x , y ) , the plane 2 = 0 , and the lines parallel to the 2 axis, through the boundary of the region 8 over which integration takes place. It, can be shown that (usually) when a double integral exists so do the repeated integrals

• • ’)fr

f dx\ff(x}y) dy\, J d y [ff(x,y ) d x | Th e succeeding coefficients are 3 /1 0 0 , 803/00,180, 2 75/24,192, . . . . Here the first line on the right is the first approxim ation to the integral— the tra­ pezium expression— which is corrected b y the second line which involves the first available forw ard differ­ ences and b y the third line which involves the last available backw ard differences. As an exam ple we consider j / /2 (1 -

z 2) 1' 2 dx = | \ / s +

-

x 2) »

I = -2 5 1 1

998749 904987

0 .1 5

988686

0 .2 0

979796

0 .2 5

968246

-2 8 -5 0 -2 5 8 9 -7 1 -2 6 6 0

-2 8 8 2

-2 9 -1 6 4

0 .1 5

803029

-1 2 -4 1

-3 0 4 6 -2 0 5

-2 0 2 3 5 916515

-3 2 5 1 -2 3 4 8 6

d x d 'J

exp ( - ? / 2) dy = / „ 2

f a / exp ( —x 2) dx

Ia =

•Ja = i f exp ( —x 2) exp ( —y 2) dx dy where the integration is to be over the first quadrant of the circle o f radius a. T o handle this, it is con ­ venient to change the variable to polar coordinates. y = r sin 8

T h e element o f area is now enclosed betw een tw o circles, radii r, r + 5/*, and tw o radii at angles 8, 6 -}- 58. It has area r 88 dr and

—2I| -6 2

/■ * /2

fa

•’ « = J t - o J r - O ™ » ( - r^ r d r d 0

" -2 6 7 =

-3 5 1 8 |-27004"|

0 .5 0

Jy = 0

x = r cos 8 -1

-1 2 5

-1 7 1 8 9

( ~ y2)

f y —a

exp ( —x 2) dx /

-2 8

-1 4 3 0 7

0 .4 0

oxp

-2

-9 7 -2 7 5 7

936750

-5 -2 6

-1 1 5 5 0

953939

+ l|

-2 1

-8 8 9 0

0 .3 5

=o cxp

Consider

— 22 1

-2 5 3 9 -6 3 0 1

0 .3 0

/

Jx = 0

where

-3 7 6 2 0 .1 0

0

where the. integration is over the square with vertices (0,0), (0,a), (a,0), (a,a). T h e limits in the correspond­ ing repeated integral are constants, and it follow s from the definition that the integral can be expressed as the p rodu ct o f simple integrals:

| -1 2 5 1 0 .0 5

fx =

fx = a

1I .000000

0

1 =

= 0.-178300 . . .

We take h — 0.05 and work to 0 /). T h e differences in volved in the correction terms are included in the boxes. x

and all three arc equal. This enables calculation of double integrals to bo reduced to that of simple ones. T h e main difficulty involved is usually the determ ina­ tion o f the appropriate limits of the simple integrals. As an exam ple o f the evaluation of m ultiple in te­ grals consider

//J o [ /rlo oxp (- r')r ,lr] '10

= f i l l I 'M ~ exp ( “ a2)ll d0

866025

= J [1 -

H ence

exp ( - a 2)l

f 0.5

/ + + + “ =

(1 - x 2) i'* dx = 0.05[0.500000 + 0 .9 9 8 7 4 9 + • • • 0.893029 + 0.433012 V\2 (-0 .0 0 1 2 5 1 ) 1/*720 (-0 .0 0 0 0 2 8 ) Vi2 (-0 .0 2 7 0 0 4 ) 19-120 (-0 .0 0 0 2 G 7 ) 0.478306

J a

- y lAt ( -0 .0 0 2 5 1 1 ) - ? i « o ( -0 .0 0 0 0 2 2 ) + - H 4 ( -0 .0 0 3 5 1 8 ) - /1 go( — 0.000062) -

J f f ( x , y ) dx dy


2 u 2)

COS J ' i x 2

2 " » (cos >2 u 2 +

\!~ xt— *7r 1 + U2 1 +

\Z2tt

Jo(t) (It = I x exp ( ~ y j x 2)

/ ;

a result n ot con venien tly obtained directly. These transform s have certain reciprocity prop­ erties, for exam ple, if P\(u) is the sine transform o f f(t), then / ( { /) is the sine transform o f Ea(t). In the com plex case we have the follow ing: 1 F {n) = - 4 = 1

then

f(x) = — —

f +°° iut d t / m ee im J ~ ÇO f + 00

I

E(i/)e'tx r ixu

d ll

This depends on the Fourier double integral form ula f(x)

1

f 4-00 f + 00 — — / e~ixu du / f ( l ) e tut dt

2 t t J - 50

J - 00

sin > j u 2)

u\

1 - U|

u exp ( - > a'u 2)

\l? - L _ ^7T 1 + X2 t M o r e ex ten sive tab les of Fourier tra n sform s can he fou n d in refs. 2, ft, 15, and 25 a t the end of the ch ap ter.

