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la a eContents: 12.1.04 Mynotes onsome math subjects (some sections include book Xerox pages andweb pages) 1. Standard Distributions Section Gaussian page, BinomiaV/Poisson notes +Proakis notes +Proakis Xerox(ProakishasBinomial,Gaussian,Chi-square, Rayleigh,Rice)+selfnotes 2.Basic Probability theory (web) +Balls inBins v3.TheLevi-Civita Symbol 4.TheMethod ofStationary Phase (forevaluating integrals) +web 5.Selectedintegrals Remainder Function, Mod ete Gauss's lawlikeintegrals in8,6space. 6.MiseSinglepageitems[vtRem.55ANGE =04HiltsSpacejmulhiinkeqrale 5 7.Asymptotic Series (web) e 8,GaussianQuadrature 9,Mean Value Theorems +web 10.ThéSimplex Method (Linear Programming Theory) 11.Rubioer Sradrecube)byLaplace. 12,Leplacee Malhod frODES 1,Non-Lunean one's Colver) 14, 15. 16. 1. ® 1 _.lina(CassinSSisldodséee bos ‘ida a 4\ , FFL aye. QD =GD=Ye Avidiecepom® ge SR=P= 30) _DBs GATS Giteet ; LEY-42Tay aesranance froienenn. iaet a if Porgy . - 7Sample > Provfata." cRO Yom PloyA————_3“SN. aw yaa. 4 Derwab noe ati =BGYl=Alapaha] A=Be >§Q=Rey2+69 -Ma)ag*8H)ae &Rewme az. R= PO). BOD» _Roly) BOS)yLag" Laat amy” 20 GConoral Araubt: Y=96=¥0 Qherawnmssarovke!(5),K2l5)----Xm(y)-Tron me is aadonFrarpled. Sagwse y=xband ~¥/(er . Boo=ge 04 ThPals)=[Palifq)+Px(-3)\=RS) ay23 . =1.1ol) aellfareky Vary? ; =aSSham Ww(21-105), “ Frongt#3 a* 2« R= er ew) LES PD bhont:(21-139) Lt=ac)’SC)v5(2)moe:IQ Ase?6.1.34 + ntl a 4Spe WL=(ae*)(erya(p28? 4 rs)Stns{z65gketVLE9=(20°):nes)=2owa=note ray Qn! yLOG =(moSE(arol=MOD0 Amd =ot[m2ar & | - seekKalfy>alee de ©Rosheds) | Ly=Saasicy=Varn2RG). Japletfy dibsde =[30295Rls) =bide Se}aa. Ll) =ab[wet-20° 2) Vous oe . ___negAteePre e caf=cummuletiia dikCdn @ é Cano.50Lan) AR=prob.dausity Sohnemocass 9 28oiarrareorwanae:vans? LCVAPTER2PRORABILTTYANDsTOCHAS! Thisconditional cdfsatisfiestheproperties previously established forthese hy r nw functions, suchas iSntCRE Ce en wa one j Fact Resins ot)0 ; StatisticallyIndependent Random Variables, Wehavealready defined * * a . Statisticalindependence oftwoormoreeventsof«samplespaceS.The “ conceptofstatistical independence canbeextended torandom variables definedonasimple spacegenerated byacombined expcriment orbyrepented Wilofasingleexperiment. theexperiments resultinmutually excuse AE “outcomes, theprobabil ofanoutsome inoneexperiment iindependent of aManoutcome inanyotherexperiment. Thatis,tRejointprobability ofthe a nooutcomes factorsinto@productoftheprobabilities corresponding toeach 4 heeycome. Consequently. therandom variables corresponding totheoutcomes fe :intheseexperiments areindependent inthesensethattheltjointpllocone ino product ofmarginal pdfs.Hence themultdimensionl random variables . , arestatistically independent ifandonlyif by COFostka)=FlaPU)+Flea) easy BBE snotadovalXondanexampleofhesept MIGURE214Atinatfomatin ofad oralternatively, ane POF)=pley)oCea)Pls.) 2138) : 21-2Functions ofRandom Variables . jferentiating (2-1-40) withrespect toy,weobtain therelationship Aroletateiseeninpracaspcatnsofprobabilityisthe Byieretnting 0)wihep following. Givenarandom variable X,which ischaracterized byitspdfp(x), q vedetermine thepdfoftherandomvariableY=g(X),whereg(X)issomegivenif dy)=4p1(=)v (24-41) functionofX.WhenthemappinggfromXto¥isoneto-ones the ; Poly)=PTsecs of70)eeiayfunptrar. Howerenthe thon(4-40)and(2-4)seytnerythrandvile¥ Deverertourfesvatonotpiyy namewienY=,west HS intemoaftheifanepaloftherandomvariablefortheTear” ie transformation in(2-1-39).Toillustratethismappingforaspecificpdf Example 24-1. ‘ x(x), consider theoneshown inFig.2-1-4(6). Thepdtpr(y) thatresults ‘Considertherandom variable ¥defined as ce fromthemapping in(2-1-39) isshown inFig.2-1-4(¢). Yaox+d (24-39) a Whereaandbareconstants, Weassumethat@>0.Ifa<0,theapproach is Example 24-2 similar(seeProblem 23).Wenotethatthismapping, ifustated inFip . Consider therandomvariable¥definedas 2-1-4(a)islinear andmonotonic. LetFy(x) andFy(y) denote theedfsfor¥ : > a andY,respectively.t Thenv — Y=axX?+b, a>0 (2-1-42) y - isone-to-one. Hence Fo)=Posy}=Pox+b<y)=A(xettY q ‘AsinExample2-1-1,themappingbetweenX~rcone.He rome -b By)=PUYsy)=Plax?+b<y) =Bac=A(2=2 . ot 7 JOpniBide=(2)Ae140)HRs oof2)"Jen(@Z)"] 4%een 1Toavoidcontusion inchanging variahtes, subscripts areusedintherespective palssnddl. - « DorC.Proakin, DaLensBerdEk,1995),MeGrou-ttl Ria Distribution --fromMathWorld file:///D:/Work/My%20Interests/Math/temp/Binomial%20Distribut... WOLFRAMRESEARCH athworid.wolfram.com THERWON ProbabilityandStatistics>StatisticalDistributions»DiscreteDistributions¥ eMMeteor» Trott - MATHEMATICS —"Binomial Distribution™ _ palatal rt,Dy FB]MattomotcoNetaboskithMathteader EiconPes5 Romie lary | pokingeee FRirnienca pio orto Sig]ThebinomialdseibutiongivesthedisreteprobabilitydistibutionFe(OI)of PAE Re: Pa Bn ei5a5YgeobtainingexactlymsuccessesoutofNBernoullitrials(wheretheresultofeach @i:erehigo") Bernoullitialitruewithprobability pandfalsewithprobability4=1~P).The Rie binomialdistributionisthereforegivenby iancous: =fovea it Roly=(Nar @ [imesommeies® fal =alway?" pe), Q) , where (%)isabinomial coefficient. Theabove plotshows thedistribution of1 successes out ofN=20trialswithP=@=1/2, The probability ofobtaining more successes than thenobserved inabinomial distribution is x P=(Dota-n¥-*=43,0—m), @) ey where B(e;a,0) a,b)= 4 t) 426)=Fay @) lof7 1/21/2005 6:34 PM Chapter 7 Probability 7.1. Experiments, Sample Spaces and Events Startwithsomedefinitions wewillneedinourstudyofprobability. AnEXPERIMENT isanactivitywithanobservable result.Tossingcoins,rollingdiceandchoosingcardsareallprobability experiments. Theresult oftheexperiment iscalled theOUTCOME orSAMPLE POINT. Sothetwopossible outcomes from tossing acoin areH(heads)andT(tails). r)‘ThesetofalloutcomesorsamplepointsiscalledtheSAMPLESPACEofthe experiment. AnEVENT isasubset ofasamplespace.Thatis,aneventcancontainoneormoreoutcomesthatareinthesamplespace. Consider tossing acoin. Thesample space isS={H,}. Theevents thatarepossible inthis experiment are@,{H}, {T"}, 8.So,while there are2outcomes inthesamplespace,thereare4differentevents. Ifa6-sided