MAJHMATIKES MEJODOI FUSIKHS. KwnstantÐnoc Sfètsoc, Kajhght c Fusik c Genikì Tm ma, Panepist mio Patr n. Perieqìmena - PDF

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MAJHMATIKES MEJODOI FUSIKHS KwnstantÐnoc Sfètsoc, Kajhght c Fusik c Genikì Tm ma, Panepist mio Patr n Apeiroseirèc kai ginìmena Perieqìmena OrismoÐ kai krit ria sôgklishc Aplèc kai diplèc seirèc MetasqhmatismoÐ

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MAJHMATIKES MEJODOI FUSIKHS KwnstantÐnoc Sfètsoc, Kajhght c Fusik c Genikì Tm ma, Panepist mio Patr n Apeiroseirèc kai ginìmena Perieqìmena OrismoÐ kai krit ria sôgklishc Aplèc kai diplèc seirèc MetasqhmatismoÐ seir n Endeiktikˆ paradeðgmata OrismoÐ kai krit ria sôgklishc Apeiroseirèc emfanðzontai suqnìtata se ìlouc stouc klˆdouc thc Fusik c kai twn efarmog n thc mèsw ˆllwn episthm n. Mia apeiroseirˆ orðzetai apì to ˆjroisma a n, με την υπόθεση ότι lim a n . (1) n n=1 H akoloujða migadik n, en gènei, arijm n a n kajorðzei tic idiìthtec sôgklis c thc. Mia apeiroseirˆ eðnai sugklðnousa an isoôtai me peperasmèno arijmì. EÐnai apolôtwc sugklðnousa an h apeiroseirˆ a n . (2) n=1 An h (2) apoklðnei en h (1) sugklðnei, tìte lème ìti h n=1 a n sugklðnei upì sunj kec. Mia seirˆ thc opoðac oi ìroi eðnai diadoqikˆ jetikoð kai arnhtikoð arijmoð onomˆzetai enallassìmenh. Tìte a n = ( 1) n b n, ìpou b n jetikoð arijmoð. An h apeisoreirˆ thn opoða meletoôme den mporeð na upologisteð epakrib c eðnai shmantikì na gnwrðzoume an h seirˆ eðnai sugklðnousa apoklðnousa. OmadopoioÔme kai parajètoume ta diˆfora krit ria sôgklishc: Krit rio 1: SÔgkrish seir n mh arnhtik n ìrwn 'Estw dôo akoloujðec tètoiec ste b n 0 kai a n 0, n N, dhlad eðnai jetikèc apì ènan ìro kai pèra. IsqÔoun ta ex c: An a n b n, n N kai h n b n sugklðnei, tìte kai h n a n epðshc sugklðnei. An antijètwc a n b n, n N kai h n b n apoklðnei tìte kai h n a n epðshc apoklðnei. Wc parˆdeigma èstw oi seirèc orizìmenec ap' tic akoloujðec a n = ln n 2n 3 1, b n = 1 n 2 (συγκλίνουσα), 'Eqoume: 1 ln n n kai 2n n 3. 'Ara a n b n kai n a n . Krit rio 2: SÔgkrish lìgou seir n mh arnhtik n ìrwn 'Estw dôo akoloujðec tètoiec ste IsqÔoun ta ex c: a n a n 0, b n 0, και lim = A. (3) n b n An A = 0,, tìte oi seirèc n a n kai n b n eðnai kai oi dôo eðte sugklðnousec eðte apoklðnousec. An A = 0, tìte an n b n sugklðnei tìte kai h n a n sugklðnei epðshc. An A =, tìte an n b n apoklðnei tìte kai h n a n apoklðnei epðshc. IdiaÐtera qr simh eðnai h sôgkrish me th seirˆ b n = 1/n p. An a n 1/n p gia n, tìte n a n sugklðnei gia p 1 kai