6.

F u n c t i o n s o f Several Heal V a r ia b le s

F u n c t i o n s o f T w o Heal V a r ia b le s. Partial Derivatives. A real function o f two real variables x,y is defined b y a correspondence between certain pairs o f real num bers x ,y (which m ay be interpreted as points in a plane) and certain real num bers. It can be represented graphically as (part o f) a surface

M ATH EM A TICS

1 -3 8 z = /(x,?y ).

Functions m ay be given explicitly

valid when f xJ v are continuous in the neighborhood o f in,b). M ore generally, assuming that / and all its partial derivatives o f order 1,2, . . . ,n 4* 1 arc continuous, we have

as

2 = sin (x2 + yy2), or 2 = (1 — x 2iy2), or im plicitly as

x 2 + 1J2 + 22 = 1 or num erically b y a double-entry table. We can define con tin uity o f 2 = /(x,?y) at (x«,//«) requiring t hat /( x ,//) shall be near / ( x (l,//«) when (x,/y) is near (x0,//o), that is, when (x — x 0) 2 - f (y —?y)2 or when |x — Xo | -F IH ~ lh\ issmall. This condition is m uch more restrictive than the con dition o f being continuous with respect to x for y — ?/o and being continuous with respect to y for x = x 0. For exam ­ ple, /(x,?y) = 2x y / ( x 2 - f ?/2) is constant for x = 0 or for y — 0 and therefore continuous with respect to each variable separately at (0,0) but, if we take 7/ = mx we have fix ,nix ) = 2 m /( l 4~ oi2) and so there are points as near as we please to (0,0) at which /(x,?y) has any value betw een — 1 and 1. We say that the partial derivative o f f ( x , y ) with respect to x at (xo,?/0) exists if

/ ( a 4- h, b 4~ k) n = / ( « 1, /0 +

where the powers o f the operators are interpreted as partial derivatives, evaluated at (a,b) and where U n+1 = - - - - - h in 4- 1)' L ox

W e denote, this limit by

A ” fxu“

Sim ilarly we define the partial derivative f y with respect to y. R epeated derivatives i f x)x = fxx, (fx)u = fxu, ( / „ L = /„* , (/,,)./ = fyyy • • • axe defined in the manner indicated. There seems to be dis­ agreement 011 this notation. Som e authors detine =

. p. M oreover, p = 1/lim (|rt»|)1/w; and p = lim (|o„|/|att+i|) if this limit e'xists. In virtue of this absolute convergence and the result on m ultiplication noted earlier, it follow s that two pow er series can be multiplied term b y term and rcarninged to give a new pow er series convergent to the corre*ct produ ct in the interior of the smaller of the two circle's of convergence. T he fam iliar serie*s for ex, sin x, . . . have radii o f con vergen ce p = oo, that for log (1 T x) has radius of convergence 1, while the scries 2w!.rn has radius of con vergen ce zero (it converges on ly for x = 0 ) . A pow er series with p > 0 defines a function which has an inverse function which can itself under certain circum stances be* represented as a pow er series. Spe­ cifically, suppose w = f (z ) = a\Z + «>~2 + • • • is cem vcrgent in \z\ < r anel that 9^ 0. Then there* exists a uniepte function 2 = (///), expressible as a pow er series 2 = b\w + b»w2 + • • • convergent in a certain circle* \ir\ < s and satisfying fU> M \ = W The* coefficients bifii, . . . can be expre*sse*d in terms e»f the* e/j,r/2, . . . ; in particular, b\ = rtj"1

b> =

— a*

=

2 (1^ — rz3 • • •

The* follow ing re'sult, elue* to Abe»l, is ofte*n useful. Suppose that is e*onverg(*nt. The*n ~(tnXn is cemve*rge*nt feir \x\ < 1. Denote* its sum by .-l(.r). A b el’s th('orc*m then state's that lim x »1 A ( j ) exists anel has the* value I n ilo r m i !on v e r g e n c c . The* ne*w epie'st ions which arise* in this se*ction are* ones e'e»nce*rning the con ­ tinuity, elifferent ¡ability, e>r integrability e»f the sum function assuming that individual terms are con ­

ANALYSTS tinuous, differentiable, or integrable. T he follow ing exam ples imlieate some o f the possibilities. Example 1. X) ^ x 2(l + x 2)~ n = 1 + z 2 = 0 Here each term is a eontinuous function, but the sum function is not eontinuous. Example 2. -

0 < x < 1

(n + 1 )'

T h e differentiated series is 2(jr“ '-1 — x n), which has sum 1 for x ^ 0 and sum 0 for j* = 0. Example S n{x) = n 2xe~ nx Jo

S,t(x) dx = =

S n(x) —> S(x) = 0

0 < z < 1

[ nxe~HXn dx Jo

J j X te~l

tc'* dt = 1 ^

JQ

S (x )

dx

T h e main new con cept required is that of uniform con vergen ce. T h e series Z /in(z)t convergent to n{z) for z in a set A , is said to be uniform ly convergent in A if given any e > 0 th en ' is a num ber AT inde­ pendent o f z (but dependent on e) such that N

u(z)

