Polinomi Lagerra ortogonalni polinomi nazvani na chest francuzkogo matematika Edmona Lagerra Zmist 1 Viznachennya 2 Prikladi 3 Uzagalneni polinomi Lagerra 3 1 Prikladi 4 Ortogonalnist 5 LiteraturaViznachennya RedaguvatiPolinomami Lagerra nazivayutsya kanonichni rozv yazki diferencijnogo rivnyannya x y 1 x y n y 0 displaystyle x y 1 x y n y 0 nbsp sho ye linijnim diferencijnim rivnyannyam drugogo poryadku i maye nesingulyarnij rozv yazok lishe dlya nevid yemnih cilih n Dlya danih polinomiv spravedliva takozh yavna formula Rodrigesa L n x e x n d n d x n e x x n displaystyle L n x frac e x n frac d n dx n left e x x n right nbsp Polinomi Lagerra mozhna zadati rekursivno Dlya cogo slid vzyati L 0 x 1 displaystyle L 0 x 1 nbsp L 1 x 1 x displaystyle L 1 x 1 x nbsp i viznachiti nastupni polinomi za dopomogoyu formuli L k 1 x 1 k 1 2 k 1 x L k x k L k 1 x displaystyle L k 1 x frac 1 k 1 left 2k 1 x L k x kL k 1 x right nbsp Prikladi RedaguvatiPrikladami polinomiv Lagerra najmenshih stepeniv ye n L n x displaystyle L n x nbsp 0 1 displaystyle 1 nbsp 1 x 1 displaystyle x 1 nbsp 2 1 2 x 2 4 x 2 displaystyle scriptstyle frac 1 2 x 2 4x 2 nbsp 3 1 6 x 3 9 x 2 18 x 6 displaystyle scriptstyle frac 1 6 x 3 9x 2 18x 6 nbsp 4 1 24 x 4 16 x 3 72 x 2 96 x 24 displaystyle scriptstyle frac 1 24 x 4 16x 3 72x 2 96x 24 nbsp 5 1 120 x 5 25 x 4 200 x 3 600 x 2 600 x 120 displaystyle scriptstyle frac 1 120 x 5 25x 4 200x 3 600x 2 600x 120 nbsp 6 1 720 x 6 36 x 5 450 x 4 2400 x 3 5400 x 2 4320 x 720 displaystyle scriptstyle frac 1 720 x 6 36x 5 450x 4 2400x 3 5400x 2 4320x 720 nbsp nbsp Grafiki polinomiv Lagerra Uzagalneni polinomi Lagerra RedaguvatiUzagalnenimi polinomami Lagerra nazivayutsya polinomi viznacheni za dopomogoyu uzagalnenoyi formuli Rodrigesa f x x a e x G 1 a if x gt 0 0 if x lt 0 displaystyle f x left begin matrix x alpha e x Gamma 1 alpha amp mbox if x gt 0 0 amp mbox if x lt 0 end matrix right nbsp Todi zvichajni polinomi Lagerra ye okremim vipadkom E L n X L m X 0 whenever n m displaystyle E left L n X L m X right 0 mbox whenever n neq m nbsp Uzagalnenij polinom Legerra stepenya n displaystyle n nbsp takozh mozhna viznachiti za dopomogoyu formuli L n a x i 0 n 1 i n a n i x i i displaystyle L n alpha x sum i 0 n 1 i n alpha choose n i frac x i i nbsp Takozh vikonuyutsya rekurentni spivvidnoshennya L n a b 1 x y i 0 n L i a x L n i b y displaystyle L n alpha beta 1 x y sum i 0 n L i alpha x L n i beta y nbsp Zokrema L n a 1 x i 0 n L i a x displaystyle L n alpha 1 x sum i 0 n L i alpha x nbsp i L n a x i 0 n a b n i 1 n i L i b x displaystyle L n alpha x sum i 0 n alpha beta n i 1 choose n i L i beta x nbsp abo L n a x i 0 n a b n n i L i b i x displaystyle L n alpha x sum i 0 n alpha beta n choose n i L i beta i x nbsp Prikladi Redaguvati Prikladami uzagalnenih polinomiv Lagerra najmenshih stepeniv ye L 0 a x 1 displaystyle L 0 alpha x 1 nbsp L 1 a x x a 1 displaystyle L 1 alpha x x alpha 1 nbsp L 2 a x x 2 2 a 2 x a 2 a 1 2 displaystyle L 2 alpha x frac x 2 2 alpha 2 x frac alpha 2 alpha 1 2 nbsp L 3 a x x 3 6 a 3 x 2 2 a 2 a 3 x 2 a 1 a 2 a 3 6 displaystyle L 3 alpha x frac x 3 6 frac alpha 3 x 2 2 frac alpha 2 alpha 3 x 2 frac alpha 1 alpha 2 alpha 3 6 nbsp Ortogonalnist RedaguvatiUzagalneni polinomi Lagerra ye ortogonalnimi na promizhku 0 z vagoyu xa e x 0 x a e x L n a x L m a x d x G n a 1 n d n m displaystyle int 0 infty x alpha e x L n alpha x L m alpha x dx frac Gamma n alpha 1 n delta n m nbsp Dlya zvichajnih polinomiv Lagerra vikonuyetsya rivnist f g 0 f x g x e x d x displaystyle langle f g rangle int 0 infty f x g x e x dx nbsp Literatura RedaguvatiAbramowitz Milton Stegun Irene A eds 1965 Chapter 22 Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables New York Dover ISBN 0 486 61272 4 B Spain M G Smith Functions of mathematical physics Van Nostrand Reinhold Company London 1970 Chapter 10 deals with Laguerre polynomials Eric W Weisstein Laguerre Polynomial Arhivovano 25 lyutogo 2010 u Wayback Machine From MathWorld A Wolfram Web Resource George Arfken and Hans Weber 2000 Mathematical Methods for Physicists Academic Press ISBN 0 12 059825 6 S S Bayin 2006 Mathematical Methods in Science and Engineering Wiley Chapter 3 Otrimano z https uk wikipedia org w index php title Polinomi Lagerra amp oldid 34949068
