Analisi due. Teoria ed esercizi

704 Pages·1999·72.29 MB·Italian

by  Giuseppe De Marco

I Decibeedli trei,Pc ado
© 1999 va
GiusepDpeMe a rco
rgoi Villella,
Decibeedli trdiicG ei o
viad elS ant3o0 ,3 512P3a dova,
I telefo0n4o9 785 65 6 049786 23 05
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decibelz©tin.it
Realizzuanlr ieb èruo n operazionec omplescshaer, i chineud­e
I ' ANALIDSUIE
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© .v
Indice
Prefazione
1
Seried ip otenzeC.o nvergenzuan iforme
1 1.1 3
12. SNeurmiederir ip e oatleei nu znmee ric omples.s i .43
Operazicoonnil e.s e.r.id•eip .o.te.n.z...e . '8
1.3
1.4 Lafu nzionees ponenzcioamlpel essa
15
1.5
Successieo sneir ideif unzioCnoin.v ergenza
uniforme 21
1.6 Passaggiaoll imistoet tiols egndoi i ntegrale;p assaggiaol sotto
limite
. . 32
ils egndoi d eriva.t . a. . . . . . . , . . . . . . . . .
. . . . .
t:.".'\'"
normatis,p azim etricit,o pologia
'spazi
21. .
Spazvie ttornioarlmia ti
2.2S pazai p rodotstcoa la.r e
2.3
Spazmie tric.i .
Spaztio pologici
22..45 L imiti
. . . . ..
2.6 Prodotdtios paztio pologici .
2.
7 Funziolniin eatrrias paznio rmtai
2.8
Normet opologicamente e.q uivalenti :.I
Continudietlàl aep plicazmiuolntii lineari
2.9
22..1110SS puaczcime estsrieio scnoiit tcoosm.upc.lc.ee.ts..is.. i .. on. i. .. .;,· ···!�;' ·
. .
2.12C ompletezdziva a rsip aznio rmat.i
22..11ES43pqa uziicv oamlpeadntezetla isl n eeoq rumeein nzd iiamlemncesonimtop.fi enan eti ttia ··"·..·-.t . ,_ ··I
2.15
Spaizc ompatet sio ttospcaozmip atti 1'·
2.16
Funziocnoin tinuec ompatti
tra
21.7P rodotdtois p azcio mpatti ··. . � .
21.8
Applicazi.o.n.i. ..... ·r �;<
21.9S pazcio nness.i . . . . . .. . ·
2.20L emmad ellceo n.t razidoiCn aic cioppeo Blain ac.h ::;···1:·� ·.
. � '
. ..� 1�
�
1···
.......
I vi INDICE © 88-08-01215-8 © 88-08-01215-8 INDICEvi i
Funzionai v alorvie ttoriali 121 Formed ifferenzidailg ir adou no
3.1 Vettodreer ivat.o. .. . �2-,L 6.1 Itnegrladii penddaep anrtamie .t ri
I 3.2 Teoremdae lv alomre dio 6.2 Formed ifferenzdiigra aldio u no.
3.3 Poligionsncarlieti rte et tificazdieolnlceeu rv.e .!_23-. 6.3 Forcmhei useefo rmees at.t.e. .
3.4 Asciscsuar vili.n e.a . . . . . .. .. . .. . . ��2 8 6.4Fo rmech iuese fo rmel ocalmeenstaet te
II 3333...58.76 IECCnesutnerenrsgricvua iecalldzu(eli itrfl' svueiisenrlstmacaiie)laslnpo gdds e nriiidiiueaff ec nze.liair l .cf oeeu.unn r.eznv.d iia z.' an a.auli rv. enoca.( isonlon. o pd iv..r aes i .ttez )o urii caloli.i d eo )11134433082 66666.....67859 ILCOFneomotmnoremttmagoreed rpo siaiiafrP azmtoas imtci .uoaeinc. ng (rc pieacaun ..mru.le.e éim.l i. t -e..iin.b p.) n avr.d. .cii.ai kum. .rn i..pati..fo ie. .ar ov.r .m nema zm oeatd ioapnftiuean ziorneig ola.r i
4 Derivatee d ifferenziali 151 Ip,tegraziomnuel tipla
I 444...321 DDT eeerorirevimadvatee easl v ez paciaolrmonozedunidornai v lo ei ttore 1-ì55'�145_ q. 7 777...321 IFInunnttzerigoorddanuao sipzl iicpi.o a.n, ..lei.. ni..t.n.e..og..r a.. b ili 22329909919 9
44..54 DDeirffievraeptnaezr izailiaedl ein ticamneunlt.le e Ì1'605 7 77..54 , IInnttegererna m-lvidiasi umlrelen. si.i. o..n. a .l i 330022
4.6 Teoremdae ld ifferenztioatlae.l e �64, 7.6 Propriedteà ll'in.t..e g rale 303
167
I 44.8. 7DRee rgiovldaeidt d eail flf'eirnevnezrisaizoinoen e 173 77..87 FDoerfimnuilzdaii ord niiied nt uegzrialo.en .me.u ..l.t.s i. epclooRn idemoa nn 330064
4.9 Funziodniic lasCse1 �5._1 7.9 Insieemil enmtearmmeinsutreab ili 309
44..101 TG1ero ardeimedine. tl.e valor , medio )�7 6..,, 77. 1110 IMnissiuerenmalo ien mm iesnutreaa rpbel.i u .l ri.i . n. t .e. rva lli 331101
nzon positivmeanteo mogenee 179 7..12·I ntegraelset easdou ns ottoinseileemmee ntarmmeinstuer abile
4.1Fu2 i i 312
4.13V ettotrain genet sie mita.n genti 180 7.1G3r afie cmiis urnau ll.a. ......... . 314
4.14 Tangei,na tdu na·c urpvaar ametrica 183 7.14A pprossimazcioonna ip eret ic ocno mpa.t. t..i. . .... . 314