apoklðnei gia p 1. p.q. gia gia th seirˆ Epeid p = 1, h seirˆ apoklðnei. n 2 + n + 3 n=1 n 3 + 2n 2 = lim na n = 1. n n 2 + n + 3 n=1 n 3 + 2n 2, (4) Krit rio 3: Krit rio tou oloklhr matoc An f (x) eðnai suneq c, jetik kai monìtona fjðnousa sunˆrthsh gia x N kai tètoia ste f (n) = a n, n N, tìte h (1) sugklðnei apoklðnei an to olokl rwma sugklðnei apoklðnei, antðstoiqa. P.q. oi seirèc n=1 N dx f (x), (5) n n 2 + 1, 1 n=2 n ln n, ne n2, n=1 apoklðnoun, apoklðnoun kai sugklðnoun, antðstoiqa, epeid to autì isqôei gia ta oloklhr mata x dx 1 x 2, + 1 }{{} = 1 2 ln(x2 +1) 1 = 1 dx, 2 x ln x }{{} =ln(ln x) 2 = dx xe x2. 0 } {{ } = 1 2 Krit rio 4: Krit rio gia enallassìmenec seirèc MÐa enallassìmenh seirˆ sugklðnei an a n+1 a n, n 1 and lim n a n = 0. (6) Wc paradeðgmata tic jewroôme seirèc (α) (β) (γ) n=1 n=2 n=1 ( 1) n n n 2 + 1, ( 1) n n ln n, ( 1) n n 4 (n 2 + 1) 5/3. Oi (a) kai (b) sugklðnoun (upì sunj kec) kai h (g) apoklðnei. Ap' ta paradeðgmata autˆ, allˆ kai genikìtera, prokôptoun: H enallassìmenh seirˆ pou antistoiqeð se miˆ apokleðnousa seirˆ jetik n ìrwn mporeð na eðnai sugklðnousa. An mia seirˆ jetik n ìrwn sugklðnei h taqôthta sôgklishc thc antðstoiqhc enallassìmenhc seirˆc auxˆnetai. Krit rio 5: Krit rio tou D Alembert An a n+1 a n r 1, n N, (7) ìpou r anexˆrthto tou n, tìte h n a n sugklðnei. An tìte h n a n apoklðnei. a n+1 a n 1, n N, (8) Mia oriak perðptwsh tou krithrðou, èqei wc ex c. 'Estw An L 1 tìte h n a n sugklðnei. An L 1 tìte h n a n apoklðnei. a lim n+1 = L. (9) n a n An L = 1 to krit rio den efarmìzetai. Σε αυτές τις περιπτώσεις ίσως έχουμε a n+1 /a n 1 για όλα τα πεπερασμένα n, αλλά r 1, ανεξάρτητο του n για αρκετά μεγάλες τιμές του, ώστε πάντα να ικανοποιείται η (7). π.χ. a n = 1/n, a n+1 /a n = n/(n + 1) 1, αλλά το όριο πάει στο 1, όταν n. Krit rio 6: Krit rio tou Cauchy krit rio thc rðzac An a 1/n n r 1, n N, (10) me r anexˆrthto tou n, tìte h n a n sugklðnei kai an a 1/n n 1, n N, (11) tìte h n a n apoklðnei. Mia oriak perðptwsh tou krithrðou, èqei wc ex c. 'Estw An L 1 tìte h n a n sugklðnei. An L 1 tìte h n a n apoklðnei. lim n a1/n n = L. (12) An L = 1 den efarmìzetai gia lìgouc parìmoiouc me autoôc pou parousiˆsthkan sto krit rio tou D Alembert. Krit rio 7: Krit rio tou Kummer 'Estw a n kai b n jetikèc akoloujðec. IsqÔoun ta ex c: An a n b n b n+1 C 0, n N, (13) a n+1 tìte to ˆjroisma n a n sugklðnei. An a n b n b n+1 0, n N, (14) a n+1 kai h n bn 1 apoklðnei, tìte to ˆjroisma n a n apoklðnei epðshc. To krit rio suqnˆ parousiˆzetai me thn oriak tou morf