V matematici formula Ejlera Maklorena viznachaye tisnij zv yazok mizh integralami i ryadami Nazvana na chest shvejcarskogo matematika Leonarda Ejlera i shotlandskogo matematika Kolina Maklorena Zmist 1 Tverdzhennya 2 Dovedennya 3 Posilannya 4 LiteraturaTverdzhennya RedaguvatiNehaj p i q dva cilih chisla Dlya 2k raziv neperervno diferencijovanoyi na promizhku p q displaystyle p q nbsp funkciyi f p f q 2 j p 1 q 1 f j p q f x d x j 1 k b 2 j 2 j f 2 j 1 q f 2 j 1 p R k displaystyle frac f left p right f left q right 2 sum j p 1 q 1 f left j right int p q f x dx sum j 1 k frac b 2j 2j left f 2j 1 q f 2j 1 p right R k nbsp de R k p q f 2 k x B 2 k x x 2 k d x displaystyle R k int p q f 2k x B 2k x lfloor x rfloor over 2k dx nbsp V danih formulah B i displaystyle B i nbsp poznachaye i j mnogochlen Bernulli B i x x displaystyle B i x lfloor x rfloor nbsp periodizovanij mnogochlen Bernulli Chisla bi poznachayut chisla Bernulli b1 1 2 b2 1 6 b3 0 b4 1 30 b5 0 b6 1 42 b7 0 b8 1 30 Zavdyaki zamini zminnih podibnu formulu mozhna oderzhati dlya intervalu mezhi yakogo ne ye cilimi chislami Dovedennya RedaguvatiDostatno dovesti spravedlivist dlya intervalu n n 1 displaystyle n n 1 nbsp de n Z displaystyle n in mathbb Z nbsp zagalna formula oderzhuyetsya za dopomogoyu sumuvannya Nehaj g funkciya neperervno diferencijovana na intervali n n 1 displaystyle n n 1 nbsp Vikoristovuyuchi vlastivist mnogochleniv Bernulli k N B k 1 k 1 B k displaystyle forall k in mathbb N B k 1 left k 1 right B k nbsp oderzhuyemo z integruvannya chastinami n n 1 g t B k t n d t g t B k 1 t n k 1 n n 1 1 k 1 n n 1 g t B k 1 t n d t displaystyle int n n 1 g left t right B k left t n right dt left frac g left t right B k 1 left t n right k 1 right n n 1 frac 1 k 1 int n n 1 g left t right B k 1 left t n right dt nbsp Oskilki dlya k 2 displaystyle k geq 2 nbsp vikonuyetsya B k 1 B k 0 b k displaystyle B k left 1 right B k left 0 right b k nbsp oderzhuyemo n n 1 g t B k t n d t b k 1 k 1 g n 1 g n 1 k 1 n n 1 g t B k 1 t n d t displaystyle int n n 1 g left t right B k left t n right dt frac b k 1 k 1 left g left n 1 right g left n right right frac 1 k 1 int n n 1 g left t right B k 1 left t n right dt nbsp Rekurentnistyu dlya k vid 0 do 2p prijmayuchi g f 2 p displaystyle g f 2p nbsp oderzhuyetsya n n 1 f t d t f n f n 1 2 k 2 2 p 1 k 1 b k k f k 1 n 1 f k 1 n 1 2 p n n 1 f 2 p t B 2 p t n d t displaystyle int n n 1 f left t right dt frac f left n right f left n 1 right 2 sum k 2 2p frac left 1 right k 1 b k k left f k 1 left n 1 right f k 1 left n right right frac 1 2p int n n 1 f 2p left t right B 2p left t n right dt nbsp Z vlastivosti k 1 b 2 k 1 0 displaystyle forall k geq 1 b 2k 1 0 nbsp oderzhuyetsya n n 1 f t d t f n f n 1 2 k 2 p 2 b 2 k 2 k f 2 k 1 n 1 f 2 k 1 n 1 2 p n n 1 f 2 p t B 2 p t n d t displaystyle int n n 1 f left t right dt frac f left n right f left n 1 right 2 sum k 2 lfloor frac p 2 rfloor frac b 2k 2k left f 2k 1 left n 1 right f 2k 1 left n right right frac 1 2p int n n 1 f 2p left t right B 2p left t n right dt nbsp Posilannya RedaguvatiWeisstein Eric W Euler Maclaurin Integration Formulas angl na sajti Wolfram MathWorld Literatura RedaguvatiCegelik G G Chiselni metodi Lviv Vidavnichij centr Lvivskogo nacionalnogo universitetu 2004 408 s Hugh L Montgomery Robert C Vaughan 2007 Multiplicative number theory I Classical theory Cambridge tracts in advanced mathematics 97 pp 495 519 ISBN 0 521 84903 9 Otrimano z https uk wikipedia org w index php title Formula Ejlera Maklorena amp oldid 38066433