4.15D ifferenze vieattboirli ittaàng enti 185 7.15I ntegrabliolciatdlàee l lfuen zioncoi nitneu ...... .... .. 315
4.16T angenet dii fferenziale 186 7.16D imostrazidoenlte e oremdaii ntegrabliolictaàdl eel lfuen zioncio ntin3u2e 3
4.17D ifferenzdiiao lrid inseu perio. re Ì87 7.17S uccessiionnvi ad..e .n .t ..i . . .. . 342
4444....221101FMFP 8oasr9u rsinmin ucmzliiedcidap imT oioi imaonnasn y isvilmiodle imro oisp.. c i sa. ùvle ai r iab. ili ..Ì..2r.2 n'00'. 5679 3'· 7777....2211180FAAE 9nffupc enoptroatnzld soiiidiu u i cecn ac fia.manzbmsi.biiuaiitoma .neenm.ndse i.tin..idvi tmia. eovi. r asmiruai.irbfi;aei.n..b l.i. ie t l. aem .enat.ded i ti.v e 333332238460
4.22 Differenz.i.a.z ione 7.22A lgebrdeip artdiiu ni nsiem.e. . 334
4444..2..223256TF LO 4eu'p oneerzqeiurmaoidaazndmi iitiidp o ffeolnnelirevrlco eieeinn td rzele.is o acilaoil ne e -�222323141 } . 77777.....2222235647 SPMAS uasduisbdbsaaiagdtdupgdidoriiviasnoitteill tiui. tniàvv umifim. iimvni.etteesit.ààrr ute . alae a.blmms .biee.ui.l in.caest.cdl ..e.edu e. .sir st ...miei ovo ne ni ot.o ne 333333333387965
5, 5V.a1r idDetiisdàpi ff affezenioro ermnmozaeirt fa.iufil. nis .z .mi .ori.n e i. . g .o d.l.e fi. an .ri. sit . ue .s .o .t.t o. i. n.s. ai.re. bm iit rari 223377 777...32209IMN 8 niu vsmuaedrrrivpeai aae bardrt inrdipaslzaielreatatad z emiàliteledv oircme nilheitl me. aàis. su .uus.ru.eraal. l.ee.im.. e...n t.a re 333444120
5.2 Funziimomnier siev fe unzsioommneirs .i.v.e ....... . 2s9: Operatordii fferenzivaeltit oriali
555...345 MVEasassreiimempddtiieià i vm aiffrneiiermvtie.idà . nin c .c zo.l li.as aMaes.te tlit.e.i .oi. nds. od p. e ai.md oz..i dl.i t i . im.p .le i.nc d.safi..it Lin oa.aior. igtn rea .n g222e4'445 64 : 8 888..1.32 FGCoarrmampdeaiindv tieeffdt eitur oernnisozaco.i la .cail a.l am..e r..p . e o .d . i.c . o. . v.. .e. . t ... t . ..o ri 333355557363
5.6O rientabiel oirtiàe ntazione 265' 8.4I ntgeraled il indeiu an c ampveot tori.a le 358
viii INDICE © 88-08-01215-8 © 88-08-01215-8 INDICE ix
88..65RF luosstoo ddiriuu e nncc aammppovve otte ttroaioller eitiat nrua anervsospoa u znoiat riudspiemercsinfiieona leor ietanto 336624 10 1Fun0z.iPo1rn iidm ienu onzai voa.nri.i. ab.icl.oem. .p.l.. e ssa 446655
887.. 8 FD uilssvoe irndgim dieeu nsninocz naa2em pv. .eo.tt . o. ir .ae. l .. .. .. .... . . .. . . . . 336791 1100..FuR23en g zoieold noiidl e roimvaozirofne.ce o. m plessa 446687
8.9I lt eoremdael lad iveregnza peru na pallae uclidea 373 10.A4p pliccaoznifoornmii 470
888.1..211 FIA01pltr p leoceaionmziartorn ediegeeoldalirderusla gi.a uav. g le.i.rn a.zgna. iee s ocnl.pase .zo ri. mpa.i e. ta r i.nca. ol. n;a fo. el rp. mi .au nloda i G reen 333777498 111000...IID675nnut edegl iraecdzimeioamn veiuv s touiliclag imi mmien.nit. o 444777941 I
88..113FC4 ao mrpmiuc ldoai Sn :ts oekrpev.esa ir .t fll. i. u .. v s. isp. oo. .;t .e ..n . vz. ei. ta. tl. oer e 338841 1100..PF98ru inmziitooindlviee lo lfmeuo en zfriufeoon lnoimz oairrfmoeon nichi e 448864
881..51 LA6al platcoirsasneero av. zi .o n.i . ... . 339921 1100..ZA11ne10arle itii p critinàcdi peioldl fuiein dzeintooilnt.oià. m orfe 448889 I
8.1E7s errcciaipitzoliati vi 393 10.T1eo2r emad elm assimom odul.o 492
8.1U8n 'appli.c.a.z ione 396 10.1S3e ridei L aurent . . 493
8,19F ormed ig raddou ee t re 397 10.S1i4n goliasroil.taàt. e 496 I
8.2C0o omol.o.g.i..a. .. 409 10.S1i5n golelamiirniabitlià 497
.6 Singoarliàtp olar.i. . 498
101
9 999M99999.i7.......s435 1 826u MEMMuUFrF isiina-uuetsssi danneuuuc inrrrilzzeendsateeigsidtermbià Clite loeidoio e'sneegnelnsnbund arrtrm.lieaetirnagiblletm iliseez. lea.ài ii iniosi siag nu . .oa . selurn..rienl. ld. .rm. L e.aela ..e ie La al.L..be. l b em.ieb.bl .i ..se.ei sa.gs ls.g gu. ieuuue.. . me r i s.aur a.bil i 444444444111112112164102891 1111111100000000...,....TLFLSSMR21222112ieuenooie39178420gontaoggnrs r z o liaeiadgimtpimoritroo.mddeàan eedd.irl d ieulirs.ii e r tasauuli o cele a nmrmnls. n z cdc o.asioii aearisld. trfeiiumr...ia à fi i.vmczc o..a..inluit....e o .n.ia g ota. r o i tmica 545545550909100029780035 II