upologðzontac pr ta ( ) a n lim b n b n+1 = C. (15) n a n+1 An C 0 tìte h n a n sugklðnei kai an C 0 kai h n b 1 n eðnai apoklðnousa, tìte kai h n a n apoklðnei. H oriak morf tou krithrðou eðnai isodônamh me tic (13) kai (14). To krit rio den efarmìzetai an C = 0. Krit rio 8: Krit rio tou Raabe To krit rio autì apoteleð eidik efarmog tou krithrðou tou Kummer an qrhsimopoi soume thn b n = n. 'Eqoume An ( ) an n 1 P 1, n N, (16) a n+1 tìte to ˆjroisma n a n sugklðnei. An ( ) an n 1 1, n N, (17) a n+1 tìte to ˆjroisma n a n apoklðnei. To krit rio suqnˆ parousiˆzetai me thn oriak tou morf upologðzontac pr ta ( ) lim n an 1 = s. (18) n a n+1 H seirˆ n a n sugklðnei an s 1 kai apoklðnei an s 1, en Krit rio 9: Krit rio tou Gauss An a n 0 gia ìla ta peperasmèna n kai a n a n+1 = 1 + h n + B n n 2, (19) sthn opoða B n eðnai fragmènh sunˆrthsh tou n ìtan to n. To ˆjroisma n a n sugklðnei an h 1 kai apoklðnei an h 1. EÐnai shmantikì ìti to krit rio autì pˆnta efarmìzetai. H apìdeix tou gia h = 1 sthrðzetai sto krit rio tou Raabe kai gia h = 1 efarmìzontac to krit rio tou Kummer gia b n = n ln n. Diplèc seirèc kai epanadiˆtax touc Oi diplèc seirèc èqoun th morf m=0 n=0 a n,m. (20) 'Opwc kai sthn perðptwsh twn dipl n oloklhrwmˆtwn pollèc forèc sumfèrei na epanadiatˆxoume touc ìrouc dipl n seir n giatð ètsi mporeð na eðnai eukolìtero na upologisteð. Parajètoume tic peript seic: H pr th dunatìthta eðnai na kˆnoume thn allag paðrnoume ìti n = q 0, m = p q 0, (21) p a n,m = a q,p q. (22) n=0 n=0 p=0 q=0 Miˆ deôterh dunatìthta eðnai na allˆxoume deðktec wc n = s 0, m = r 2s 0. (23) Tìte n=0 n=0 a n,m = [r /2] a s,r 2s. (24) r=0 s=0 O metasqhmatismìc Sommerfeld Watson 'Enac trìpoc upologismoô ˆpeirwn seir n eðnai kai mèsw thc metatrop c touc se migadikˆ oloklhr mata. Paramorf nontac ton antðstoiqo drìmo mporoôme na upologðsoume to olokl rwma kai kai thn ˆpeirh seirˆ. Autìc eðnai o metasqhmatismìc twn Sommerfeld Watson. 'Estw h sunˆrthsh f (z) h opoða: MhdenÐzetai grhgorìtera apì 1/ z ìtan z. 'Eqei peperasmèno arijmì pìlwn se shmeða z k sto migadikì epðpedo makriˆ ap' ton pragmatikì ˆxona kai èstw R k ta antðstoiqa oloklhrwtikˆ upìloipa thc f (z). MporoÔme na upologðsoume ta ajroðsmata: n= 'Eqoume jèsei f n = f (n). f n = π R k cot πz k. (25) k ( 1) n f n = π n= k R k sin πz k. (26) Parˆdeigma upologismoô dunamoseirˆc UpologÐste to ˆpeiro ˆjroisma f (x) = ( 1) n+1 n 2 x 2n 1 n=1 (2n 1)!. (27) LÔsh Ja qrhsimopoi soume thn anˆptuxh tou sin x se seirˆ gôrw apì to x = 0, h opoða eðnai sin x = ( 1) k x 2k+1 (2k + 