(z) I

1 -41

A power series 2 a Hzn with a radius of convergence p 9^ 0 can be differentiated at any point inside its circle of convergence and integrated over any range inside this circle. Sufficient conditions for tin* convergence or uniform convergence of Fourier series, or of other orthogonal series can be found in the literature. T h e on ly general result we m ention is the fact that a Fourier series can always be integrated term b y term. M u lt ip le S eries. A dou b ly infinite sequence Sm.n of com plex num bers (possibly depending on a param eter) is said to be convergent, as mpi —► to a limit S if, given any e > 0, then* are num bers m — //i0(i), n 0 = a 0(€) such that -

S\

< e

if m > Mo, n > no. If the choice1 o f M0,/fo 0 (for sufficiently small t). The follow ing theorem s are often sufficient to ju stify m anipulations with series. Theorem 1. u (x ) = ~ u H(x) is continuous for x in A if S ii„(x ) is u niform ly convergent in A and each ?/„(z) is continuous in A. Theorem 2. u n( x ) is uniform ly con vergen t in A and may be differentiated term b y term if each term n tt(x) is differentiable in A and if the differentiated series 27/'„(z) is convergent at a single point in A. Theorem 3. ?/(z) = 2 a M(z) may be integrated term by term, that is, b 11(x) dx = i

a

v n(x) dx

/ .

if the series 2m„(x) is uniform ly convergent in the interval \a,h].

= then

lim

(M

If — 71)

(m + w)

S m,n does not exist but

i n , n —> »

m —n ( lim V/Í—» X) m + n m — n lim |( lim " )/ m + n v-* » lim

=

Th e definition of the sum of a dou ble series is re­ duced to that of the limit o f a double sequence thus: Hi

ft

lim VI, ttl —i

A

l

I

w

" x- 0 = 1

*

1

^

Corresponding to the repeated limits we have the sums by rows and the sums by colum ns o f the double series. T he exam ple given a bove can be m odified to show that the sum b y rows and the sum b y col­ umns can be different; other exam ples show that even when they are equal the (proper) double sum m ay not exist. H ow ever, the follow ing result is available: if all terms of the series are positive and if one m ethod of sum m ation (by rows, or by colum ns, or by diagonals) gives a finite sum, the series has this value for its proper double sum. T h e same remains true if the series is absolutely convergent, i.e., if the series whose terms are \a\,v\ is convergent.

1 -4 2

M ATH EM A TICS

T he questions o f analytical m anipulation, e.g., passage to a limit, differentiation, integration of a double series of terms depending on one or more param­ eters, are often quite delicate. Very often, however, these processes can be justified by appeals to ob viou s extensions o f the M test (see a b ove). An im portant exam ple of a double series is the ease where a\,y — (z + Acoi + w j) ° , It can be proved that wl'ax.v is convergent if — a > 2, provided à (coi/co-*) 7* 0 and that z + X«i + ^ 0. A s y m p t o t i c Series. Another type o f series of con ­ siderable interest is asym p totic series. These ean be introduced as follows. Consider that solution y = u (x ) of x i f — x y -f* 1 = 0 which vanishes for x = + 3c. B y using an integrating factor, choosing the constant of integration appropriately and in te­ grating b y parts, we find У =

f * e* 1 Jx X

- ,

i

tit

+ ( - 1) »■ 1 r n\, or for T(•**)• In the (a bove sense it e*an be shown that In

when u ( x fy) and v(x,y) are real functions o f the real variables x and y . T h e definition o f differentiability of a function o f a com plex variable is the same as that for a real variable. W e require the existence of Inn h^o

f j z + h) - } {z )

where the approach to the lim it m ay be in any manner. M ore precisely, given any t > 0 there must be ail 5 > 0, such that for som e com plex constant 4 , which we den ote b y f '(z ), |/i| < 5 implies f ( z + h ) j - f (z ) _ A

< €

In particular if we let /i —> 0 along the x and y axes the lim its o f tin* iiiorem ontary ratio must exist, and be equal. From this follow s the fact, that if f { z ) = u fx,y) + iv(xty) then u and v satisfy the C auchy-R iem ann differential equations ux = vy u u = —vx From this it follow s that if / is differentiable at all points o f a neighborhood of z then u and v are har­ m onic, that is, satisfy the (poten tial) equation wxx + Wy„ = 0

■ 1 In 2ж + Пг 2 I 2 r + :i

¡U • \ X

in that n eighborh ood. It follow s, furtherm ore, that the curves u — const and v = const- cut orthogonally.

1 -4 3

A N A LYSIS These curves correspond to equipotentials and stream ­ lines or lines of force in certain realizations. A fun c­ tion /( z ) that is differentiable at all points of a region is said to be regular in it. In certain respects the theory o f harm onic functions and the theory o f regular functions o f a com plex variable are equivalent : any harm onic function can be regarded as the real (or im aginary) part o f a regu­ lar function. Indeed if u is harm onic in the interior o f a curve C\ then v =

lishes Cauchy7a formula: If £ is a point inside a closed curve w ithin and on which / ( 2) is regular, then

jc f ( z ) ( z

-

£ ) - ' dz = 2 * i m

From this it follow s that / ( 2) is indefinitely differ­ entiable.