99..91AI 0nn csorisnaeu olmnmlfuiei n zsiuornmi.ias .ubr.aib.i.lli i 442242 1100..IT22elt56o ree meoamr dadeel llmlaa' pipnaad pieclotraagat r.oitr mei .oc . . . . . . 551113 I
9.1S1p aczoimn i sulart aerm;.i nologa. i "quasoiv unque" 425 10.C2a7lco lod ii ntegrcaoliin lm etoddeoir esidui 514
991..31 IAA2dln dcutntiiivef igadtfirteàdnaitl fubi lilenate en zsa ui.e mmo .iens riu.ar a bbpiiosllitieiv e. 4442262 98 11100.029.. EV23osaLeg80lrcar o iizrpitirvi m iad nriuci .i n. paf.a .lui.n. ..z. .i.... o ne 555222827 I
99..511 4 JR). ....... 429
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I
.JX INDICE © 88-08-01215-8 © 88-08-01215-8 1
11.R1i5s oludziei qounaez ailodlniee r ipvaraziatli:eil m eotdo delle
caratter.i .s. t. .i ch.e. .. . . .. . . . ... .. . . . . . . 578
I
Equaziondii fferenzilaliin eari
12 583
12.G1e nearlitàs ulleeq uazdiioffneir elniznieaalrii 583
12.E2q uazionil inedaiorr dini en . . . . . . . . . 587
I 12.S3i tsemid ifferelinneazriia ac oleiffi ciceonsttia nti 593
12.4E quazdiioffnneizr ielailanirde iio rdnian c eoe fficientic ostanti 603
Seridei F ourier
13 13.L1em mad iR iemann-L.e besgue 661155
1 13.U2n 'appli.ca..z..i.on.. e 618 Prefazione
Introduzionea lsleeer d iiF ourier 619
1133..3S4 er iet rigonome.t r.i c.h e. . . 620
13.C5o efficideiFn otuir ie..r . .. . 621
I 13.pN6uun tculdeaei Dl.o i.ri hc,el te. .d ...mi.os.tra..zi.o.en.d .e.tl.e .o.r de.imc ao nvergenz6a2 8 Quesètu an nau oevdai zriiodnoeect otna denisnua nut nai vcool udmemeli por e­
13.L7' intdeigD riarlie.c .h.l.e.t. .,. ..... 631 cedetnetseAt noa lDius/e1i,A nalDius/e2i.M olctaip istoonsloti a itnit eramente
13.S8u ltelndae nzaa z erdoec io efficideiFn otuir ier 633 riscreiv taterisi,e rscoindzoii v earlsmiei;nls oe sspaencrte an dteoml a terèi ale
I 13.S9e rdiiFe o ureis eerrdi iLe a ure.n. t. . . 634 nuovos.c odpeolp lrae ceeddeinzteiero qanu ee lfolroni rete stoc hpeo tesse
Lo di un
13.I1lt 0e ordeiFm éaj .e.r. ........,. . 635 servçiormgeeu iedr ai feraimnecdnhotepo iot l e r.dmeigsnlteiu edfo is,sc eo asmì­
13.S1p1a azp ir odostctaoleas rpead ziii l ebrt. 639 pio permetutnesarc eea ld toac ednigt uis atnic mhoel dtiov ersspie.r anza
13.T1e2o rdeimP ai tag.o ra H 640 disd ear vciormteee sdtiro i fertiomr einmaannec pheeir pl r esveonLltaue lm ase c;e lta
1133..SP11reo34eri zdiioiFene oo urrtiioneg ro. n..a.l.e.. . . 664402 degalrig omèe pnetrsiòt altiam iatq autlealat radiziHoona anlcech.ee r cdait o
L�. ...... cambilarose t idlipe r esentea rzdiei nodnpeeir "ùld oe mocor"a;ta ighcgoi udnit­o
13.T1o5t adleilt às tirsitegmoan o.m etrico 643
13.I1d6e ndtiiP taàr seval ... 645 mostradzoivpoern iilm aas cliapa rvooav lla e tteodhr oee ,l imilnepa atrcoth ise i
13.T1e7o redmica o nve.ru.gnifoe.rn,m. ezp .ae furn zionait ratti rivelpaivoùas ntoia clhmeed,no ovc eie ròa p ossisbeinclzoeam promneotztieornei
13.I1n8t egrtearbminiela tiertmiàne d elsleerd iiFe Co 1u rier 664467 essenLz'iianlsii.ès qtuepini szùula l 'duellseoc o sel;ed imostrsaipz oisosnoin o
13.U1s9do e lcolmap eltezza:s padziHi il bert 649 quasseim pormee ttaeder see;m qpuieosl ullel 'idniLt eebgersasguloeend oe dicate
1133..2IS2s0oe1m ro drfiiisvm eeto tod riiLo r�ct ootn2g( o.Z.n ) .a .l.i ... 665531 savlsoltelos ut odapie rnnattteeptr puieelactc sturtisasooa oos, tclm aaooo ra n il tceudchnneegqefi ou n eissizvtnioatoplt eneteribreoa e p r noeiets ae,es s oasr eermei
sool nuen iat. i
Appendice c
À.lT eoredimo am omorfipserm sop avzeit ita)ol ri 665555 Hom antenlu'tiom pcioengtoei dne usos endzeilnaloleze i doisn pia vzeitot oriale
A.2A pplicmauzlitoinli(i neari 656 efu nziloinnee alrien:g udaeglgli'oA.a ntatludiaeslvuiees airc eo ncdeetltli' alge­
il
A.3S erpireo doetc tnovoolu zione. . . .. . . .. . 659 brlai neèai rneu;tg illiess sualdrleeafin izdiiofo nremd ai fferednigz riaaudlnoeo ,
evitdaind daori lnma o deos pliecsisèt uaon c: a mpdoic ovettori.