dieisrolled, thesample space isS={1,2,3,4,5, 6}. Sometimes weuseatree diagram tofind allthepossible outcomes ofanexperiment. Consider tossing acoin 3times andnoting the resultofeach toss. fint second tind tome HHHHre 7 HHT H H HTHTKTHIT H<CHTHH T T THT rSsTIT [email protected] S$=(HHH, HHT, HTH, HTT, THH,THT,TTH,TTT) Y 7Gaice Bpscin 1998, 1999, 2000. These notes maynotbedistributed forprofit Teroa AEM »LevieCivita symbol andcross product vector/tensor Page Iof5 Levi-Civita symbol and cross product e vector/tensor Patrick Guio $Id: levi-civita.tex,v 1.9 2001/08/29 15:53:02 patricg Exp $ Definitions TheLevi-Civita symbol¢,,,isatensorofrankthreeandisdefinedby 0, ifany twolabels arethesame ijk=1,if,9,&isanevenpermutation of1,2,3 () 1, ifi,j, bisanoddpermutation of1,2,3 TheLevi-Civita symbol ¢,,,isanti-symmetric oneachpairofindexes. Thedeterminant ofamatrix 4withelements q,,canbewritten intermof¢,,,as e i42a3 3.33det]a2i22ans|=>>>) eyaariaryacy =eynariarjase =<— (2) 431 32 G3 isi j=1 k= Note thecompact notation where thesummation over thespatial directions isdropped. Itisthisone that isinuse. Note thattheLevi-Civita symbol cantherefore beexpressed asthedeterminant, ormixed tripleproduct, ofanyoftheunitvectors (6,é,,é,)ofanormalised anddirectorthogonal frameof reference. =det(é;,8,84)=8+(8)Xe, @) eajk=dot(&i, €;,2x)=B:-(8;xen) Now wecandefine byanalogy tothedefinition ofthedeterminant anadditional type ofproduct, the vector product orsimply cross product a: @ ésaxb=det} ara2a3|=eyedsayy ® by ba bs or eachcoordinate (aXb)i=eijnasbe (5) http://folk.uio.no/patricg/teaching/al 12/levi-civita/ 2/11/03 .LevisCivita symbol andcross product vector/tensor Page 2of5 . Properties ©.TheLevi-Civita tensorej,has3x3x3=27components. ©3x(641) =21Components areequal tog. 3components areequal to1. 3components areequal to1. Identities TheproductoftwoLevi-Civita symbols canbeexpressed asafunction oftheKronecker's symbol,, €ijeEimn =t6djmbin +bimdjndit +Sindee bindjtbun—6:18jndkm —Sindjmndet (6) Setting ;=1gives igh€imn=5jmdtn—5jn5km (1) or CijkCimn =S44(SjmOun —5jn5km) +dimdjndei+bindjidem—SimdjiSkn —SindjmSki =3(8jm —Sjndim) +Stmdjn +5jndtm —5jmbtn —Stndj =Sjmbin —5jn5km Setting ;=;andj=mgives ajkeijn =2kn @) Setting ;=1,j=mandx=n gives encys =6 ® Therefore ax(bxe) =b(a-¢) —e(a-b) (10) proof http://folk.uio.no/patricg/teaching/al 12/levi-civita/ Jrontiunsin, dasaf2/11/03 co Roark Liteyohoy UCB. e . xPhysics 209 oavadsm Raaey. Fall 2002 Notes 3 The Levi-Civita Symbol ‘The Levi-Civita symbol isuseful forconverting cross products and curls into thelan- guage oftensor analysis, and formany other purposes. The following isasummary ofits most useful properties inthree-dimensional Euclidean space. ‘The Levi-Civita symbol isdefined by 1,_if(ijk)isanevenpermutation of(123); cise=41,if(ik)isanoddpermutation of(123); (3a)0,|otherwise. Tthas 27components, ofwhich only 6arenonzero. Itfollows directly from this definition that e;js,changes sign ifanytwoofitsindices areexchanged, ijk=jak=Chey=—Ejik =ekg =Ege (3.2) e ‘TheLevi-Civita symbolisconvenient forexpressing crossproducts andcurlsintensornotation. For example, ifAand Baretwo vectors, then (AxB)i =eyeAjBry (3.3) and opNk (xB). =canFE. (3.4) ‘Anycombination ofanevennumberofLevi-Civita symbols(oranevennumberofcross. products andcurls) canbereduced todotproducts with thefollowing system ofidentities. Similarly, any combination ofanodd number ofLevi-Civita symbols (oranodd number of cross products andcurls) canbereduced toasingle Levi-Civita symbol (oracross product oracurl) plus dotproducts. The first isthemost general: Se Sim Sin ijkCtmn=|55eSimbin|+ (3.5)Sie Sim dn Notice that theindices (ijk) label therows, while (¢mn) label thecolumns. Ifthis is contracted oniand J,weobtain 83min cansinn=|GS|=bmban~BnSi (6) 2 This identity istheoneused most often, forboiling down twocross products that have one index incommon, such asVx(AxB). Bycontracting Eq.(3.6) injandmweobtain ijk€ijn=en (3.7) Finally, contracting onkand nweobtain ijkijn=8. (3.8) Itshould beclear how togeneralize these identities tohigher dimensions. IfAy=—Ay isanantisymmetric, 3x3tensor, ithas3independent components that wecan associate with a3-vector A,asfollows: 0 Ay Ap Ag={-As 0Ar|seinAn. (3.9) Ag -A, 0 ‘The inverse ofthis is 1Ais=5eunAte (3.10) ‘Using thes identities, themultiplication ofanantisymmetric matrix times avector can be reexpressed interms ofacross product. That is,if rd Xi=Ay¥j (3.1) then X=YxA. (3.12) Similarly, ifAand Baretwovectors, then A;B; ~AjBi =jk(AXB)x, (3.18) and OB,OB;FeBe=OH(TXBDe (3.14) Finally, ifMi; isa3x3matrix (ortensor), then 1 detM=eijxMaiMajMax =GeijhetmnMieMjmMen- (3.15) ‘TheLevi-Civita symbol hasbeen defined here only onR°,butmost oftheproperties above areeasily generalized toR"(including thecase n=2). Itonly transforms asa tensor under proper orthogonal changes ofcoordinates, which iswhy weare calling ita “symbol” instead ofa“tensor.” Itcan, however, beused tocreate so-called tensor densities onarbitrary manifolds with ametric, and hasfascinating applications inHodge-de Rham theory indifferential geometry. 7- Simon Malhams wwwma hwiac.uk/~ simon ™ | Wholesotdirwrtoodad \. CHAPTER 5 Method ofstationary phase ‘This method was originally developed byStokes and Kelvin inthe 19th century intheir study ofwater waves. Consider thegeneral Laplace integral inwhich 4(¢) ispure imaginary, i.4(¢) =iv(t) sothat 5 I(2)=[F(t)=Oat, where a,b,2,f(t) and Y(t) areallreal. Inthis case, I(2) iscalled the generalized Fourier integral. Note thattheterm eV!) ispurely oscillatory and sowecannot exploit theexponential decay oftheintegrand away from a maximum aswas done inWatson’s lemma and Laplace’s method. However, ifxislarge, theintegrand inI(2) oscillates rapidly and sowemight expect. approximate cancellationofpositiveandnegativecontributions fromadjacent t) intervals, leadingtoasmallnetcontribution totheintegral. Infactwehavethefollowing result. RIEMANN-LEBESGUE LEMMA. If|f(t)| isintegrable andy(t) iscontinu ously differentiable over a<t<b, but(t) isnotconstant over any subin- terval