Ortogonalni polinomiLezhandraVidkriti Adriyen Mari LezhandrFormula P n x 1 2 n n d n d x n x 2 1 n displaystyle P n x 1 over 2 n n d n over dx n left x 2 1 n right Diferencialne rivnyannya d d x 1 x 2 d d x P n x n n 1 P n x 0 displaystyle d over dx left 1 x 2 d over dx P n x right n n 1 P n x 0 Viznacheni na 1 1 displaystyle 1 1 Vaga 1Norma 2 2 n 1 displaystyle 2 over 2n 1 PrimitkiPolinomi Lezhandra ortogonalni polinomi na intervali 1 1 displaystyle 1 1 Polinomi Lezhandra mozhna otrimati z sistemi polinomiv 1 x x 2 x 3 displaystyle 1 x x 2 x 3 ldots za dopomogoyu ortogonalizaciyi Grama Shmidta Mozhut buti obchisleni za dopomogoyu pryamih formul P n x 1 2 n n d n d x n x 2 1 n displaystyle P n x 1 over 2 n n d n over dx n left x 2 1 n right abo za rekurentnimi P n 1 x 2 n 1 n 1 x P n x n n 1 P n 1 x displaystyle P n 1 x 2n 1 over n 1 xP n x n over n 1 P n 1 x Voni ye rozv yazkami diferencijnogo rivnyannya Lezhandra d d x 1 x 2 d d x P n x n n 1 P n x 0 displaystyle d over dx left 1 x 2 d over dx P n x right n n 1 P n x 0 Grafiki polinomiv Lezhandra poryadku n 0 1 5 displaystyle n 0 1 5 Generatrisa dlya mnogochleniv Lezhandra dorivnyuye n 0 P n z x n 1 1 2 x z x 2 displaystyle sum n 0 infty P n z x n 1 over sqrt 1 2xz x 2 Pershi 9 polinomiv Lezhandra P 0 x 1 displaystyle P 0 x 1 P 1 x x displaystyle P 1 x x P 2 x 1 2 3 x 2 1 displaystyle P 2 x begin matrix frac 1 2 end matrix 3x 2 1 P 3 x 1 2 5 x 3 3 x displaystyle P 3 x begin matrix frac 1 2 end matrix 5x 3 3x P 4 x 1 8 35 x 4 30 x 2 3 displaystyle P 4 x begin matrix frac 1 8 end matrix 35x 4 30x 2 3 P 5 x 1 8 63 x 5 70 x 3 15 x displaystyle P 5 x begin matrix frac 1 8 end matrix 63x 5 70x 3 15x P 6 x 1 16 231 x 6 315 x 4 105 x 2 5 displaystyle P 6 x begin matrix frac 1 16 end matrix 231x 6 315x 4 105x 2 5 P 7 x 1 16 429 x 7 693 x 5 315 x 3 35 x displaystyle P 7 x begin matrix frac 1 16 end matrix 429x 7 693x 5 315x 3 35x P 8 x 1 128 6435 x 8 12012 x 6 6930 x 4 1260 x 2 35 displaystyle P 8 x begin matrix frac 1 128 end matrix 6435x 8 12012x 6 6930x 4 1260x 2 35 P 9 x 1 128 12155 x 9 25740 x 7 18018 x 5 4620 x 3 315 x displaystyle P 9 x begin matrix frac 1 128 end matrix 12155x 9 25740x 7 18018x 5 4620x 3 315x Zmist 1 Ortogonalnist 2 Priyednani funkciyi Lezhandra 3 Zastosuvannya 4 Div takozhOrtogonalnist RedaguvatiUmova ortogonalnosti spravdzhuyetsya na intervali 1 1 displaystyle 1 1 nbsp 1 1 P m x P n x d x 2 2 n 1 d m n displaystyle int limits 1 1 P m x P n x dx 2 over 2n 1 delta mn nbsp de d m n displaystyle delta mn nbsp delta simvol Kronekera Priyednani funkciyi Lezhandra RedaguvatiPriyednani funkciyi Lezhandra viznachayutsya za formuloyu P n m x 1 x 2 m 2 d m d x m P n x displaystyle P n m x 1 x 2 m 2 frac d m dx m P n x nbsp yaku mozhna takozh predstaviti u viglyadi P n m cos 8 sin m 8 d m d cos 8 m P n cos 8 displaystyle P n m cos theta sin m theta frac d m d cos theta m P n cos theta nbsp Pri m 0 displaystyle m 0 nbsp funkciya P n m displaystyle P n m nbsp zbigayetsya z P n displaystyle P n nbsp Yih chasto nazivayut priyednanimi polinomami Lezhandra hocha naspravdi ci funkciyi ne polinomi Priyednani funkciyi Lezhandra ye rozv yazkami diferencialnogo rivnyannya 1 x 2 y 2 x y n n 1 m 2 1 x 2 y 0 displaystyle 1 x 2 y 2xy left n n 1 frac m 2 1 x 2 right y 0 nbsp abo ekvivalentnogo jomu 1 x 2 y n n 1 m 2 1 x 2 y 0 displaystyle 1 x 2 y left n n 1 frac m 2 1 x 2 right y 0 nbsp Zastosuvannya RedaguvatiPolinomi Lezhandra shiroko zastosovuyutsya u fizici Zazvichaj argumentom polinomiv ye kosinus polyarnogo kuta 8 displaystyle theta nbsp yakij zminyuyetsya vid 1 pri 8 p displaystyle theta pi nbsp do 1 pri 8 0 displaystyle theta 0 nbsp Zokrema dlya otrimannya multipolnogo rozkladu elektrostatichnih poliv 1 r r 0 1 r 2 2 r r 0 cos 8 r 0 2 1 r 0 1 1 2 x cos 8 x 2 1 r 0 n P n cos 8 x n displaystyle frac 1 mathbf r mathbf r 0 frac 1 sqrt r 2 2rr 0 cos theta r 0 2 frac 1 r 0 frac 1 sqrt 1 2x cos theta x 2 frac 1 r 0 sum n P n cos theta x n nbsp de x r r 0 displaystyle x r r 0 nbsp a cos 8 displaystyle cos theta nbsp kut mizh vektorami r displaystyle mathbf r nbsp ta r 0 displaystyle mathbf r 0 nbsp Inshe vazhlive zastosuvannya rozklad poliv na parcialni hvili Napriklad ploska hvilya rozkladayetsya za dopomogoyu formuli e i k r l 0 2 l 1 i l j l k r P l cos 8 displaystyle e i mathbf k cdot mathbf r sum l 0 infty 2l 1 i l j l kr P l cos theta nbsp de j l x displaystyle j l x nbsp sferichni funkciyi Besselya Div takozh RedaguvatiSferichni garmonikiCya stattya ne mistit posilan na dzherela Vi mozhete dopomogti polipshiti cyu stattyu dodavshi posilannya na nadijni avtoritetni dzherela Material bez dzherel mozhe buti piddano sumnivu ta vilucheno zhovten 2015 Otrimano z https uk wikipedia org w index php title Polinomi Lezhandra amp oldid 20536984