( — iiy dx -f- ux dy) J(ro,yo)

is a one-valued harm onic function, the conjugate of u and u + tv is regular in /). The path o f integration can be any curve within C joining a fixed point to the current point. T h e fundam ental exam ple of a function regular in a dom ain is a pow er series ~ anzn in its circle of con ­ vergence. This can be established directly or a proof can be based on the fact that zn is differentiable and that a scries of functions, each regular in dom ain, which is uniform ly con vergen t there, is itself regular and can be differentiated term b y term. P r o p e r t ie s o f R e g u la r F u n c t io n s . It is essential to realize the follow ing difference between a real func­ tion F (x ) differentiable in an interval and a com plex function f (z ) regular in a region. In the latter case the existence of / '( z ) implies the existence of all suc­ ceeding derivatives / " ( 2), / " ' ( 2), . . . , while the existence o f F'(x) has no such im plication, for exam ple, if F ( x ) = x 2, x < 0 and F (x ) — 2 j 2, x > 0 , then F ' ( 0) exists but F " ( x ) does not exist at x = 0. One m ethod o f establishing this fundam ental property of regular functions is the follow ing. One first shows that if / ( 2) is a (com plex) function continuous on a curve C\ then the function

F 10. 3.3 It can then be shown that if / ( 2) is regular in a region l ) y then f(z ) can be represented as a T a ylor series about each point of D, convergent in som e circle, the radius o f which depends on the position of the point, w

n!

This is in contrast with the real case when the ex­ istence o f one derivative, or even all and the con ver­ gence of the T a ylor series, does not suffice to ensure that the T a ylor scries converges to the right sum. For exam ple, consider the function C(x) = e~x2} x 7^ 0, C(0) = 0. All derivatives of C(x) at x = 0 exist and arc zero so that the form al T a ylor scries is identically zero. W e notice also, as another con se­ quence of C a u ch y’s form ula, that a regular function is com pletely determ ined inside a closed curve, b y its boundary values on the curve. This is also true for harm onic functions. In particular, if C isthe circle |z — «| = r, we have the Poimson integral formula P(a - f pd*) = ~

J()

]>(a 4- pcie)

is a regular function o f £ for all £ not on (7, and its derivatives can be obtained b y successive differentia­ tion under the sign o f integration: /■'“ (£) = » ! j c f (z ) (z -

{ ) - » - ' dz

Next one establishes ('auehifs theorem: if F(z) regular within and on a closed curve C, then

is

1/2 “ p2 do r2 + p2 — 2rp cos (0 — 4>) Another consequence of C a u ch y’s form ula is Liouvillc7i> theorem: if f (z ) is regular in the whole plane .and bounded, then it. is a constant. W e prove /(«■) = /№ ) f ° r iiny W c have

Sc’F(z) dz = 0

sb M

In other w ords the definite integral o f a regular fun c­ tion is independent of the path w henever the paths form a d o se d curve within which the function is regular. T he last con dition is essential for, for exam ple,

_ a - h l

/

z~] dz — 7ri

] ACB

while

/

z~l dz =

- 7 ri

JADB

In fact, if C is the circle |z| = r, r > 0 (Fig- 3.3), then Jc2_1 dz = 2tri. Using this last result, one estab­

27ti

r - - : - 7 ^ ) /w f (z ) dz

J (z — a) (z — h)

where the con tou r integral is taken round a large circle, center Yi( x n o m a t t e r h o w 2 —> a. T h e r e are, h o w e v e r , o t h e r t y p e s o f s i n g u l a r i t y ; f o r e x a m p l e , 2 = 0 f o r / ( 2 ) = c l/* a t w h i c h t h e b e h a v i o r o f t h e f u n c t i o n is v i o l e n t ; in t h i s c a s e w e c a n f in d zn —> 0 s u c h t h a t j f{z„) | h a s a n y a s s i g n e d l i m i t a n d w e c a n fi n d p o i n t s a s c l o s e a s w e p l e a s e t o 2 = 0 f o r w h i c h / ( 2) a s s u m e s a n y c o m p l e x va lu e different from zero. T h i s p o i n t is c a l l e d a n isolated essential singularity: it is n o t a p o l e b u t t h e r e a r c n o o t h e r s i n g u l a r i t i e s in t h e n e i g h b o r h o o d o f 2 = 0 . T h e p o i n t 2 = 0 is a n e s s e n t i a l s i n g u l a r i t y f o r c o s c c 2 - 1 , f o r t h e r e a r e p o l e s at 2 = ( m r ) - 1 , n = 1 , 2 , . . . . W h i l e t h e T a y l o r e x p a n s i o n o f a f u n c t i o n is n o t a v a i l a b l e in i h e n e i g h b o r h o o d o f a p o i n t a t w h i c h it is n o t r e g u l a r , t h e r e is a g e n e r a l i z a t i o n , t h e Laurent

expansion, v a l i d in a n n u l a r r e g i o n s .