ofa<t<b, then igap 6/—~—~I(x)=[FeO dt0as2+00. ——- Toobtain anasymptotic expansion ofI(x) asz—+00, weintegrate by parts asbefore. This isvalid provided theboundary terms arefinite(Snd\the gsulting integral exists; forexample, provided that f(1)/(t) issmooth over @<t<b and non-zero at:either boundary, thenao du=&Al)wen FO) 1 paso ae ie Ta)=|2O_.wo-ifALO) oveayYW TSGal ee (@)(oo bf$(SB)ee Sarsinv aiee diret-08%F ah ‘The2ndintegral iso(2)asz++00bytheRiemann-Lebesgue lemma and44%= ¥~ ct at 3wk_2Cal f(t) .1 ~ 0) ~er 1)~[gy OPO]a8e400 ysIntegration bypartsmayfailifUt)hasastationary pointintherangeofintegration, ic.W/(c)=0forsomea<¢<b.Supposethatindeedy'(c)=0 e@forsome a<c<band W(Q) £0 forallf¥cin[a,b]. Then proceed as follows: 1 Step 1.Forasmall ¢>0wedecompose I(z) toI(x;€)justlikein(77) forLaplace’s method. The remainder terms weneglect, forexample inthe casewhena<c<b,are [p(y at+fFM at A ove and these vanish like 1/z asz—++00, as(t) hasnostationary points in either interval, and wecan integrate byparts and apply theRiemann-Lebesgue Jemma. Step 2.With ¢small enough, inorder toobtain theleading order be- haviour, wereplace f(#) and(t) bytheapproximations He FOXFOandy(e)rw(o)+O. @-ep This assumes y"(c) #0,otherwise wemust. consider higher order terms. Hence weget, I(2)~fFoF(MOSES) aeas2400. Step3.Wereplace¢byinfinity—this againintroduces termswhichvan- ishlike1/asx—++00andwhichwillbeasymptotically smallertermsthat t) canbeneglected. Thensetting(achangeofvariables duetoMorse) a2PO 6sen(y"@) yields 2 a m2 I(x)~f(oe¥..f [ersen(¥")? dgas2too.(5.1) (2)~flo) aot te 6.) ‘Toevaluate theintegral ontheright, usethatf°,e##ds=Ve#*/4 so that eventually weget Tax)~fle)=WO+ieenco"(Oe/4. |27 ng, 400, aly"(c)| Note. Ifc= aor c=b, thecontribution from theintegral, which isonly overasemi-infinite interval,meprtsthattheasymptotic resultabovemust. bemultiplied byafactor of4.VIf p(t)hasmany stationary points in(a,6], then wesplit uptheintegral into intervals containing only one stationary pointandein eachoneindependently (thoughtheircontributions are additive). “Again, therelative size off(t) atthestationary points of(2) willnow beimportant. Ifthestationary point issuch that ¥/(c) =W"(c) = s+=W™1(c) =0andyc) #0then make theapproximation W(t) ~ ¥(c)+2212.-(t—c)™ instead.InthiscaseI(z)behaveslike2~1/"as«++00. r) Note.Alternatively, after(5.1),wecouldhavemadethesubstitution5=vel") */4andusedthatf°,e-dv=VF. Example. Tofindtheleading order asymptotic behaviour of A I(2)-[cosz(t—t) dtas2+00, 0 wwefirst write theintegral intheform 1I(2)=Ref[eX?)at}f sothat wecanidentify f(t) =1andy(t) =#—¢. Hence w(t) =31?— 1and w"(é) =6tsothat yhastwostationary points, ofwhich only the positive oneliesintherange ofintegration atto=1/V3. Since y(t) = 6t>0for¢>0,this isalocal and global minimum; and finally note that +b(to) =—2/(39/*). Hence truncating ourinterval ofintegration toasmall neighbourhood oftg(this only introduces asymptotically smaller terms), we then make the approximation (to HO=vtto)+O) igy? and then extending theendpoints ofintegration offtoinfinity (again only introduces asymptotically smaller terms), wefinally get H(z)~eevee)fotI-10)? dt,as2+00, @ Makingthesubstitution s?=2-fe)+(t—to)?,generates 2 ooig? Qn T(z)~o2¥(to). [eit?dg=elle¥(to)+a/4) — © aT) Ie a) Now taking thereal part ofthis, weget ee10)~[aqex(Z- 2%)eo4 Tttacos(t-B) © “om tiacos(¢-$"divesie=Jigos(oseteed)=TO — Jedkag =9d.(8) D@=aL Tlel=o SOs =ad(*) fun3D=e a Te(be)de =alto(ab) i js” Set sina .a={ouedx=2° 7 SELECTED DEFINITE INTEGRALS r v T Jsomedx=LoosGrn)dx=asS3 0 JSMx4,== J)sanedeaCe(4mreBy x Q .x “Utlamdo OR Oomee ~—<K 1jedx=a >o) J@ J5“ 1 \Se%2 aJeKXdn= Jeede=28 < \ >oxat 2. Jéxdx=ye Pad==)!° Co 2 so 2~oX ™ =XX, ave\z Jeo =& exdx =Qua coy “eeeoSRSSE Ub§~ex"an 4gS xdy=(Qn-1)}! iE(Qa) cvs oo Ssitean=EF 5teake2S@-2) “e . Sedk=UT _ Jore 3 Fesimgnae=caeOa . :fExXdx= 2 sq = : ike ‘ Cs}See =7,So. ey0ok on! JeKan=+Veh)Dash)RE 3 JePane Te. 3 ; 7 Wore,sieteinlegaly S: Be eeee Bo\neret- od Fle SES ga - =Seen) TAP eT a SasSMb@eGos] =RO. aT \ Gea Vb) - oe e 4aye . @Sova(x= BCat-\, Ht)—Jpaerok . LC ae WA) es ~ ~ ° RoR SOR. ™:- - ©Shen) =Vhe hal- a. IZ e re 7a ‘ a ed a ET NE oS oo ee>re Skese, desizds. corm2$43 ——-Sae x7+5eo :CECatsolare ot o Se a de ne 9ae — +Gb)©|ets2aeet]__ig).9 G@-ay@-8)___ ws Ae8-1 ES ae gua!TkeS sexe Gannon X=ngaSaye —Wal thS22 es ty AG)Zooe ee RecutSonosfreen\,Muteaoct tea _ 7 en —_ B=eset Feel, seman wan he BD BL (na VONRUCYCn —Sayonammet =82OA Oe _ a _S<ttoe Sareea ee_ oek2.G-Ale&) .-4G2). @-8) BAN : . ~ =\ See - 2see eta 43> et.WAGER Spi aeS A,lehoweSommeSuga_ on _ . Oe Sou_— See WAS Se : <>)oss7i ns —Sag-\asoaud>, sweAaahippoe _—WE —dimwldaddsodelithdoasieloo. ; 7 oo ee — _ p> oakoa ae are2 ortosesx a mbit BoasadalesaSWOT Rikpewadsoo0+S-.Ys sek Ss a a = re — a a ee, a —. Soxlososx iaB-akwOOsK ea eo [ netik Saa>Vel Oaanal —_ th Matecenfinrnns972,70, 02D _-—_—_ >_ __ ( WOR oe8A RR do ee, ——vo aesfla|~~ ChanC8 re Coe . . UO — Me --— ——- ——-- -..-— aS Ayat8 —--a. || ee2aklasosK dee _ moda: a>lbl>o- a_asall || (NT _Nat vr ttt—CSSCOCSESS - ox Sinx a, 3 i oO Site TeaS =Bears_= =a = Se a ee en——— ee ee — Mpecern —Aa oo ee oe aee gtd, a>\blyo; abval| ( a ee re oboe. dL alecosa. anben8% = _ se 3.na ~~ !!lUen ere ab ade LL>\b\0.A oe ne ——\ 8s. = \+—t—\ (2 6—bhepse, 2 atbeosy axbeose |\**/ 3: — OA cos) TT a wR aie oA at es (Sy=—oos() Hr OO we oeje emacetNET Bran—nee 2GE & = \_ - . orakesucy za bsmy anbsna| - ° eea a ne =x a>\bl20a Ss coos aa a ~\ode a\gx \ens cs ar2 9)ofeodn__alat=ee /: a -nn eSmT oe yPra Ska .jot fe _ |. a an — 1.Ne =TL an i _J2 waBeosx aaa oe |ammiadtky jaseo 6abek eT Ss ( ——voitEstee. -- —— ——srlpakaa0)Qossancfdh:Dru 2-2sfRaber a —Yaelasesite]=aIdeSlonse,subel_ aa Clea, suc catloan) =side ghoe tC ers) ete — a en | dae psene,sel=Jaet\swle,cave| . a - aBe fo —Gonsinre, Votamala..._____- a= — WoCine spreaDekeSpnyhPABdepods —@Meo. ee ee _ Tes NaotCo TT SAT. bs)=26 ys) TGs) ==Tecate x(ble) @tan2+Tea}Toca,b,s)»} (20)+Cola) _ 2-8-03 There>Fanweaedaanad fieFe,8),poetics¢ : wehawSarl§d)=o Prt: San(s.8) =rahe eGR) wub=AfrdG+oSy +Ae(s) OHaggseGE)=(Se) =Oswe$=fucka — @=Sho Ladube) . =Sol, =OSwe amodac SOSA aae=)MV\a0+SV-(FM)aL _>oo TrudefF=f, seTe=tpg. J . \Sh-GyaL=\8VydL aayorwre“chotngpacts?oleBu'V. mi} /0-3|-08 Kaopth ~ nn «si. LS at yy peWd | an ry (vtTHEVALUATIONOYCERTAININTEGRALS :3 TnChapterweencounteredinegralsovertheintermediate satesofthe :aarenpytina oc saante hi {5thecompatatinn be a raieatty tatrHt, “eden ntofrinhanngaen1Pet(ASH)reytineoteflfr, : yPoonaTieatoll nicoRay 1 fae : |semenPacemrteerntate tn artietn[ts :Tiewiaipritamienteccmad gpctinegaiewicen ft i Slybedeattheslothteleptinretheeasy x 1 uss i"raewconshemaatdthatlcompton ‘TagowaFowl a)Fa i‘Tharaethessdraningmtohsprocedure,Oneeons ‘Theintegrationsoethenewvariablesy+taaeite SOad baitaptctytatopalsoraaeIGAsmthatthedeny(A5-2)aadtherfore(AG)teeforSpe | SrolSmeelsdadedpsnog fogssc.Thenwcnwatefarpaola‘Gp icp eioe ‘ ee iifarte [eet wees NAwesteseamacntgnesitss. natn ifAeterna eetetrad =fSeborsbywetheeatin i FStepsbnacal oles iat! eae 1eetntatharyvbwhinedeeyen-~,YoOh —— wn SESaepoets By «x narymac.Wefdwh+= 1 fenf FBS ees(aes) Poreriad fe ,fe tftdoofgg i= : ae oof SRrere yneersrnee aoarbeae poo:‘EliasAfaSentepoety naomi FernT4RYbp,5HARBASDfrapieThyCkthySahywisthandriolommetion ewewheeGR k te i qo-oo cogs7feHiberrtenet an2inthatthe oe egoe oy«hsslipthenteeateeetesaaeame, TFRPream,Pi1780 we “aaeparnetydineuoed inthoflowingection. hae ee Croanbeckgerrdyah5 AolagePm4M)gi J aleaoe Yt. be». Brag. —Pfn\POn) Vm) Dede dye 1 ee NANet Mle: BEAK e ma See Oy qn"..Qe Df)=.Pw) x§dadededpe pe e heat We Arte leek dobh %whl Deginneralf® Taywik=2. e agp Mn). Pw) Ah Ool,2> 7x§dxdaeAeeCriPa foal Ghar We Ares og iegek sdedh SGraeooty~2)du. Jet4e=Y% Asymptotic series: Amathematical aside http://farside.ph.utexas,edu/teaching/jk |/lectures/node76.html [af [previc @_Nev"TeWKB.solutionsasUp:Elestomagnetic wavepropagation inPrevious:Reytracingin the Asymptotic series: Amathematical aside Itisoften convenient toexpand afunction ofthecomplex variable /(#)ininverse powers ofz: As,Ai 1)=$0)[tots Sa], (890) where (2) isafunction whose behaviour forlarge values of#isknown. If/(2)/#(#) issingular as |2|-+00thentheabove series diverges. Nevertheless, under certain circumstances, theseries may Stillbeuseful, Thecircumstance needed tomake thispossible isthatthedifference between /(#)/#(#) andthefirst n+1 terms oftheseries beoforder 1/#"*? ,sothatforsufficiently large 2thisdifference becomes tdvanishingly small’Moreprecisely,theseriesissaidtorepresentJ(#)/$(#) asymptotically, thatis ° Heeey, (891) pao provided that Hs)_ssAp Tim{29/272 _Sy“?1}40, 892) Inotherwords,foragivenvalueofn,thefirst"+1termsofthe series may bemade asclose as desired totheratio#(#)/6(#) bymaking #sufficiently large/For eachvalueof#andmthereisan erroroforder1/#%**“Sincetheseriesactuallydiverges, thereisanoptimum numberoftermsinthe series usedtorepresent J(2)/#(2) foragiven value of#.Associated withthisisanunavoidable error,As2increases,theoptimalnumberoftermsincreasesandtheerrordecreases.[] eConsider asimple example. Theexponential integral isdefined lofé 11/22/2004 10:09 AM Asymptotic series:Amathematical aside http://farside.ph.utexas.edu/teaching/jk /lectures/node76.htm! get Bi(s)=fSa (893) e‘ Theasymptotic seriesforthisfunction canbegenerated viaaseriesofpartial integrations. For example, ov Pet ale) = - [Fae ($94) ( xContinuingthisprocedureyields ZSalt) 46=BE,&) * ot =1)Pnt Bs)=—[p-2+3-344.sagf Fa a —— ster Peetn gayefo 3 +(-ytingne[Sa\eo (895) * Ss$6)-SaG) =xé2%. ‘Theinfinite series obtained bytaking thelimit n+00diverges, since theCauchy convergence test yields im[2]—tien[2 e ite|=fn,[Ela y (896) Notethattwosuccessive terms intheseries become equalinmagnitude forn==,indicating thattheoptimumnumberoftermsforagiven#isroughlytheintegernearest=.Toprovethattheseriesis asymptotic,we need toshow that cogot iyotet(—1)™4(n 41}!fazo. (897) =~ we=RRO)~SJ] 2.LSS This immediately follows, since 0grt 1of? oF ffart<omfett=55.SedeeL(898) owe Aene£0)-5.l9<CTGon!ER” Thus,theerrorinvolved inusingthefirstmtermsislessthan(n+1)!e* /s"*? ,which isexactly thenexttermintheseries, Wecanseethatasnincreases, thisestimate oftheerrorfirstdecreases andthen increases without limit. Inorder tovisualize thisphenomenon more exactly, let J(s)= soxp(s) 2{s) ,andlet es(-1)Ppt ®Jafz)=>Ure (699)y=0 20f6 11/22/2004 10:09 AM Asymptotic series:Amathematical aside hutp:/farside.ph.utexas.edu/teaching/jk /lectures/node76.htm) @_theasympotc seriesrepresentation ofthisfunctionwhichcontains*-+1terms.Figure16shows therelative error intheasymptotic series |/n(#) ~J()]//(=) plotted asafunction ofthe approximate numberoftermsintheseriesnfor#=10.Itcanbeseenthatasnincreases theerror initially falls, reaches aminimum value atabout n=10,andthen increases rapidly. Clearly, the optimum number ofterms intheasymptotic series usedtorepresent /(10) isabout 10. Figure 16: The relative error ina typical asymptotic -). seriesplottedas¥/ afunetion ofthe number of terms in the series @ Asmpioric seriesarefundamentally differenttoconventional powerlawexpansions, suchas et oat ings pe ye, (900) sases—ata at 0) This series representation ofsine converges absolutely forallfinite values of=.Thus, atany#the error associated with theseries canbemade assmall asisdesired byincluding asufficiently large numberofterms. Inother words, unlike anasymptotic series, there isnointrinsic, orunavoidable, error associated with aconvergent series, Itfollows thataconvergent power lawseries representation ofafunctionisuniqueinsidethedomainofconvergence ofthe series. Ontheother hand, an asymptotic series representation ofafunction isnotunique. Itisperfectly possible tohave twodifferent asymptotic seriesrepresentations ofthe same function, aslong asthedifference between the two series isless than theintrinsic error associated with each series. Furthermore, itisoften thecase that different asymptotic series areused torepresent thesame single-valued analytic function in different regions ofthecomplex plane. Forexample, consider theasymptotic expansion oftheconfluent hypergeometric function #(a,¢,2) . Thisfunctionisthesolutionofthe differential equation e aP"4(c—2)P’-aP =0 (901) 30f6 11/22/2004 10:09AM_ Asymptotic series: Amathematical aside http://farside.ph.utexas.edu/teaching/jk |/lectures/node76.html which isanalytic atx=0[infact,¥(,¢0) =1).Here,‘ denotes ¢/42.Theasymptotic expansion r)ofP(a,e,2)takestheform: roem) Flojee) &Te—a)s*oF[1 +O(1/2)] +0(a)z-*o™ [1+O(t/a)] (902) for—™< arg(z) <0, and Fee=2)aha,c2)~De~o)e**ef[1 +O(1f2)} +0(a)a-*e'™ [1+O(1/2)] (903) for0<arg(2)<m ,and Horena) Payee)©Ele—a)atem)oF[14O(1/2)] t) 40(a)e-*e!