Formulario
663 Hoa ncrhiet ednidu otvode irs cultace ornet idneulfuilntezà i loinnite rasarp ia zi
Bibliografia 677 normaelt 'ie,q uitvaolpeonlzdoaeg linlcoear miend imensfiinoinètq eau :e spteor
meu nfa ttfoon damendtgealslipe aa zp iid ùi menscihnoeenr ii,vl eacl aan onicità.
Listdai s imboli
679 Peqru estteos tgoia àd i soiszinoeu nl ibdrieo s ersccirziiitnc, to ol laborazione
è p
Indicaen alitico 681 coCna rMlaor icoenp duabli,bc oai tnq uessttae csoslal aensas;iro i ferimenti
in
sonaol vleac cehdiiaz iinod nueve o lummia,i lli bèro ov viamaepnptreo priato
ancpheeqr u esetdai zione.
· Purn edle isedrio dia lleggemrasisriemi olv oluemdea ,n csheei ll ibro
al
die esrizèi g idàisp oibnil,en onh or tieuntoo pprtuonot oligeetr ropepsie rcizi:
c
2 88-08-01215-8 88-08-01215-8 3
© ©
I
ritengo ecshesese ssniiza isnavolo ecl otnit emporanaelaslmtoeu nddtieeoln loez ioni
teoriqcuheess;tp oi elgama o ldee lilb rcoh,pe e ir4l 0p ecre nètc oo stiteusietro­
di
ciezc io mplemLeanm toild.ee lli bèrp oe raòn cdhoev uatdua n ac errtiad ondanza
die sposiczhiieoo rn iet einngdoi spepnesirap briilnec ipLiea netsip.o sstirziino­ni
gate imprepsesrlia lo onrmaoan gog ieolreeg amnazs ao;n aop prezdzaagiità e
espepritcùih dea galpip rendisti.
Anchseel ei mminernitfoirp moer taanduo n ad rastirciad uzdieolnqleua a ntità
dim atemaitnisceag nneapitr ai amnin lios, tudceunriteoec s aop adcee vavee re
ad isposailzmieolnnapoe o ssibdiigl uiatràd galraisr ig omesnatlitd aatdlio cente
CAPITOLO
pemra ncadnzita e mpo. 1
Quelsitbcoro on ticnoumau nar qiufieu tla'ricedh elea' AnaMlaitseim aptoisc­a
san one ssecraep imtaas ,o ltanto acphpes lisi tdcuiac tiaos,èo luon ar icetta Serdiiep otenze. Convergenza uniforme
risolsuetniezvnaat ,r naermlee rito.
***
Nesli twoe b 1.1.N umerir ealie numeric omplessIilc. o rpdoep. iu merreia èli in dicato
coJRn ,q ueldlenoiu mecroim plceosCsn;i indicihnse ergàu uintoqo u alsdieais i
OC
http://space.tin.it/scienza/gvil.lel.l/decibel.htmdlu ceo rpi.
Dat>.o E conl>- 1i ndichiilva amloo arses olou mtood,u ldoi>., . ha
nsiatrepàeve ernitoaudgaiglciaiagual mgn teitenoestrn elteap. 'o tue aror evacsitoe ar rrdci'diigee zq siua emsveto.ol uems ea rafonrn­o olg>-=ncm1iaax s{o-E OC>.,JR, >.e s} e >.2: èO:r . e a,ll Ie>.= (a+2b 22)s 1e>.1 = a+i bc ona, b E SJRi;i n
Quanl>-ds1oc r ilv1e>-d riesmuogu agltiraean lzeem deiOCn ,èts io tticnhtteea seloli e ­
***
mentsio nroe acliio,è EOC , dir>.e,: E JR (çOC )e >. $ (iJRn,
Alcuneis ercmiatz aliv,o alntcahi en tesreez iodoa p.licu,tne io resmoins,oe gnati conl 'orudsiunae.l R >.ei,µc ord>.ci haµ$ae mvu l ooc l o rpcoo µe soC novni eanµse s egnato
cognl "io cchiQeutetsit"i. simboli significano: alcournd ienc eh,)èe i mposilsedi ebfinsui reom rpldsitnoet aclhreee ndac orpo
QQ seisgedlsrifait craal itunizttd iuia;tr ipdle,iir cl o ntrdoicl olmop rendseilotlneaeo rsiica o;­n oprrdidivisne(v ea entdso1io . .1 l.ed3i )s:u guagfrlanieu ma une nrzice o mplneosnrs eia sloein p oe rciò I
()() materiimaploer tdaanl teeg,g ee rcea piarfo en don:o nt utitlmo ateriale Dat>.oE C ilc oniudgia>. st oii n dicco_a>.n ;r icorcdhoeig an>.miE o C s is crive
importèa cnotsseìe gnaatlotv,ro el itcleo ntedsotvor efabrb cea pilr'ei m­ inu noe du ns olmoo doc om>.e = a + iba,,b E JR,d ovae è d etptaar rteea dlie I
�� pmoartetradpineiazllaùla'se at rrgaaotc mtaeorn,at ttote;eo rreoi g ceon,e rcaolmep;l ementi; D>.,ea t=ct Rioe(pò >.e,,dr )be eè fid neitzctèioo ffio=e Xc nai e-e indb=te Rele l(>.')-i imI mm(a>.d)g.i>.i ,en s airiion dciocInam ( .>. )
esercizio difficile.
®® 1.1.()1() ESERCIZIVOe.r ifilcesa ergeu foernmtuil e:
I
Re(>.+µ=)R e(>.)+R eµ() Im(>.µ )=+ I m>.()+ I m(µ) \;/>.,µE C,
Re()c >.c Re>.() Im(c= >.c)i m>.() V>. E Ce, E R ,
=
Rei{>. ) I=m (->.) Im()i= >.Re(>.) \;/>.EC , I
=>. >: V>.EC,
1>-12
>.-l V>. CE' {O.}
=>: 112>- 1
1.1.E2S ERCIZIDOa.t o >.p rEo vcahree >.-1s ees osloe
IC, è>:= l>-11 .