W e have, at a

p o l e o f o r d e r r,

/(2 ) = A_r(2 -

a )" ' +

+

A - i (2 -

a )“ 1 +

I f f (z ) — g(z)/h(z), w h e r e h(z) h a s a s i m p l e z e r o a t p a n d g{p) 9^ 0 a n d b o t h a r e r e g u l a r in t h e n e i g h b o r ­ h o o d o f z = p, t h e n

(2 )

r e s i d u e o f / ( 2) a t p =

Examples of Contour I nicy rations

l,

2n

t)

s i n 2 0 dO

2 tt(a — \ / a2 — b2)

a 4~ b c o s 0

b 2

cou ld

! = JQT

^

An(z -

^

J

-

—

+

1 ) 2 dz 2 az

w h e n * t h e c o n t o u r i n t e g r a l is t a k e n r o u n d c i r c l e C. Using the residue th e o re m ,

4>(z)

therefore be

|

s u m o f residues o f

(2 2 -

-b b)

the u n it

l)2

— 2 iz2(bz2 4 - 2az

4~ b)

a t its p o l e s i n s i d e C’ J

0

the p o le s in sid e C are a t

0,

rt)n

n= 0

4-

[-a

T h e sum

\ /(a

2

b2))/b

-

t h e f o r m e r b e i n g a d o u b l e p o l e , t h e lat t e r a s i m p l e o n e . W e use rela tion (1) a b o v e to ca lc u la t e th e re s id u e

r

I

22 — 2iz2(bz2 2i. (

ed0

a 4~ b c o s 0

W hen a > b >

b > 0

P u t 2 = eid, t h e n t h e i n t e g r a n d c a n b e e x p r e s s e d a s a r a t i o n a l f u n c t i o n o f 2,

= 2wi w h e r e 4>(z) is r e g u l a r a t a, a n d e x p a n d e d a s a T a y l o r s er i e s

h (p)

A -(z -

o )‘

[ — a 4- \ / ( z 2 — b2)]/b.

at p i =

I t is

s= 1 is c i il le d ( h e principal part o f / ( 2 ) ill a. T h e A 9 arc u n i q u e l y d e t e r m i n e d : A _ 1 is c a l l e d t h e residue o f / ( 2 ) at A . T h e f o l l o w i n g re s u l t , w h i c h is a n e x t e n s i o n o f C a u c h y ’s f o r m u l a , is o f c o n s i d e r a b l e p r a c t i c a l i m ­ portance. i f / ( 2) is r e g u l a r i n s i d e ( a n d o n ) a d o s e d

Ihn

-ose; the. evaluation of J(Z)

j _ f " x6 ” io x8 + 1 26 e/2 Here we consider ( D — > whe?re the con tou r is the J s* + 1 real axis from — /£ to /¿, and the upper half o f the sem icircle o f radius It, whore R is a (largo) param eter. T he absolute value o f the integral along the sem icircle does not exceed: ttR

It6 /¿8 -

1

and this tends to zero as H increases indefinitely. W e therefore have, as It--* co,

f+/e 7 z* + 1

1 -R

.r8 + 1

(mi,m2) for nil = 0 , ± 1, ± 2 , . . . , m 2 = 0 , ± 1, ± 2 , . . . as periods. T h e periods 2a>i, 2a>2, are called prim itive periods. ^ ( 2) has dou ble poles at all the points ) +

where the prime indicates sum m ation over all pairs of (integral) indices m i,///> except mi = 0 , m 2 — 0 , where = 2 mio>i + 2 /n 2co2 and where (toi/u>2) > 0 . Such functions are called meromorphic. A function f (z ) , such that for som e constant, to we have f(z ) = f ( z + to) for all 2, is said to have period to. For instance c2viz has period 1. It can be shown that, in general, any function with period 1 is a rational function o f c2wi: = uy for exam ple,

,

T o ?/ — r, v — cf correspond (parts of) the rectangu­ lar h yperbolas j*2 — y 2 — c, 2x y — c* (see Fig. 3.1r and 0 on to |?c| < 1 and the fourth the most, general m apping of 4 (2) > 0 on to #(w) > 0. M ost of these results can be established geom etrically. T h e case w = Vi{z 4- 2-1 ) is o f interest in classi­ cal aerodynam ics. T h e fact that w has a pole at the origin and that w f has zeros at + 1 make this cast; more com plicated. If we write 2 = reiey we find и — Yi{r 4~ ?,_1) cos 0, v — } 4 { r — r-1) sin 0 so that to a circle in 2 plane there corresponds an ellipse on the w plane, and the same ellipse corresponds to reciprocal values o f r. Confining attention to |z| < 1, we see that as r —* 0 the ellipse expands indefinitely becom in g more circular, while as r —►1 the ellipse shrinks to the line — 1 < и < 1. T h e behavior of |г| > 1 is similar. F or 0 < d < }4, the m ap of the interior o f the circle [2 4“ d\ — 1 — d or the exterior of the circle |2 - d{\ - 2d)~'\ = (1 - d){ 1 - 2d)~ x is a sy m ­ m etric “ a irfo il” having a zero angle at w = d. If we choose to m ap a circle with center not 011 the real axis, we obtain an unsym m etrical airfoil. Generalization of this Jankowski transformation, and the specialization o f the various param eters introduced, make possible the conform al m apping of airfoils o f realistic shape. S u p p o s e /( 2) is regular in J) and that / never assumes the saint; value т о к * than once in I). Then it can be show n that / ' ( 2) does not vanish in D . Let £) denote

MATHEMATICS

1 -4 8

tin; set of poin ts w = f ( z ) for z in D . T h e n / generates a o n e -to -o n e m a p p in g o f D on to 5). Let (w) den ote the inverse function of this m a p p in g . T h is is a regu­ lar fu n ction of tv in virtue of the existence o f

lim

= lim

U ' - ' U ’0

XV —

U'O

= h im i U U L * ) - * XV —

XV0

V 2 — 20 /

the critical stop follow ing since f ' ( z 0) ^

0.