1)! k=0 Allˆzontac deðkth wc k = n 1 h seirˆ (27) grˆfetai f (x) = k=0 ( 1) k x 2k+1 (1 + 2k + k 2 ). (2k + 1)!. Grˆfontac touc treðc ìrouc xeqwristˆ èqoume f (x) = ( 1) k x 2k+1 (2k + 1)! k=0 +x 2 d ( 1) dx k x 2k (2k + 1)! k=0 + x2 4 d dx x d dx = sin x + x 2 d dx ( 1) k x 2k (2k + 1)! k=0 sin x x + x2 4 = 3 4 x cos x (1 x2 ) sin x, pou eðnai kai to telikì apotèlesma. d dx x d sin x dx x Parˆdeigma 1o seirˆc me oloklhrwtikˆ upìloipa DeÐxte ìti: n= 1 n 4 + n = π tanh π 3. (28) 3 2 LÔsh Oi pìloi dðnontai ap' tic rðzec thc z 4 + z = 0 kai eðnai oi z 1 = e πi/3, z 2 = e 2πi/3, kaj c kai oi migadikoð suzhgoð touc. 'Ara qrhsimopoi ntac to metasqhmatismì Sommerfeld Watson kai thn (25) èqoume n= 1 n 4 + n = π ( 2 cot πz k k=1 4zk 3 + 2z + cot πz k k 4zk 3 + 2z k = = π 3 tanh π 3 2 Oi leptomèriec af nontai wc ['Askhsh].. ) Parˆdeigma 2o seirˆc me oloklhrwtikˆ upìloipa DeÐxte ìti: n=0 1 n 2 + a 2 = 1 (1 + πa coth πa), (29) 2a2 ìpou a pragmatik stajerˆ. Pˆrte prosektikˆ to ìrio a 0. LÔsh Oi pìloi dðnontai ap' tic rðzec thc z 2 + a 2 = 0 kai eðnai aploð pìloi sta shmeða z = ±ia. 'Ara qrhsimopoi ntac to metasqhmatismì Sommerfeld Watson kai thn (25) èqoume n= 1 n 2 + a 2 = π ia cot iπa = π coth πa. a Grˆfoume ( ) = 2 n=0( ) 1/a 2 kai apodeiknôoume to zhtoômeno giatð 1 ( ) = 1 ( ). Gia na pˆroume to ìrio a 0 anaptôsoume pr ta to dexð mèloc 1 2a 2 (1 + πa coth πa) = 1 a 2 + π2 6 + O(a2 ). Afair ntac ton ìro me n = 0 ap' to aristerì mèloc brðskoume ìti 1 n=1 n 2 = π2 6. (30) Oi leptomèriec af nontai wc ['Askhsh]. Parˆdeigma 3o seirˆc me oloklhrwtikˆ upìloipa DeÐxte ìti: ( 1) n n=0 n 2 + a 2 = 1 ( 2a πa ) sinh πa. (31) Pˆrte prosektikˆ to ìrio a 0. LÔsh Oi pìloi dðnontai ap' tic rðzec thc z 2 + a 2 = 0 kai eðnai aploð pìloi sta shmeða z = ±ia. 'Ara qrhsimopoi ntac to metasqhmatismì Sommerfeld Watson kai thn (26) èqoume n= ( 1) n n 2 + a 2 = π 1 ia sin iπa = π a 1 sinh πa. 'Opwc kai prðn grˆfoume ( ) = 2 n=0( ) 1/a 2 kai apodeiknôoume to zhtoômeno giatð 1 ( ) = 1 ( ). Gia na pˆroume to ìrio a 0 anaptôsoume pr ta to dexð mèloc 1 ( 2a πa ) = 1 sinh πa a 2 π O(a2 ). Afair ntac de ton ìro me n = 0 ap' to aristerì mèloc brðskoume ìti ( 1) n n=1 n 2 = π2 12. (32) Oi leptomèriec af nontai wc ['Askhsh]. O tôpoc twn Euler Maclaurin O tôpoc tou Euler Maclaurin anadeiknôei me susthmatikì trìpo th diaforˆ metaxô enìc diakritoô ajroðsmatoc kai tou antðstoiqou oloklhr matoc. 