=
sisN cortitavzaeiuls oavnatoinec c:lx h§t=1eea;IC ' {xèi O lgu} r=; u{ zpmE e po o:l \ tz=i\ lp d}lce,iini ruc cmauoentclrioioitm vapcorlhi eeos ,s iI. .
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I 4 1.S ERDIIEP OTENCZOEVN.E RGZEANU NIFMOER 88-081-50-182 88-08-01215-81 .S ERIDEI P OTENCZOEN.V GEERNZUAN IFORME5
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Con incteinòdam ioc hes ew ,z E c x,a llwozrE ac x;i nolltamr oel tiplicazio­ Pers tudilaarc eo nvergedni(z * a)p ossiliamitmaor coiv viamenstteu dai iarle
I wne-è= 1as s 1/owc Eic xah,tca uii nova euè,nl n eummeenrentout altocerco h h,weeè w 1 -,e=1 d ow g-nwi1= Ew c xch eha cu noi mnvevie srstoo casoe= O,c iloèas erie
nell'e1s1.e1.rè dc aitzdoiaw o -1 = w/lwl2. 1,
I (=LInoaalmc, rtooe lmU,a teè psi popetlvnèitiacs ocnt ogaeodmrlz emlugril'upuoe tppnsaceope1ctro o. xci1m ic,s.vsez2i aaie a.,osp o, Eeè pa. 1!.J , i {3)sa E tmUiaoa n1!.J .l. l a.o{3rE aU ed a.-1 DiciamCoa l 'insdiice omnev erdig (en*z*(a)a =( na)neNè l as uccesdseiio ne
I csiePÀr =e c oourOnlg ,itaoan>. liriEo t1!.J Cr.s ii dse mfigneinsn=c(ti e.X.Xisl )/ e IgEdn.Xl!.J >.oil;, s s gg(n.Xn()èn.X,le l) amp ordoois eezgiruoeanndseti egd:ain>. l(=s e.XOu ) l vc1uo.ole2t Lffi.oE.c MicMehDAnieIA tn B iEdL,i.S v liiwasadE e u ratCic ae hc)lih;eaie r tl ae OmiErm enC ngeatec ,enh eeq aurainlnnwedn dio e nè lsm leaari i e
(* *§)i lai miPteaortgo zn. i C taclhee < l lwzll las eri*(e* )a sousltamente
E è
congveenrte.
I DIMOSTINROEAS.Ze la nwnl per osgnihi a
� M n,
n n n n n
lanz=n llan (z/w)wl= l anwlz /wlln lz/wl
� M
n ·
Las erdiit ee rmgiennee rlazl/lew conveerdgatcoh !ez /<w 1l; critereli o d
I M , il
confromnotsota rlal cohrleaas e r ie !naznclon verge
FIGURA1 .1F.o rmap oldaernieu mecroim plessi. I::=o . O
particsoewl E aC raea,,l ltourtato d iscoa pertcoen tOro rea gg!iwèo!
I tÀO= eg rrnnumiuc,i om nneuc iEro.l!.J om , rp > l >.Oe E;cs i scòsx osoi ist ctriciieonuvnrn nee e o= Id u .X ensl u o= mlsoog>.d nn,oe es lolclaoo q nfu oermsati polaralceonseI eents,sn o ustolei iunnCntdc fiaoao(sn mniire vtineeèctu rsoècengichr a semeo sdu onenciu ,tcv ne eesas rse iliglrèoie i ncmneoei nt vtseeair)ridit .gc ea ion;rsaltdul e iott, r e chersemiìun gen esaen reirea ­
I (è1n2 x.=e 1xgO.xSa3=StE( M iR-aVv-x AoZ=1)I. ( Oie IN-d2èEèri xo u.x )ng >q ncOuoioa - d rxrtp> aoiotOtn,Co s .ae x Ple=/.omrr Oetidn)ana1ntn:to=toie 1 on2q e èu s piaosidstsrnCeioa uatntnpiloo oisvirl -iovtdo1,iir n ae cdBie(Pc ncee=tc,crn E rer teBCr o e[r(, ar r o c ga>= ,ggrOCrgis r).]iiPop).e oe o B rrn(B= e cO(= ,, sc[ rirp{= ]o z{E nC zBEe: C( l :,cr- z![=c �z 0�lc l<,r B r}((d} ci(d ,s=iccsr{oo,o]e ,oa lps} pleara;c lh a=il+pu ooasde ,air ta
totacloe mpatcionbl iaslt reu ttdurica oord piC . Datas eripeo tenzean zne,d etCtaoi s lu ioni esme conv_e rgenza,
I 1..14E SERCIZIDOia.m op rilmasaeg uneetd efinizuinso ontet:eom ieAn sCKi sdii ce sip onel a di L�=O di
Mbiolsatncrcitaehaaprt loee,uvr ano s nlocttehaqteu o iilbinbiisralnKitao es nmepce deio irKgan cita sioE o Kns,coo o lljnoesa ::; ej1 g ,su aiehal n attaAeiç tr A in.v e: Ra=s up{!w!:w E C a},
A
I t> AA== 0 ;K ; estrseumpoe rdiemoior deu dliwi p i e cru lias ercioen veRra. ègu en;n u merroe ale
t>t> esirs tO2:e,r E JR taclhese iA a= {xE K :! xl<o pprur}eA, = {xE K :! xlr::;} . dnotanne gatoipvpouè,r +eoo R; aè d etrtaog dgici oon verdegleslneazrdi ia peo t enze
a .