™ [1+0(1/2)} (904) for*<arg(z)<2m,eic.Itcanbeseenthattheexpansion consists ofalinearcombination oftwo asymptotic series (only thefirsttermincachseries isshown). For|#|>1,thefirstseries is exponentially larger thanthesecond whenever Re{) >0.Wesaythatthefirstseries isdominant in thisregion, whereas thesecond series issubdominant. Likewise, thefirstseries isexponentially smaller thanthesecond whenever Re(z) <0,Wesaythatthefirstseries issubdominant andthe second series isdominant inthisregion. Consider aregioninwhichoneorotherofthe series isdominant. Strictly speaking, itisnot mathematically consistent toinclude thesubdominant series intheasymptotic expansion because its contribution isactually less than theintrinsic error associated with thedominant series [this error is (1/2)timesthedominant series,sinceweareonlyincluding thefirstterminthisseries].Thus,ata general point inthecomplex plane theasymptotic expansion simply consists ofthedominant series. However, thisisnotthecaseintheimmediate vicinity ofthelinesRe{z)=0:thesearecalledthe anti-Stokes lines.Whenananti-Stokes lineiscrossed,adominant seriesbecomessubdominant and @viceversa,Intheimmediate vicinityofananti-Stokes lineneitherseriesisdominant,soitis mathematically consistent toincludebothseriesintheasymptotic expansion. 40f6 11/22/2004 10:09 AM Asymptotic series: Amathematical aside bttp:/farside.ph.utexas.edu/teaching/jk/ectures/node76 html ‘Thehypergeometric function F(4,¢,) isaperfectly wellbehaved, single-valued, analytic function in thecomplexplane.However, ourtwoasymptotic seriesare,ingeneral,multi-valued functionsinthe e complex plane [seeEq.(4.252a)]. Canasingle-valued function berepresented asymptotically bya multi-valued function? Theshortanswerisno.Wehavetoemploydifferent combinations ofthe two seriesindifferent regionsofthecomplex planeinordertoensurethatF(a,¢)2) remains single-valued. Equations (4.252)showhowthisisachieved. Basically, thecoefficient infrontofthe subdominant series changes discontinuously atcertain values ofarg{#) .Thisisperfectly consistent with F(@,¢,2) being ananalytic function because thesubdominant series is“invisible”; i.c.,the contribution ofthesubdominant seriestotheasymptotic solutionfallsbelowtheintrinsic errorassociated withthedominant series,soitdoesnotreallymatterifthecoefficient infrontoftheformerserieschangesdiscontinuously. Imaginetracingalarge circle, centred ontheorigin, inthecomplex plane. Close toananti-Stokes line, neither series isdominant, sowemust include both series inthe asymptotic expansion. Aswemove away from theanti-Stokes line, oneseries becomes dominant, which means thattheother series becomes subdominant and, therefore, drops outofourasymptotic expansion. Eventually, weapproach asecond anti-Stokes line, andthesubdominant series reappears. inourasymptotic expansion. However, thecoefficient infront ofthesubdominant series when it reappears isdifferent tothatwhich ithadwhen itdisappeared. This new coefficient iscarried across thesecond anti-Stokes line into theregion where thesubdominant series becomes dominant. Inthis new region, thedominant series becomes subdominant anddisappears from ourasymptotic expansion. Eventually, athird anti-Stokes lineisapproached andtheseries reappears, but,again, with adifferent coefficientinfront.Thejumpsinthecoefficientsofthesubdominant seriesarechoseninsucha e@ mannerthatifweperformacomplete circuitinthecomplex planethenthevalueofthe asymptotic expansion isthesame atthebeginning andtheendpoints. Inother words, theasymptotic expansion is single-valued, despite thefactthatitisbuilt upoutoftwoasymptotic series which arenot single-valued. Thejumpsinthecoefficient ofthe subdominant series, which ateneeded tokeep the asymptotic expansion single-valued, arecalled Stokes phenomena, after thecelebrated nineteenth century British mathematician SirGeorge Gabriel Stokes, who first drew attention tothiseffect. Whereexactlydoesthejumpinthecoefficient ofthesubdominant seriesoccur?Allwecanreallysay is“somewhere intheregion between twoanti-Stokes lines where theseries inquestion is subdominant." Theproblem isthatweonly retain thefirstterm ineach asymptotic series. Consequently, theintrinsic errorinthedominant seriesisrelatively largeandwelosetrackofthe subdominant series very quickly after moving away from ananti-Stokes line. However, wecould include more terms ineach asymptotic series. This would enable ustoreduce theintrinsic error inthe dominant seriesand,thereby, expandtheregionofthe complex plane inthevicinity oftheanti-Stokes lineswherewecanseeboththedominant andsubdominant series.Ifweweretokeepaddingtermsto ourasymptotic series, soastominimize theerror inthedominant solution, wewould eventually beforcedtoconclude thatajumpinthecoefficient ofthesubdominant seriescanonlytakeplaceon those lines inthecomplex plane onwhich Im{s} =0;these arecalled Stokes lines. Thisresult was firstprovedbyStokesin1857.150n aStokeslinethemagnitude ofthedominant seriesachieves its ‘maximum value with respect tothat ofthesubdominant series. Once weknow thatajump inthe coefficientofthesubdominantseriescanonlytakeplaceataStokesline,wecanretainthe t subdominant seriesinourasymptotic expansion inallregionsofthecomplex plane.Whatweare basically saying isthat, although, inpractice, wecannot actually seethesubdominant series very far away from ananti-Stokes linebecause weareonly retaining thefirstterm ineach asymptotic series, 50f6 11/22/2004 10:09AM Asymptotic series:Amathematical aside http://farside.ph.utexas.edu/teaching/jk I/lectures/node76.htm! wecould, inprinciple, seethesubdominant series atallvalues ofarg(2)} provided thatweretained a esufficientnumberoftermsinourasymptotic series. anti-Stokes tine nD SE branch cut Figure17:ThelocationoftheStokeslines(dashed), theanti-Stokes lines (solid), andthebranch cut(wavy) inthecomplex plane fortheasymptotic expansion of thehypergeometric function Figure17showsthelocationinthecomplexplaneoftheStokesandanti-Stokes linesforthe @_esymptotic expansion ofthehypergeometric function.Alsoshownisabranchcut,whichisneededto make #single-valued. Thebranch cutischosen suchthatag(2}=0 onthepositive realaxis.Every time wecross ananti-Stokes linethedominant series becomes subdominant andviceversa. EverytimewecrossaStokeslinethecoefficient infrontofthe dominant series stays thesame, butthat infrontofthesubdominant seriesjumpsdiscontinuously [seeEqs.