[ / '( 20) ]^

T h e stu d y of fu n ction s } { z ) which are regular and o n e-v a lu e d in \z\ < 1 has been carried a considerable d istan ce. Such fu n ctio n s can be expressed as a pow er series w ith radius o f convergence u n ity , and it is con ­ venient to n orm alize them b y se ttin g / ( 0 ) = 0 , /'( ( ) ) == 1. T h e pow er series is therefore = 2 + Oj2 2 4- u32 3 +

• • •

T h e q uestion arises as to what, restriction 011 the coeflieionts are im plied b y our h y p o th e sis; it has been show n that | three? give*n points 011 the* circle \z\ =

2

T h e re is

therefore com p le te s y m m e tr y b etw een the b eh avior of / in I) and that of

« b e ­ c o m e s t h e e x t e r i o r o f v2 = — 4 « 2(?/ — o 2). Take = I; t h e n M ( 2) > I g o e s o n t o t h e e x t e r i o r o f v1 __ — |(,/ — i ) , a n d t h e r e f o r e o n t o t h e e x t e r i o r o f

v2 =

— 4m. u n d e r xv — z 2 — 1.

F urth er, with

xv — 2 2 — I sh a ll let 2 = 2 c o r r e s p o n d t o xv — 3 a n d , s i n c e ivr — 1 t h e r e , t h e p o s i t i v e re a l d i r e c t i o n s a t t h e p o i n t s

we

correspond. W e a r e left t h e r e f o r e w i t h t h e d e t e r ­ m in a tio n o f a (b ilin ear) tra n s fo rm a tio n w h ic h m a p s \z\ < 1 o n t o (11 ( 2 ) > 1 in s u c h a w a v t h a t 2 = 0 g o e s i n t o 2 = 2 , p o s i t i v e r ea l d i r e c t i o n s b e i n g p r e s e r v e d at t h e s e p o i n t s . Or, w ith the ch a n g e 2 = f 4 1 w e h a v e t o m a p \z\ < 1 o n t o ( R ( f ) > 0 in s u c h a w a y t h a t z = 0 b e c o m e s f = 1. T h i s is a c c o m ­ plished b y

ANALYSIS and, in order to preserve the positive direction, w c take 5 = 0 . Hence, collecting the results, we have

1 -4 9

If f i x ) has period 2tr and f i x ) = (ir — x)/2, 0 < x < 2ir then \ /( * )■

11.

Orthogonality

(General. T w o (possibly) com plex-valued fu n c­ tions / , defined on a real interval (n,6) (possibly infinite) are said to be orthogonal if

f 1*

I

( /,* ) where 4> denotes tem o f functions pair o f different to be normal, if,

f4>dx = 0

the com plex con ju gate o f 4>. A sys­ \n\ is said to be orthogonal if every functions art; orth ogon a l; it is said in addition

for all n . functions

Tt

CTr

where the cTare any constants. for sim plicity,

Taking the real case Tl

j n

^

4>n4>n dx = 1

T h e classical exam ple is that o f the set of

enix

k

W hen does the series ~ a nn converge to the sum fix)? If f { x ) and the system are reasonably behaved, the series will con verge to the correct sum. T he second exam ple a b ove indicates an exceptional case which occurs frequen tly: when f ( x ) has a simple ju m p at a point {x = 0, in this case) the scries co n ­ verges to the average o f the tw o lim iting values, — in this case. Even if the Fourier series does not converge, the Fourier polynom ials give the best inean-squarc ap­ proxim ation to f ( x ) am ong all the “ trig on om etric” polynom ials, i.e., expressions

Sn = (4>n,4>n) = f "

sin kx

L k= 1

i f — s«)2 dx = J

p dx + ^

Tt

|cr — ar|2 — ^ \ar}~

= o? ± \ t + 2 , . . . In particular, taking cT = aT so that sr = / r,

in the interval (0,2tt).

