U matematici Integral Borvejna integral nezvichajni vlastivosti yakogo buli vpershe predstavleni matematikami Devidom Borvejnom en ta Dzhonatanom Borvejnom v 2001 roci 1 Integral Borvejna vklyuchaye v sebe dobutki funkcij s i n c x displaystyle mathrm sinc x Funkciya sinc viznachayetsya yak s i n c x sin x x displaystyle mathrm sinc x frac sin x x de x 0 displaystyle x neq 0 ta s i n c 0 1 displaystyle mathrm sinc 0 1 1 2 Ci integrali chudovi tim sho demonstruyut yavni zakonomirnosti yaki v kincevomu pidsumku rujnuyutsya Navedemo nastupnij priklad 0 sin x x d x p 2 0 sin x x sin x 3 x 3 d x p 2 0 sin x x sin x 3 x 3 sin x 5 x 5 d x p 2 displaystyle begin aligned amp int 0 infty frac sin x x rm d x frac pi 2 amp int 0 infty frac sin x x frac sin x 3 x 3 rm d x frac pi 2 amp int 0 infty frac sin x x frac sin x 3 x 3 frac sin x 5 x 5 rm d x frac pi 2 end aligned Cya zakonomirnist prodovzhuyetsya do 0 sin x x sin x 3 x 3 sin x 13 x 13 d x p 2 displaystyle int 0 infty frac sin x x frac sin x 3 x 3 cdots frac sin x 13 x 13 rm d x frac pi 2 Ale na nastupnomu kroci ochevidna zakonomirnist ne spracovuye 0 sin x x sin x 3 x 3 sin x 15 x 15 d x 467807924713440738696537864469 935615849440640907310521750000 p p 2 6879714958723010531 935615849440640907310521750000 p p 2 2 31 10 11 displaystyle begin aligned int 0 infty frac sin x x frac sin x 3 x 3 cdots frac sin x 15 x 15 rm d x amp frac 467807924713440738696537864469 935615849440640907310521750000 pi amp frac pi 2 frac 6879714958723010531 935615849440640907310521750000 pi amp approx frac pi 2 2 31 times 10 11 end aligned U zagalnomu vipadku podibni integrali nabuvayut znachennya p 2 displaystyle frac pi 2 yaksho chisla 3 5 7 displaystyle 3 5 7 ldots zaminyuyutsya na dodatni dijsni chisla taki sho suma yih obernenih znachen mensha za 1 displaystyle 1 U navedenomu vishe prikladi 1 3 1 5 1 13 lt 1 displaystyle frac 1 3 frac 1 5 dots frac 1 13 lt 1 ale 1 3 1 5 1 15 gt 1 displaystyle frac 1 3 frac 1 5 ldots frac 1 15 gt 1 Z vklyuchennyam dodatkovogo mnozhnika 2 cos x displaystyle 2 cos x zakonomirnist vitrimuye bilsh dovshij ryad 0 2 cos x sin x x sin x 3 x 3 sin x 111 x 111 d x p 2 displaystyle int 0 infty 2 cos x frac sin x x frac sin x 3 x 3 cdots frac sin x 111 x 111 rm d x frac pi 2 ale 0 2 cos x sin x x sin x 3 x 3 sin x 111 x 111 sin x 113 113 d x lt p 2 displaystyle int 0 infty 2 cos x frac sin x x frac sin x 3 x 3 cdots frac sin x 111 x 111 frac sin x 113 113 rm d x lt frac pi 2 U comu vipadku 1 3 1 5 1 111 lt 2 displaystyle frac 1 3 frac 1 5 dots frac 1 111 lt 2 ale 1 3 1 5 1 113 gt 2 displaystyle frac 1 3 frac 1 5 dots frac 1 113 gt 2 Prichina porushennya zakonomirnosti ta rozshirennya ryadu prodemonstrovana za dopomogoyu intuyitivnogo matematichnogo poyasnennya 3 4 Zokrema pereformulyuvannya u terminah vipadkovih blukan z argumentom prichinnosti prolivaye svitlo na porushennya zakonomirnosti ta vidkrivaye shlyah dlya ryadu uzagalnen 5 Zagalna formula RedaguvatiDlya zadanoyi poslidovnosti nenulovih dijsnih chisel a 0 a 1 a 2 displaystyle a 0 a 1 a 2 ldots nbsp mozhna predstaviti zagalnu formulu dlya integrala 1 0 k 0 n sin a k x a k x d x displaystyle int 0 infty prod k 0 n frac sin a k x a k x rm d x nbsp Dlya vivedennya formuli potribno rozglyanuti sumi sho vklyuchayut a k displaystyle a k nbsp Zokrema yaksho g g 1 g 2 g n 1 n displaystyle gamma gamma 1 gamma 2 ldots gamma n in pm 1 n nbsp nabir z n displaystyle n nbsp chisel de kozhne 1 displaystyle pm 1 nbsp to todi zapishemo b g a 0 g 1 a 1 g 2 a 2 g n a n displaystyle b gamma a 0 gamma 1 a 1 gamma 2 a 2 cdots gamma n a n nbsp sho ye pevnim zkakozminnim ryadom dekilkoh pershih a k displaystyle a k nbsp