I 1.2.S erideip otnezeS.i (ana )neNs uccessdiion nuem ercio mpleee sE sC i. 1.2T.E2O RAE.MS ih aB ( OR,a ç[ C aç B( OR,a ]I·on ltsruBe ( OR,a ([ chèie l
(Lnaas) reNniè edle ipa s o enetrz eideip, u nitnizoali e ee c oeffic(ioe sunctcie ssdii ocnoee fficienti) gdeinsaazcpsoaoesl druictt ooaen .rv geonl z'aei,rnn odt ecle rcdhici oon vergseihn acz oan)­v er
DIMOSITNOREASZ.ez E (0R,a [e,s iws EtC ea t aclhe!e z < !! wa!n ;o rmdae l
:Z:00 :.:an(:z-ct lemmdaiA bellas, e rI:iB� e=O an znè laolar assolutamente incnp o anrvteir­gente,
n=O colare coSnepv oewir E g e'-CnB t( eOR.,a c]e,r tamleasn ertLie�e nO= an w non
èc onverigllee nmtmedia:A bemlo starnzaci h el as ucces(sain)onwenNen oèn
(ès otticnhtpeee zsr= o e edn = Os iha o 0= 1). nemmeno limitata. O
6 1. SERIE DIP OTENZE.C ONVERGENZAU NIFOMRE @) 88-0182--1085 @) 88-0182--1085 1.SE RIED IP OTNEZE.C ONVERGENZAU NIFORME 7
Il ragdig cisooiiuin { rpv 2:e r:ROlg a aspe uunacòznc acee eh(s asnes sn i)cern oearnNrèel a itimttcea,ort: sia ìz}zato sspnoeeihzrn h;/:ca c léo 1/ inmm,vzm-ee moor/nls+g metezollz+ =;r=v zl +e=1 vll ll,isme i ,ams( m msiszm=me a() cllm-n) l hz=oo+t sme m eeO+ 1=d + i < vl + 1 reoo g.S <,re pe a 1/ +e r,zoo<rtnm; +/ 1a tsi c,-+aeoi nn hO ficrinph ealaee lmm mr(zs e= -+llI ni1 zm-+ +m tn-+aoo)e ooooz ; s,iz e e m/, +z1s> =lzlm+ e1Oal ,,rl i e
surp2: O L::: =aonrnèc onve}r,g ente nonh al imiItnefsa.etc o tsfoiìs sea,n czhmeav rebebl os teslsiom idteet;t >.ot alceo mune
iii{ zl=i zmm+isatizre;emsb, cba ehr ie1abc,rb oaelnm t=le>-a rn'1ss1ot,eu e dna lzliionloriiamnzm -ieooa lz em.+ /zI m = >./>. =l ;m aa llora
surp OL::: =ola nlrn è c onver;g ente
en Lo'nènsir eisetdrmiciie tpo {trni2:evv one CrdCagena ersR nei+raz inadz uiCcscae:ohl léaco o}s sèeRua an fa= c iOèel;t se uez triC.tco soie cr1o2e.an. olc6Esr tS ciieiE.rAuR iddZtCo eIiIC Os maP.epue ticr-vroh Hodoy,almav ieaarltnarro,r gpdvdea gooie dirii co clr sho anpégvge eidnssioricnzo t og ane av emlnrgèoe son dzensaomi d ip erncelocs nloevanrel ntiiirerema niptincocetoo o i r re­
llRaaasc= iu +crafooch ;ersoie neOunB sz<R (ua aO rR< ,aa +oo ]B, è ( u= ORn i,a[ nU( SsOR(ic,aaalReso [)mdim ,oe pc {v ShrofreO(aena R }sv d=a oei i{) rlasxaE g sc peC lesorun:o jzBtztjs ( oOaa uSR R ,, =R(a}a [ aèe) nLoo =-nln .I n z n ' I
csi onvoa triidp icci oo nmdvpeirmo geresnontuzBt ;caa e ; rten oscneo rnivee mragioc,ne ord taepe pr­ ilc ritdeerlrilaoad ciocdnuece a lc alcodlio
tuotc,te ratnec oirnaa l csuontit o;il nac soinevmeirgenszea i,cng ' aeènn,ce hrea le malxinmCj
noènas soal.u t n-.oo.: n:: :.!;!: :..
Sussiislt e orano,tn u ttsia ncnhoeè (isu sia de sempio AnalUinso1i,91 . 1.1(6i i))i
I
ri12.ed. i3Cp o RtIenTzEeRL: ID:O=Io CaA nUznCiH -lYraH gADgiAdoMi Ac RoDPn EvReI rLgR eAèGn GzIaO D I CVOENRGENZA. Pero gnsie ­ nli-moovrnr n = .!.e ;
1 map eqru eess tialm siie riiecl,ri tedrieoral p portcoo nduicem metdaimaeanlrti esu tlot:a
Ra =. . sai zi O ssato;c onsideriamo
max lim\f'ia,;j fi
n-oo an+1n+zl (n 1
èmDdm(iI eilasÈls�iccns tISomhetTOisatqitRteAl romesiut.Ze ieo hiILne ascm aaO °'èsd co Nhia/ toEisF set\f'ia,;jnw.es >it ec o/ sl1la l ucsuèe,mwset +a è aiEraaoofatom ct sm.ccneoexi)sol,eenn n i:eil ltvmltCc mei earais ro2i tegièt BeeereOn d(n ro p,tse,Oin1eoco ilho r s/ lvtnasa°'eRaoemra dag=,e[e na= i m t,,prp+cae eoonplxe i,rmel nmic-saieslhoosnlcwut é so\f'ia,;jru atlE eèenIte a, R Cr 'L:ro a iB°'m: = g= /e (eiOnoaw Oen1ns<r wa/,e /e 1lnie ,al ] pvineeSfinrirne-+mg i s mitsei+elmoonenIte.o.- e tN )a p,eLezenz1s /a n <ert s g roe 11vIurac--,erniL:e h o-ei ::"L:ne = -rc+nl i- :!ao= a-l gn)!0gv(ndie, (+ordnicszn g lone o ee) v)"- !ann h1enz1v,,za;./n=-• I r + e na>--n gnI lrg1d ! (ioeg ( ln/+ ,e1pl rèsnc�d n ea oceizern phrrii tvalIea+e éetlns e t r/ rpozm-->go asiennesnee�eo.zgn l lz aezlu nit eaarlmaenel cnoeotè nen ­ II
stian
d1.ic2 o.EC4naS .vERe ICrZOpIog.sIi etivnpnoa rzsoelaee p s oovlseoer ei escl,ir sittone1nurom.i eo2dr ri.ieac l3cies h >le aOs t earL:clih:i=ees oia na/ zn na/h S as rna gp gOie r? sifa nI:oo =:On! I
nl a'bebsastitrnargefenizrmoadao eritn ,ce ldai siopRè e. ernn d2: res nic ,.oA nntdzo,e ii ellrs alga'gtidtodic eiozn vzqeauar geffeèsenal trzza/em i aidoo,.ven °' è nl-ioojm�( n+ l)znn!! I= nl-co jn i+z J-·1=m- ;O
12..(J5(J ESEMPIOS. erigeoe metari,c gleasne zr+aièoo eq;u lindasia u ssaos momlpaue atrcnatotmenoesv nieptsrpeetgreor e ego g nnznitzEi eEC C ,e, qd eufiinnhisladc aipe ri iaùdmg ipcogor ntivaoen rte­ I
funziondee llmaat ematica,l afun zioneeso pnneziacloem plessa,
n
Usanldaboe nn otiad etnitzàm +-l1 (=z- 1)+(z 1 + +z m)v,a lipdae org nmi E epxz=ez : =�� 1z · 1
!Il e zE C ,l ar idotmt-ae simsam( zd)it a el seriries c· sr· ii· v e n=On .