These form an orthogon al, b u t

not a normal set. T h e set einx/\/2Tr, n = 0, ± 1 , ± 2 , . . . , is normal and orthogonal. Given a norm al orthogonal system «, we define the Fourier coefficients an o f a function / with respect to \n\ and the interval (a,h) b y the equations an = (/,„ dx

In order that these should exist, / and n m ust be suitably restricted, e.g., it is sufficient to assume f and each n are o f integrable square. W e call the series

i

a nn(x)

the Fourier series o f / (with respect to M) and the successive partial sums f T o f this series are called the Fourier polynomials o f / . W e indicate this form al relation by

f ab i f ~ f » ) * d x = Jab p d x and thus

J

i f — f n) 2 dx < J h ( f — s n) 2 dx

with equality occurring if and on ly if each aT = eT (r « 1,2, . . . ,w), that is, the Fourier polynom ial gives the best m ean-square approxim ation. It is clear that the left-hand side is non-negative and this means p dx so that, if / is o f integrable square, wOr2 is convergent. An im portant property o f an orthogonal system is that o f com pleteness: {„ | is said to be complete if there is no (nontrivial) function orthogonal to each 4>n. It can be shown that if } „ | is a com plete normal orthogonal system , then if / is of integrable square,

oo

J

i f — fn) 2 dx —*■ 0

as n —> oo

Onn

№

and

(There will be no confusion with the sym bol for asym ptotic equality.) Exam ples in the classical case arc the follow ing: If f ix ) has period 2tt and f i x ) = [(tt — jr)/2J2, 0 < x < 2 jr, then

p dx

The latter result is known as IbirsevaVs theorem. Application o f it t o / + g and to / — g gives the follow ­ ing m ore general form : ^

№

^ a r2

7J-2

V

cos kx

12

L

k^~

a nhn =

J

fg dx

where b n are the Fourier coefficients of g.

1 -5 0

M A T H E M A TIC S

T h e f o l l o w i n g c o n v e r s e o f P a r s e v a l ’s t h e o r e m is true. I f t h e s e r i e s ~er2 is c o n v e r g e n t , t h e r e e x i s t s a f u n c t i o n / o f i n t e g r a b l c s(|iiare w h i c h h a s c r f o r its F o u r ie r c o efficien ts w it h respect to i ^ « } .

T h i s is t h e

Riesz-F ischer theorem. G i v e n a n y s y s t e m o f f u n c t i o n s ( 0 «|, it is p o s s i l ) l e t o c o n s t r u c t a s y s t e m o f f in it e l i n e a r c o m b i n a t i o n s o f t h e m w h i c h c o n s t i t u t e a n o r t h o g o n a l s y s t e m |/„ |. T h e p rocess ca n b e m a d e clear b y d iscussing the case w h e n t h e g i v e n s e q u e n c e is 1 , x, x 2, . . . a n d t h e i n t e r v a l is — 1 , + 1 . W e c h o o s e f\ = 1 . W e try / . = x a n d f i n d t h a t t h i s is s a t i s f a c t o r y s i n c e

(/1 , / 3)

/3

=

= x 2 and

J

h a,ß t o

and choose t h a t is,

ßh

satisfy

( / , , / , ) + « ■0 + 0 + / 3 ( / 2J s) +

and

+

(/i,/a )

= 0,

(/-.»,/ 3) = 0 ;

( f hx 2) = 0 ( / 2, x 2) = 0

and

dividing

each / „

by

a = — 1,6 = 1

Tn(x)

w(x) —

a = — 1 ,6 = 1 V i -

w (x) = e

a — 0,

w(x) = e

a = —

b —

00

H erm ite: 00 ,

b =

00

M a n y p rop erties are c o m m o n to a n y s y s te m o f o r t h o g o n a l p o l y n o m i a l s , e . g . , t h e f a c t t h a t all r o o t s o f s u c h p o l y n o m i a l s a r e re a l a n d s e p a r a t e d b y r o o t s o f the p re c e d in g p o ly n o m ia l.

Legendre P olyn o m ials

This g i v e s a = — }■$, ß — 0 a n d t h u s /3 — x 2— C o n tin u in g , w e t r y / — nf\ - f ßf* + 7/3 + x 3 and fin d /4 = x 3 — Y\X. T h i s p r o c e s s c a n b e c o n t i n u e d a n d w e o b t a i n a n o r t h o g o n a l s y s t e m |/„| w h i c h a r e essentia lly th e L e g e n d r e p o ly n o m ia ls . W e obtain a norm al orth o g o n a l system b y tak in g the j/„| ju st con structed

w(x) = 1

T ch ebych cf:

H n(x)

W e the re fo re t ry

= « /1 +

/\ .(x )

Ln(x)

fi nd th is is u n s a t i s f a c t o r y s i n c e

x 2 dx 7^ 0 .

Legendre:

L aguerre:

f +1 ( / „ / * ) = / _ , x dx = 0 W e try

P a r tic u la r ly i m p o r t a n t are the s y s te m s o f p o l y ­ n o m ia ls o b t a i n e d b y o r t h o g o n a l iz i n g w ith r e s p e c t to c e r t a i n ? c ( x ) , in t h e m a n n e r j u s t d e s c r i b e d , t h e s et o f p o l y n o m i a l s 1, x , x 2, . . . . W e s ha ll d i s c u s s t h e f o l l o w i n g c a s e s in s o m e d e t a i l .

the

Representations l~Tt [(jr* ~ 1)1,1 2'*n! 0

This function is regular for all 2 except 2 = 0 , — 1, — 2, . . . where it has simple polos with residue ( — 1) l/l\ at 2 = —I. It is easy to verify that

kK

an equation first studied b y C auchy. Graphical con ­ sideration indicates that this has an infinite set o f roots an —► cc and that an ~ nir as n —> oo.

dt

= z ]c y

K (cos air - f e sin air) = (a sin air — ca COS air)

T h is means that solutions are possible on ly if a is a root o f the transcendental equation

12.

f *

Jo

Y v (2) =

1 r\\'(r 4- r 4- 1)

1 . (cos vir.f v(z) Sill vir

- J „ ( 2)]

V ^ 0, ± 1, ± 2 , . . .