(4.252)]. Finally,thejumpsinthe coefficient ofthesubdominant series aresuch astoensure thattheasymptotic expansion is single-valued. all |previc Next: TheW.K.B. solutions asUp: Electromagnetic wave propagation inPrevious: Raytracing in the Richard Fitzpatrick 2002-05-18 60f6 11/22/2004 10:09 AM. WVAVG 28 Onedss+Aepait(-~05,Diyrora Socker| Gaussian Quadrature. Theorem: suppose youhave apolynomial f(x) ofdegree 2n-1. Then theintegral ofthis function over aninterval (a,b) against aweight w(x) isgiven exactly bythisformular » ™ Jawa fe=SLAsm ¥= orl Te, you get the integral exactly byadding upnnumbers. The function must beevaluated atnmagic points (not2n-1points). Thex4arethezeros of thepolynomial B(x) which isdetermined bytheweight function andinterval (a,b). TheAj,called weights, aregiven bytheformula: aASp783 Kiha(i)Bo. V(2-22) Kner SOD LyCHOY? > b. This formula is discussed and derived below. e ProofoftheTheorem: Trivial. Allyoudoiswritethepolynomial f(x)inLagrangian form andatonce yougetthat A;=INT( w(x)L,(x),x). Bychoosing thexjinLy(x)tobethezerosofB(x), youmaketheevaluation ofA,very easy, yielding the above simple formula. This is, however, only aproof for f(x) being apolynomial ofdegree n. Since theentire polynomial isspefified bynmumbers, itisnotverysurprising that the integral can befound byevaluating the function atnplaces. Far , more interesting isthefactthat theformula isexact even iff(x) israised . tobeapolynomial ofdegree 2n-1. What makes that true? Myway ofseeing this \ isnot the best but here itisanyway: Certainly everything isOKiff(x) is ofdegree n.Imagine then that youhave found theweights A,andthezeros x; using theabove derivation. Then ifyouexpand f(x) into itspowers, here iswhat you have infact shown: a , a y | = "'yo= wR: Ke * MeBewyko> Spot AAYe de,weknownowthat(*)isvalidforJ-0,1,2....n. Ifyoulike,thiscanbe @ taken asthe equation you solve for the Ajinamatrix sense. Ifwecan show that (*)isinfactvalidforjintheextendedrafomthenwehave' ‘oven.what preatwewant BAzsw decause then wejustmultiply both sides bybyanddotheextended sumto conclude that theintegral formula isexact forapolynomial ofdegree 2n-1. ~‘Theproofthatformula (#)canbeextended inthiswayisgiven @ below. Itisbased onthe recursion relation for the orthogonal funetions. Comments: thus wehave avery powerful result. Byknowing thevalue, sdy, ofa7th degree polynomial atonly 4points, weknow the integral exactly. There are ofcourse infinitely many 7th degree polynomials that run through these four points, but they must all have the same integral. Notice that these arenot4arbitrary points, butaréinfact 4very special points. Accuracy: Suppose weworkwiththeneGause¥-Legendre“formila. Howdowe estimate theaccuracy ofanintegral? Theaccuré¢y isperfect forpolynomial ofdegree uptoand including 7. Where are the gaussian points? | |L ' } \ H nl ~~ | 5 ' Pye , \ '> i 1 - I ~ i jv “i R= (@) -2 44=381430190) $(s)=[NSxr0 fe+(8)(a):(6.6¥/0%) This last shows an estimate ofthe remainder. You see that if the function has no8th derivative formula isexact. BUt iffunction has deriviatives that tend toincrease insize asyou gotohigher derivatives, then your error may beimportant. The thing about apolynomial isthat its derivatives keep getting smoother. Suppose you apply toan8th order polynomial. The 8ith derivative isperhaps oforder 1,more likely oforder 81!=40,000, so error isatmost 2parts in 1000. Arbbvious drawback ofthegoussien formulas isthetyoucannot easily increase nasyoucaninSimpson's ruletoseeifresult issteble. @ ( veuydoseuratactJennehuhhowe,ts? ———Swisrovkiwrs—____Dewdingy Ure.svnsateforenoon ste2mnh @oS ak Rot De eeZoom,BQ) oo EE NE mnheRyPomshsTBtyonehatSn=ZamAnls«Vowdo nniia eepA a ane i(wieGnee:) et Qi. aa.Canes) 1ampCiotGwar), Te,Loebocoscksaleespidh Ooo, nkbeeQU‘ XAEO1Cun (aes) =(mn) and (anr)mo” sme gm Ont) =o. _ YorQsVinWare).Gat2)=Cnet)arfete(me3~ 4 aun)Cmar so=1@(wert). Bukfaaie1@[lmtmtee)=269-004+16(9 A\csgorigDaearipelicnch Wenn. Gchmeeand1O(wei)Som,wintnmt £0sa ee me) =VOQu) Cues)gummy: ruleLatoolFheCsi)seen) 2 Ces) +LOCme! KFdn(Qm~3). Fustedsanchs$3.0. Qs)42OCms) Naar Ascend ehrancher WONT 50 a Loe 2 OK,NON otetercomeneemailer EDont nVOns) gyreaedOF©astOY Maa BetakQuelaseeeatDee dm =Ces)(amen)——iecamerADegaldinn. FA @ =2(Um~2)HO Pwr)|MO pewihelroane Arne BOGm4OC NONevar iyghtenn nome +eae) 2 Ve m= 2eneosin see a Oe ao 2bye QeOQ. Md Oe 08 ooCe0 ee ig ee -a&laesde st]=_¥da _ - .-@\. wg a —lseez+ier 2One, Za) AofoT — a poe * --_Fonwule. fo.Nswisighle,inquvasalGaussian. inlayetsinn 1,You can get all these weights. byJust-inverting atertain matrixwhoseelements ;are various functions f, evaluated atthe zeros of one of the functions. It is better andmoredirect though tousetheformula derived here. _____2, Thenotation isacombination ofthatofScheidandAbram-Stegun. Iwilltryto doeverything generally for allthe orthogonal polynomial systemscoveredinChapter 22oftheASbook, ____Im particular, herearesomesymbole Iwilluse: . n= the order you are choosing. This isthe numberofserosxj.