ta poklademo e g g 1 g 2 g n displaystyle varepsilon gamma gamma 1 gamma 2 cdots gamma n nbsp de 1 displaystyle pm 1 nbsp U cih poznachennyah znachennya vishevkazanogo integrala dorivnyuye 0 k 0 n sin a k x a k x d x p 2 a 0 C n displaystyle int 0 infty prod k 0 n frac sin a k x a k x rm d x frac pi 2a 0 C n nbsp de C n 1 2 n n k 1 n a k g 1 n e g b g n sgn b g displaystyle C n frac 1 2 n n prod limits k 1 n a k sum gamma in pm 1 n varepsilon gamma b gamma n operatorname sgn b gamma nbsp U vipadku yaksho a 0 gt a 1 a 2 a n displaystyle a 0 gt a 1 a 2 cdots a n nbsp to C n 1 displaystyle C n 1 nbsp Krim togo yaksho isnuye n displaystyle n nbsp sho dlya kozhnogo k 0 n 1 displaystyle k 0 ldots n 1 nbsp vikonuyutsya umovi 0 lt a n lt 2 a k displaystyle 0 lt a n lt 2a k nbsp ta a 1 a 2 a n 1 lt a 0 lt a 1 a 2 a n 1 a n displaystyle a 1 a 2 cdots a n 1 lt a 0 lt a 1 a 2 cdots a n 1 a n nbsp tobto n displaystyle n nbsp pershe znachennya za yakogo chastkova suma pershih n displaystyle n nbsp elementiv poslidovnosti perevishuye a 0 displaystyle a 0 nbsp todi C k 1 displaystyle C k 1 nbsp dlya kozhnogo k 0 n 1 displaystyle k 0 ldots n 1 nbsp ale C n 1 a 1 a 2 a n a 0 n 2 n n k 1 n a k displaystyle C n 1 frac a 1 a 2 cdots a n a 0 n 2 n n prod limits k 1 n a k nbsp Rozglyanemo vipadok koli a k 1 2 k 1 displaystyle a k frac 1 2k 1 nbsp Yaksho n 7 displaystyle n 7 nbsp to a 7 1 15 displaystyle a 7 frac 1 15 nbsp ta 1 3 1 5 1 7 1 9 1 11 1 13 0 955 displaystyle frac 1 3 frac 1 5 frac 1 7 frac 1 9 frac 1 11 frac 1 13 approx 0 955 nbsp ale 1 3 1 5 1 7 1 9 1 11 1 13 1 15 1 02 displaystyle frac 1 3 frac 1 5 frac 1 7 frac 1 9 frac 1 11 frac 1 13 frac 1 15 approx 1 02 nbsp Oskilki a 0 1 displaystyle a 0 1 nbsp to otrimuyemo formulu 0 sin x x sin x 3 x 3 sin x 13 x 13 d x p 2 displaystyle int 0 infty frac sin x x frac sin x 3 x 3 cdots frac sin x 13 x 13 rm d x frac pi 2 nbsp yaka virna pri viklyuchenni bud yakogo z mnozhnikiv ale 0 sin x x sin x 3 x 3 sin x 15 x 15 d x p 2 1 3 1 5 1 7 1 9 1 11 1 13 1 15 1 1 7 2 6 7 1 3 1 5 1 7 1 9 1 11 1 13 1 15 displaystyle begin aligned amp int 0 infty frac sin x x frac sin x 3 x 3 cdots frac sin x 15 x 15 rm d x frac pi 2 left 1 frac 3 1 5 1 7 1 9 1 11 1 13 1 15 1 1 7 2 6 cdot 7 cdot 1 3 cdot 1 5 cdot 1 7 cdot 1 9 cdot 1 11 cdot 1 13 cdot 1 15 right end aligned nbsp sho dorivnyuye znachennyu zadanomu vishe Literatura Redaguvati a b v Borwein David Borwein Jonathan M 2001 Some remarkable properties of sinc and related integrals The Ramanujan Journal 5 1 73 89 ISSN 1382 4090 MR 1829810 doi 10 1023 A 1011497229317 Baillie Robert 2011 Fun With Very Large Numbers arXiv 1105 3943 math NT Schmid Hanspeter 2014 Two curious integrals and a graphic proof Elemente der Mathematik 69 1 11 17 ISSN 0013 6018 doi 10 4171 EM 239 Arhiv originalu za 5 bereznya 2020 Procitovano 28 travnya 2020 Baez John 20 veresnya 2018 Patterns That Eventually Fail Azimuth Arhiv originalu za 21 travnya 2019 Satya Majumdar Emmanuel Trizac 2019 When random walkers help solving intriguing integrals Physical Review Letters 123 2 020201 Bibcode 2019arXiv190604545M ISSN 1079 7114 arXiv 1906 04545 doi 10 1103 PhysRevLett 123 020201 Posilannya RedaguvatiPatterns That Eventually Fail Arhivovano 21 travnya 2019 u Wayback Machine 20 September 2018 Breakdown Arhivovano 2 lipnya 2020 u Wayback Machine 2 February 2012 Illusive patterns in math explained by ideas in physics Arhivovano 27 veresnya 2020 u Wayback Machine 19 July 2019 video When random walkers help solving intriguing integrals Arhivovano 5 veresnya 2020 u Wayback Machine 19 July 2019 Otrimano z https uk wikipedia org w index php title Integral Borvejna amp oldid 35810042