2

Y n (2 ) = - ./m(2 ) I 7 4- In m-

1 ir

V

1

L r 0)

(" -

r “ r\

1)! \2/

_ 1 v ir

¿—i

/’ ! ( / /

■ + I + . . . 1 2

f- /■ )!

+

J— )

r 4- n/

+ 1 r

n = 0 ,1 ,2 , . .

A N A LYSIS are independent solutions o f the differential equation z2e " + C' + (z2 -

1 -5 3

Spherical Bessel Functions.

= 0

p 2) Q

\ /

Thes^ functions are one-valued integral functions for n = 0,1,2, . . . , but for other indices are m anyvalued and have a branch point at z = 0, in view of the factor z v. T h e so-called Ilankel functions //„< » (* ) = J v(z) + i Y v(z), U ,™ ( z ) = Jy(z) -

t t/

x

J

v (x

)

v = ± ] 2 ( 2/1 4- 1), som etim es called Stokes functions, are independent solutions of the equation x 2y " + 2x y f 4- [x2 -

(c2 -

M )]// = 0

These are elem entary functions, for exam ple

iYp(z)

are also independent solutions o f the* Bessel equation.

provided |2 | »

|i/|, \z\ »

e,d constants

A ir y Integrals

4- sin ^

F 4- x/^ ] dt

1.

Zeros. J n(z) has an in finite set of simple zeros jn.i < j «.2 < * * * . It is known that j^.r —► and that j„.i > n.

are independent solutions o f the equation

Functions Related Bessel Functions

T h ey arise, for instance, in the theory o f diffraction. T h ey can be expressed in terms of Bessel functions of order ± Li mid ± Y i and of argument ri.r3/2.

to

Bessel

/„ ( * ) = e-y*i/*J¥(ix)

Functions.

Modified

K , ( jt) = H i r i c ^ ' - H p ^ U x )

These are independent solutions of the equation x 2y " + x y ' Kelvin Functions.

y " 4- x y = 0

Struve Functions

(x2 - f v2)l! = 0

T h e Bessel functions with argu­

ment r \ /7 or r y/i* occu r in problem s in electrical engineering. T h e follow ing are the usual definitions:

■

kei*, x = }^ tt her, x

—

P 2 ) /V

=

-

----------------

4- L 'H »

It ciin be represented as

In particular, we note that for x > 0 H p(z) = Jo(x \/F ■= ber x — i bei x

n (m , 4* L i O V 4- m 4- '}2 )

4- ZWr 4~ (z2

her* x ± i h civ x = 7/,,(1) [( — 1 ± i)x/ \ /2 ] heiy x

=0

is a solution of

b er„ x ± i bei„ x — ./„ [( — 1 ± O x / V ^ ]

k er, x = —

I

-

2 0 oz)v ^ “ —

f /

IV 4- l ->)r0 >) j {)

sin (,z eos 0) sin2»' 0 dx

1 -5 4

M A T H E M A TIC S

As in the ease of Bessel functions, the Struve fu n c­ tions are elem entary when v — n - f j *>, for example,

H\/Az) =

Orthogonality + 1 / V / V - . / x ---------2— 2,t + 1 (n -

/ . -1

j ~ (I — cos z)

n = n

m) !

= 0

n 7* n'

Recurrence Relations and

+ Fixed ///, 0 < m < n — 1,

Riccali-Bessel Functions.

These are defined as (2// 4- 1) j7 V

S .(x ) = y / } f a j „ , /t (x) C\.(x) = ( - 1 ) ’* v /'v ir x J - , . - l / 2(x)

Fixed / / , ( ) < m < n -

and are indepeiulcnt solutions of the equation x h j " - f x 2{n2 -

p um+2 + 2 (/// 4- 1) cot e l \ m+x 4- (n - m)(n 4- m 4- 1) P nm = 0

}4 ) y = 0

T h ey occu r in the theory o f scattering. For the standard work in this field, see ref. 25. Legendre F u n ction s. Properties o f the Legendre polynom ials have been already given. It was pointed out, in particular, that /\*(r) satisfies the differential equation (1 -

x 2) y n -

2 x y ’ + n(n 4- 1)!/ = 0

General Legendre Functions. W e have so far con ­ fined our attention m ainly to real argum ents, x arbi­ trary in the case o f P nm and — 1 < x < 1 in the case of Qn. T h e question o f the definition o f these quantities for general com plex argum ents and indices is rather intricate (see refs, G and 15). Error F u n c t i o n . It seems convenient to use the notations

.

- i

j

_2 —V i'

erf x =

An independent solution of this is