S(z)·1 -zm+l Los tesmsoe tomdoosr tas uibtcohe in veceL: ;:°=n0!z n"onc onverpgeera l ucnz i' O,e d
m = --- 1-z hap ertarnatog dgiic oon vergeOn.z a
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8 1. SERIED IP OTENCZOENV.E RGENZAU NFIORME © 88-08-01215-8 © 88-0182--1085 1.S ERIDEI P OTENZEC.O VNERGENUNZIAF ORME 9
1..72 �� ESERICOIS.Ze rileo garitmica, NOTA.C onq uesotpee radzii oianzdiid omnoel,t ipliemc oalztiiopnleit crcaaoz sitaonntei
e serliese e,r diip eo tefonrzmea unnoea - algcemobmurtatai vadetatlag edberals leeri e
I formaliinu nian determ(isncuaoltr capom op lseos),i ndicc, oaIC[nt za ].
Unp olinoimo(a c eoffiicenctoimle ps)sp iueòs seersneat o pcoumnesaer ied ip otenze
I cUperstrziae ornd1i idiClci,o a rn ui ctbaHhdelascayed rreir-ltaie ar mioopar d ptcdiLooo sern,itid rbauvnlocnbao ceein hn nz eioc, rlto lona v ig,.e-emgr.droog iicez yinoo n -v=pe1.r g1S e.i)zpTna uòaè d lsie(1epm roecrasilotrvrteoenecrrg ,hiaee l idemc ele-ue aicop liloprnogeomddffieioiCo,cbt d c[tiairhpiz,.e aos se.n].l i o sti cnntiroosu mL:i toinv;:'t =ncue0hioali aa r lkpai.azmL nr ekt'etid piinaomesr l me+oiedm n ei1iipq, npuo ii olnniidiontpi; sma o liiicli n andsmos icooèmncod ioCanue ( tn;gztssa r]ooo a tmdt­oo
I taÈlsae =le trrecsìiofa ecn ivlvseeed uretreg u cteth1U oe' { -12oo} :.n2 � = plnaefuDn s.atzaiotcuno oaanf smae e eeC:la rclie-+ i pm eeoted d eniCetnn) ,filozida as oetd uv aCf!aae aun çz i(z C=)o è lin s limemo'sn -+mooim I:eodasm::' =,iceeoa o m nz npnv l(=ei I:rcds�ge=emoOemalms nnelzaez"'nèa,rat , i e e
n=O data.
II chaoSr ntavugedmrgdiogiica ieeos olcsn oz nnmedapvoo, rlet "arz/m1 gens;1ntte2=a v ond lol ztena/al s2e li L: ac:;"=h noa1svl e erg/2rennèiz fia et n oitctsoaomulte,s eu a tplpataicma oic .ro nefenrzdai 1.(3i.PIt)1lraRraO Rg PagOedi SdRicob I; o .Zi Sn nIgivnoOeeaeoln.L;nN rR tezE�a=+rlabaOdl zs e'zelinsoLilnn�en=atdriesob eeen o,z grn imls nmèie sina riedo diei pnicnm eo ofiriten ogervaAnrnellzeoz sla roemih .i a n imo
faz()f b(z).
Loo -nn.I"'z n (ifiraa)(g+b gz=di) ico o +n gveenrRzaa•d be lsrleiape r otdtèoon oinne friaolmri en imo
fl
n=l tra edR bi;n lotnree'l ilnsteiezordnleeil n'terncoe rddceichi oi en grvenza
Ra
sulsluaca i rcondfeirc oenvenrgzeazna S(:es )elz =le s ih a sih af abz•()f a(zfb()z) .