++++x, ,andthe number ofweights Ay»The x;are zeros ofthe polynomial g(x). w(x)is theweight function, andtheinterval is(a,b). cy=defined onpage 773 - =defined onpage 778,given p-77 hy=defined onpage 77h _. =-coeff ofx”inB(n) ‘ Bpandgyrefer totable onpage 783which gives df,/dx. v PI(x)means(x-x,)(x-x2)-+++++(x-%,) wherex,arethenzerosofA(x)_Yo. e1,(x)isdefined onpage53ofNumAnalSchwam.ThisisLagrangia® multiplier function whichellowsyoutowriteapolynomial p,(x)ofdegreeninterms ofthevalue ofthepolynomial atnarbitrary points p,(xj). Note that there isnox,inourpresefn usage, butthereisonpage53. _____(a) First,youhavetorealize thattheAjaregivenbytrivial integrals ofthe ______Legrangefunction, Thisiscompletely obviouse ——_____»—__ —____—_4 =»———-- ——___———— ---+--+ --- cy cory , lm - Bk $e)=2hmie) ess -@— -->Ae Nano SS . = __ biNext,“youobservethatthefunctions PLandPrt“whiehcanbeusedtogive _____1y(x) exesimply related-to J,asfollows:6 {] ob. =IK) oT oe &@ =keOy= EWekwTaY Bk oom9248/7" ALG=gee/g.00d}FG)Luca _. TED =|.9.05)/08)] Sane 1RY ee Tu TO bale yO ~___ (e)Weingtheformof1y(x),wethenwriteAyasintegral off.(x)/(G-x) Likesor _. Leys gy Bato GS) oe - a IO) OD KD ~~ |goSneak eeee HD KD a (a) thenextproblemietofigureoutthisintegral. ‘hetrick1stotekethe—christoffel-Darbous formulasonpage785andepectalize toyex;.Tougets _. ea(8redoC4))[in={-¥\md.(Ss}Yo —- . we Kr Me wo a _ WecansimplifytheHHSbyreplacing J...by.accordingto22Jh,since _____ dsaserooffj»Thus: ee -- oecaLn=2hBaal) aaesae _—FS AT Ne>nein — oe a Gees (.~ younowintegrate bothsidesofthissunruleagainst.the weightw(x),only Wore 3atornmurvtzenbecauseofrthogoneltty.’ Orthoggenerator tywhich_____-sancellle the_h,,ontheleftsoresultter ee ee ALDe OR HD Nin ButJ,isapolyofdegree sero,ie,aconstant, socallit$(2). Theseyoucan_ read off the table onpage 774, called stenderdization. Thus, when x,isone ofthe zeros ofZyou may conclude thats a ——— $e —— — ee K~Ki) Ki Kaa oo Vi PN OO) L x a 7 es erOenePe NS beAR oo—_& @./g) 2 feaea OO a mg —}o)wd GS)kK ag Fk]bn \ Ro Lew i:es et—Loe)-aGan TO) AS Faw[a Pee ied|| 20 edeee eee _ —OL ood)oy|(VEN afeOO)ate _ a eee en ald eece —— -We Le css) | 2d gewea A _ 0 yt -@ eee / pakke Bok! f . _____Hivst-Kind Gauss-Chubby Quadratame formlas; eC TES SE Te WaneeRe the e@[vite ten aeGetita | oS | -—--_--Gauss-begendre-Quadazature-Formulast———— ~~~ ------_-___. RY =Bed, BkMasaroseonsdkNaesaadforeTaLenng Tb19,OrnscsieenDarganasyyniarate ee : _= - __© - » a ____ @Vrows_ 4|--—_. o5 ——__-—_-| ani = ATO ot ____Goments:_IthinkthisisthemostfamousGaussianintegration scheme.Aslight ‘oblemisthatyouhavetolookuporcompute theweights andzeros,sincethere donotexist closed formulas for'these things asoccurs intheChubby cases. But. these arelisted andyouonlyhavetoenter themonceinaprogram todo-sueh integration. ee Acouracys see elsewhere. _ . ae $$ eeSFeseeeeseee ( 4 pommel he2gftw GUsaatbo. oassaaikfoalagaake,S\ayta\ Skbs Vlya Dincak erent ee : =2 Carb) = -a-b ee =eyN yz §a= FCShkSVC] ae _-—whi =Gora)x+(otal oo : i ee — eeN ~-= fk aeeeFTQatse SLAs dAS|deere — TS eA Fond) +Boos e- = | -—©+¥/_a ALCRnd aFex,4d) —.-..~Rsipn bate ASAKa odBODoS (oma) 2 A= Noca)/2 ——$— =co:A&A+xsx \ A~cx _ ee a page 6ANS: oecAltcen)+Ha-cxd] oo _—A=.65214 SISSE 625% wy=BBANRIONS_RHEE —__Ag= 287854895) BUSY =ROU GBS OSS \3- '. . Puasa SO)watemsChae)Gurchion.wartleerdregs. tonaaonceoeoeeee eeQi).Ghen i \ --Ah ~ - -|SM@ax= ZuiSq)=Im i)|saTeDHSaeDeeomegamesehPC),anclolan «ifwdoneTecemadewte whisksotue:Io. -MeLS - aa |e Sao=2BACT) m€(,™) <3) © NeaTwbeautlan WinalesyosdeaddoLinkug. YnmsesCheened)Techedegoaance-codeudode TintCoypsumead20) andnumlasre aro), ‘nodeDra,wil Ad.ALsndreRoagen ArmAge mm€OmI},greainnagintSomeqinuniaghhe.+axteud!|Seo=z&:Ga) meG@,rm-1) WoGrce(mt,tur) byeee onaeat.ag” tm “" powake 5CrN[Bbclay=fw)2roRcx]cawoptalely ask. Swuamokiawel Srvieg(Mises) /TegatBdecunile 1g, | Seegpmagreweekdetebeamtee.18S e weBe < aadofN)dewuato m\edu deQrae,radd -_2“ masacid” =>n>ie® A=24. Raphore wily awdmu: . ose" A8stomCrn) asain, UrtierDiemarr!SWEwdCOW)rel ; Masaural Rwayyweertyy(a).OK,wad greey(tal),MansAxYAoenGnida =XG.) fousCarn)dg y-ie-e Mort)waseecmtandqerenadvated: AxJHCinsd)benho TH)den26Ref(rtsHQT eats et Fryte e A®ayJS,eatass] seok(1)dw (weee° “20 : OndDassenogeedo18phenen&eeRN FA)ekdn. 4 ‘ L¥ 40s)anVva wens Chapter 1 a f£ b Introduction 1.1 The mean value theorems e First,weintroducethemeanvalueforintegrals.Assumethatfiscontinuous,then = Ls(tdta=[sou=Parv This 1iscalled thearithmetic average orthemean value offintheinterval {a,2]. Westate themean value theorem forintegrals: Theorem 1.1([1,p.258] Mean value theorem forintegrals). Iffisacontinuous function on(a,2],then there exists anumber éin[a,z] such that ffHod=HeNa-a). a) This isasimple butvery important mean value theorem ofintegral calculus. |Itstatesthatthemeanvaluethoyger.ofacontinuous functioninaninterval‘a\belongstotherangeofthefunction, Anditassertsonlytheexistence ofatleast 1 one€,intheinterval forwhich f(€.) isequal totheaverage value offbutgives nofurther information about thelocation .Inthisthesis, wecall&anintegral mean point offon (a,x}. Instead ofthesimple arithmetic average wecanform theweighted averages: peLEf(tg(t)dtSeg(t)at gistheweight function andg(t)>0.Wegivethegeneralized mean value theorem forintegrals: Theorem 1.2([1,p.257] Generalized mean value theorem forintegrals). Iffis acontinuous function on(a,,gisintegrable on[a,2] and9>0,then there exists anumber &.in(a,2]suchthat [s@ateae= 106.)fFola. (12) @ Obviously, Theorem 1.1isthespecialcasewheng(t)=1. ‘Theorem 1.3 ([1,p-197] Classical mean value theorem). Suppose fiscontinuous ontheclosed interval [a,2]anddifferentiable on(a,x), thenforsomeczbetween [a,2],wehave 1p.) —£62)=f(a) iq) EX, (as) Wecould estimate czas€inthesame wayasin(1.2). And wecallcea . differential mean point offon(a,2] ‘Theorem1.4([1,p-198]Cauchy’smeanvaluetheorem).SeepropA Suppose f,9arecontinuous on[a,2], differentiable on(a,2).Ifg(t) £0forany t€(a,), then there exists cz€(a,x) such that F(z)=f(a)_fC)= . 