'Estw h akoloujða f k kai h antðstoiqh sunˆrthsh f (k). To peperasmèno ˆjroisma èqei thn asumptotik anˆptuxh b f k k=a b f (a) + f (b) dk f (k) + a 2 + k=1 B [ 2k f (2k 1) (b) f (2k 1) (a) (2k)! ], (33) ìpou a kai b eðnai akèraioi kai B 2k eðnai arijmoð Bernoulli. Se pollèc peript seic to olokl rwma mporeð na upologisjeð an kai autì mporeð na mhn eðnai dunatì gia th seirˆ. 'Ara tìte h asumptotik seirˆ mporeð na upologisjeð me stoiqei deic sunart seic. Gia z = 1 prìkeitai gia thn asumptotik anˆptuxh thc seirˆc k=1 1/k2 = π 2 /6 (Euler 1735). K. SFETSOS Majhmatikèc Mèjodoi Fusik c Ta polu numa Bernoulli orðzontai ap' thn anˆptuxh te xt e t 1 = B n (x) tn n=0 n!. (34) Oi arijmoð Bernoulli orðzontai ap' th sqèsh Oi pr toi ex' aut n eðnai B n = B n (0) = B n (1). (35) B 0 = 1, B 1 = 1 2, B 2 = 1 6, B 3 = 0, B 4 = 1 30, B 6 = Wc parˆdeigma èqoume n=0 1 (z + n) 2 dn (z + n) 2 0 } {{ } = 1/z + 1 2z 2 + n=1 B 2n. (36) z2n+1 'Apeira ginìmena 'Ena ˆpeiro ginìmeno orðzetai wc a n, (37) n=1 emfanðzontai de sth Statistik Fusik, se gewmetrikˆ probl mata majhmatik n, se jewrðec pedðou kai qord n kai ˆllouc klˆdouc twn jetik n episthm n. Mia sunˆrthsh analutik se ìlo to migadikì epðpedo onomˆzetai olik. Sto ˆpeiro den qreiˆzetai na eðnai analutik giatð tìte ap' to je rhma tou Liouville eðnai stajer. Genikˆ sto ˆpeiro mporeð na eðnai eðte pìloc eðte ousi dhc anwmalða (ìpwc oi e z kai sin z). 'Estw f (z) olik sunˆrthsh me mhdenikˆ pou dðnontai ap' thn ˆpeirh, mh sugklðsousa, akoloujða a n, n = 1, 2,..., to kajèna me pollaplìthta m n. Tìte isqôei: f (z) = f (0)e f (0) f (0) ) z [(1 zan mn e n] z/a, (38) n=1 h opoða bohjˆ na ekfrˆsoume apeira ginìmena me gnwstèc sunart seic. To shmeðo z = 0 eðnai aôjaðreto arkeð na mhn antistoiqeð se mhdenikì. An h f (z) eðnai meromorfik sunˆrthsh tìte mporeð na ekfrasteð wc o lìgoc dôo olik n sunart sewn, gia kajemiˆ ap' tic opoðec mporoôme na qrhsimopoi soume thn (38). Wc parˆdeigma jewroôme thn f (z) = cos z. 'Eqoume f (0) = 1, f (0) = 0. Ta mhdenikˆ thc dðnontai ap' thn a n = (n 1 2 )π, n =,..., 0, 1,. Qrhsimopoi ntac ìti a n+1 = a n brðskoume ìti cos z = ( 4z 1 2 ) n=1 (2n 1) 2 π 2 'Alla parìmoia paradeðgmata (['Askhsh]) eðnai: n 2 1 n=2 n = π sinh π, n=2 n 3 1 n = 2 3.. Parˆdeigma upologismoô seirˆ apì ˆpeiro ginìmeno Qrhsimopoi ntac thn anaparˆstash tou uperbolikoô hmitìnou se ˆpeiro ginìmeno, na apodeiqteð ìti sinh x x = coth x 1 x = n=1 ) (1 + x2 n=1 n 2 π 2, (39) 2x x 2 + n 2 π 2. (40) LÔsh PaÐrnontac to logˆrijmo twn dôo mer n thc (39), katal goume sthn ) ln sinh x ln x = ln (1 + x2 n=1 n 2 π 2. ParagwgÐzontac thn teleutaða sqèsh apodeiknôetai to zhtoômeno.
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