=
esezri.co i
DIMOSTRZAIONE(.iF )a cile
(icio)n segdueelnlzaa coansvsetoralg,ue nza
È A.2.3 . o
sc(rreoioirnèdceqv ai uemiroctnsg hadte(selirn+1 ee ez1r nta/iot encnacTro m)esn e,ee"cvcn onetn etvrmei ergangive,e da cilotaoèoasrcc rse eho scileentul dtseo o aerz d Em egS ie;)in(ne rt;aleeslee )u rn lomclinèiagne ri encne rofianlnefid etlerdleiesa niz mao . p13re.Dd. aEe2Stufi EaCRnnI saiZOleIz .Ma rsido iesoitprrenao oei rett ce htue eI:etn d�Rnsaz=eeaO ee a r nd Rzin b,vs olfonan oasrdd umoisav ae eislrreidhimsat e Ri eear +nrib=mt I: vdmie�ae=ianl{ ltnRn i lan,,. sRz bde}aar1..tè"'ia e. -
Unsare iep oetnzele as usae ridee rivhaatnaln oso t erasgsgoi o
p1.u3nO.itp neori azzeiio=nOaci l)o cne l o esm eprlideei p toiatnnmezdieeidn.Uvut ndaeaa t lassdl ueipacr ot cieeens( zsseio otnt:eid nit eso 1d.ic33 o .LnEgMvMeeAn.rz a. di
cNd esiu cooi effiètc iiA.el nloep eratzrisaou ncic ecsossrirpoidononai=n q(o un a)innoEdpE­Ni e DIMSOTARZIODNetEt.i R �i converdgie nza
razitorsnaei er :di atlees erdiìpe ot enze I:�=O la sseormimea Ra, ragcligi
dditpi ao nltziehecli a c ps ooetmerseniuè zecl e cas e esdrI:sei�c=iieo(0o aennI:ffi+ e bc�n= i)Oiapnlen z;zr ln o,nsatd e ior pctibrootzendno ,nouv d tzoetilldool=un se eee rie
di e
delsluec cessdieofinnidi at o a*b
ab, em ostriamo convePrrgees.co o n <
silaRw< ,a/sR iha ,a chI:e�= ln nl/lwn- lla z, Jw/
Ln <
Jz/
Cn= an-bkk,
ogniEN (veadnic Ahpeep ndk=iAOc. e.3, ) E sid efinainscchee n Ja,.Jl. wn-ll=n !an/iw //-"1zz/nl-l= (, 1:-1�.�T)/1nzn la.
per n Data oIC , Las erI:i�=eO / ,.za.I"'c onvepregrecllh <zé R a,e de ssendo< 1,s hia
00 00 Jw/z/
aI(>nzn)= L (ana)zn. li.!:.m j?:j:-nl=O
n=O n=O n-oo
/zzJ
10 1.S ERIDEI P OTENCZOEN.V ERGEUNNZIARF MOE 88-0081--2-185 88--0182---1085 1.S ERIDEI P OTENCZOEN.V EGRENZAU NIFORM1E 1
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(cièò c non sidebreantn oo toè; a de sempcioon seguednezlfaa t toc hel as erie Sih a! a(w)f (za)-z :=lan (w-nzn) ;us andlio'd enttài
dI:e:=cllr ni zthedarer irlaoa g pgdpiiooc r. oNt neovs )eerggu1ee,ne zq au inidniu ltiamnaa lciisèòic onsegzuae n wn-z =n ( w- z)(nw-+ w1n z-+ . 2..+ w z-2n+ zn)- 1
=
-n-1 w1n-1< 1 (w- )Lzn wn-kk1-z
lzlz = k=l
qpeurin naadb nibc ashtI:ea ng:;zrnaan anwdnM-ea.è la asl sloojlruannwt naa-mlIce$ on lntnvezaen lrpeg er.eQn nu atibnbdRasiat$ a Rng�zr aa n,d e sit rova �
riLea r ecjinnpwrnaR-o1�cl lai $ m pèRlai is mcumabe iadqttiauo:e s ledll ia w< Rl�l, nl waac njoen,svse erngldednonewa zlnalls ae ­ lf(a=wla�(z)_� nanzn-1= 1l a�n�(U:n-kzk-1) -nnazn-11
jwjlI:nw :a=nl- ll jnnawn-lnlona ppenan l.wI: l�=O =
$ 2: O
Comen elc asor eallead, e riviants ae nscoo mplessuon af : $ l(t�anWn-kzk_-na1znn- 11)
di O -+ C, CO, ç
èd efininteapl u ntz�E O ,di a ccumulazpiero 0n,ec ome
J('z= 'Ww)l-+EiZC m f (W --wZf) ( z) +n=fNI+l( a n kt =lwn - zkk--1nan zn-1)I· t
set alel imietsei sitnICe . L an oziodnied erivcaotmap lessasrasà t udiaa fotan ­ Siao ral <z r <lRa; s upposltwo$ rl,s io titene
don elc apit1ol0oq;u ic ia ccontentdeiro esmsoe rvcahrele e r egodlied eriva­
ziondee lsloem mee dperio dovtatlii ndeelc asor elaes,i e stendono senza
diffi­
colatlàc asoc ompslseoi;np articoslepar (ez,) z ;:oaz èku np olinoimos,ih a
p('zI.:)= ;;"kzalk.- vedaen chseu bict=h oe : k
= 1k Si
1.3.P4RO POSZIIONES.f e:0 -> C dèi evriaz nEb 0 ,iè a lnecc ohnienzt . i nua < n=fNl +nla wln-zlk-1k1tn=f+Nl j+n anzn-11
DIMOSTARZIONE.h a k=l
Si
oo n oo
lim(f (w-)f(z ))l im( -wz )f w()-w- J(zz)= O J (z)'= O. O < L !aiLn ln-wklzllk-l+ L n!a nrni-l
�E6 = �E6 n=Nl+ k=I n=N+I I
00 n
Dimostrioarmaloi ' mportante L nalnl-r1 .
$2
1.3T.EO5RE MA.A 'lildnceteledr rncicoo h nivlosea org dmeuimnns aaze d arii,e · n=Nl+
peonètd zeebr ieill dvaeea ,rid vesalotlcmaaom iacn loscanmoi mddaese le lriae e poihcél as erzie:= ln lnlna-rlèc onveernegt,q usetl'tuimtoe rmipnueòe ssere I
derivata. resmoi nordeie /2s ceglienadbob astnazagr andes;ih aq uindsic,e lutnto a le
N Ne,
(Brveemeen:ta l'ilnterdneolc erchdiico o nvergelnasz oam,m ad i usnear diei
potenèzd ee rivatbeirlmei an eetr mine.) I
DIMOSTRAIZONES.i a iRrlaa ggdiioc onvergdein za anzn,c hec omea ppena
I::=o
vitsoè a nchielr aggdiioc onvergdeinz z:a =1nan zn-1fis;sa zt coo nj j<z Ra,
dobbiapmroo vacrheep ero gnei > O e sis8t> eO t alceh es eO < j-wz $l s8ih a sej w$j r . Èc hiacrhoea lt endedriwe a z shia 'I
n
L wnkz-k 1-- nz-n1--+ O; I
k=l
I