1.4) (2)—9(@)~ae) o# 2 e | HerewecallczaCauchydifferential meanpointorsimplyCauchymeanpoint, Recently, there areanumber ofstudies onthelocation oftheintegral mean point £.,asz+ a.B.Zhang (7],improving B.Jacobson’s result [4],showed that, iffisr(r€N)times differentiable ata,withf’(a) =f"(a) =... =fO-Y(a) =0 butf(a) #0,then lim#—2-_1 | eeG-a (+1)? Schwind-Ji-Koditschek (5]went onfurther. They allow fandgtohave a'singu- larity’ ata,Namely they showed that if and @ wherer£0,8>1,r-+3>—1,Ci,Cy£0,then =a Ontheother hand, ‘Tong andBraza {6}studied theconverse oftheclassical mean value theorem. They showed that iff"(c) isnotalocal extremum, that ¢ isadifferential mean point. Itseems tobewell-known that the four mean value theorems above are in- terrelated. Infact, Theorem 1.3implies Theorem 1.1. Under some additional assumptions, Theorems 1.1also implies Theorem 1.3. Similarly, Theorem 1.2 and Theorem 1.4areroughly equivalent. ‘Thus itisnatural tostudy theesti- mates ofthe differential mean point and the converse ofthe mean value theorem forintegrals. 3 e Inthenext section, weshall discuss therelationships among themean value theorems. InChapter 2,weshall firstdiscuss theworks ofB.Jacobson, B,Zhang and Schwind-Ji-Koditschek onthelimiting position oftheintegral mean point &.Then weshall prove corresponding theorems forthedifferential mean point CzOftheclassical mean value theorem and Cauchy mean value theorem. InChapter 3,weshall discuss theconverse oftheclassical mean value theorem studied byTong andBraza. Then, weprove aparallel theorem forthemean value theorem forintegrals. 1.2 Relationships among the mean value theo- rems @ ItisknownthatTheorem 1.3impliesTheorem 1.1.Thereisaproofinthe Calculus book written byCampbell andDierker (3}. Proof ofTheorem 1.1([3,p.209)). Suppose f:{a,2] +Riscontinuous on[a,z]. Define F(t) =fff(s)ds. Then F iscontinuous on(a,2]andF’=fon{a,x}. Hence byTheorem 1.3,there issome cz€(a,2) such that F(z) -F(a) =F(ce)(z- a). That means, [s@a=He\e-2). of(95U.) Conversely, suppose FisC!and F’=f.So F(e)—Pla)=f°float 4 ByTheorem 1.1and above formula, weconclude that | F(e)~Flo)=[sat=1(6)(a-0) Since Fisaprimitive off,wecould replace f(E2) byF"(E-) andobtain mie) =Fla)=Fla)PQ) Itisthesame form as(1.4). Thus ifFisC',then Theorem 1.1implies Theorem 1.3. Yo(11.3) Assume giscontinuous andg>0,then Theorem 1.4implies Theorem 1.2. ‘Theproofisasfollows: eDaf FO=[fe)als)ds, Ge)=['o(s)ds. FandGarecontinuous on(a,z]anddifferentiable on(a,z).Also G(e)=[a(thdt>Gla)=0 gare Hence weapply Theorem 1.4toobtain some c,€(a,z)such that Fe)-Fle)=FIG) -(a) ou | \ ==HMeoles) ft froana=FESPoeae V =He)food. y= (tay 5 of(4>2) Conversely, under additional assumption onfandg,then Theorem 1.2implies Theorem 1.4.IffandgarebothCl,let6=£continuous, @=g’>0for simplicity, thenbyTheorem 1.2,thereexistsc,€(a,x)suchthat ® [seateyat =oc.)[Poenar. ‘Therefore, L{{ [$0 FS[stow So Se)=F(a)_f'(ce) (2)—9a)(Ce)* / : of(esas) 6 PlanetMath: proofofextendedmean-value theorem hitp:/planetmath.org/?op=getobj&efrom=objectsid=3589 (moreinfo}Differential Equations Calculus SolverePlaneiMath O@rgaaS2-~- Sa Mathforthepeople,bythepeople._Encilosedia|Requests|Forums|Docs|Random—E BuERG Login KorooPhf‘extendedmean-value theorem (Proof) sreatenewuser |etf:[a,b]>Randg:[4,6]—Rbecontinuous on[4,8]and name: [|==")itferentiable on(2;8),Definethefunction pass:v-beetvooworey|Pt)=£(2)(96)—9(@))—fz)(40)—£(@))—Fla)a®) +fOg(a)- MainMenu Because fand9arecontinuous on[2,4]anddifferentiable on(2;8),sois Wa Enoveloneodtah.Furthermore,h(a)=(6)=O80byRolle'stheoremthereexistsa Eapers€€(a,b) suchthat4'(€)=0.Thisimplies that Expositions ©(9-9(@))-9GFO)—f(a)=0 meta Requests 71) and,it9(b)#9(2), Orphanage (10)Unclass'd FQ)_fO-f@ Unproven270) 6)—g(a)” e@Corrections(120) 7)a)a) vO talkback Polls Forums “proofofextended mean-value theorem” isownedbypbruin.Feedback (viewpreamble)Bug Reports View style: nmmagesBal somjewstyle:[HTCwismoosTlERI Snapshots.Snapshot SeeAlso:mean-value theorem intormation othernames:proofofCauchy'smean-value theoremDocs ssobject’Classification Thisobject'sparent,News Cross-references: Rolle'stheorem,function,differentiable, continuous, Legalese History Thisisversion3ofproofofextended mean-value theorem, bornon2002-11-12, modifiedChangeLog 2003-01-27. TODO List Object idis3589, canonical name isProofOfExtendedMeanValueTheorem. Accessed 1690 times total. Classification: “AMS MSC: 26A06 (Real functions ::Functions ofonevariable ::One-variable calculus) e Pending ErrataandAddendaNone. [View all4] lof2 sad‘11:31AM. PlanetMath: mean-value theorem http://planetmath.org/?0p=getobj &from=objects&name=MeanValu... 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Crosseferencs:nera tangent, heorm, dierenble ten contouous, fc ‘Thereare9references tothisobject. ‘Thisisversion6ofmean-value theorem, bornon2002-02-15, modified 2004-07-21. Objectdis1890,canoricalnameisMeanValueThecrem. ryRoteasod8090tenestt Classitcaton‘MSMSC"25406(eltmconsFincafnvalOnects) 1of2 10/14/2004 11:36 AM. PlanetMath: proofofmeanvaluetheorem http://planetmath.org/?op=getobjéefrom=objects&id=2960 2)prtauton atausae2422 PlanetMath:@rgese r Be mae e‘Mathforthepeople,bythepeople. Encylopedia|Requests|Forums|ace|aandom[jie Login Aproofofmeanvaluetheorem (Prot) sraatenewsser pene A() on[68]by gg SE| poss:———} - $0)=Ha) = AG)=s(0)~re)-(AIL)(oy Peeters 7 Clearly, Aiscontinuous on[4,5], differentiable on(4,6), and ‘Main Menu ssoctions h(a)=f(a)—f(a)=0Fistopeeda AO)=0)~$(a)—(AE)a)=0 Bases Expositions NoticethatAsatisfies theconditions ofRolle'sTheorem. 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ForumsFeedback ; ‘nthe z-e>0, ‘BugReports because thenumerator inthelimitisnonpositive intheinterval J,yet |,88xapproaches c eSeana romtheright.Similar, ; domnioadsa,f2)-#0 ‘Snapshots L=limAII)59 anan yoe Information SinceFisdifferentiable atc,theleftandrightlimitsmustcoincide,so9SL=RS0,thatistosay,Doos ¥Gassifcatin —F()=0.News Legalese HistoryShapaetoa “prottRoteesteomedbyenon[aur|unesey1)FOROst (viewpreambi View style: [ivwininooosaBENRAE ‘This objects parent Cross-references: differentiable, limit,numerator, one-sided mits,derivative, opehinterval, extremum, constant function,interval bounded, dosed. compact tantinoussetae ‘This version2ofpooofResthaoem,bornon2002-06-27,modified2004-0226. Onjectid's2047,cnonialnamesProofSRotes haorem. Releoved 1310 tines tl Classification:AMSMSC:25408(Rete: Functofneva:Oarcai) Pending Errata andAddenda None. e@ [Viewall2] lof2 10/14/2004 11:44AM.