U matematici stepenevim ryadom odniyeyi zminnoyi nazivayetsya neskinchennij ryad vidu f x n 0 a n x c n a 0 a 1 x c 1 a 2 x c 2 a 3 x c 3 displaystyle f x sum n 0 infty a n left x c right n a 0 a 1 x c 1 a 2 x c 2 a 3 x c 3 cdots de an koeficiyenti n go dodanku c deyaka konstanta a x zminna viznachena v deyakij oblasti sho mistit c Na praktici chasto c rivne nulyu i stepenevi ryadi mayut prostishij vid f x n 0 a n x n a 0 a 1 x a 2 x 2 a 3 x 3 displaystyle f x sum n 0 infty a n x n a 0 a 1 x a 2 x 2 a 3 x 3 ldots Stepenevi ryadi shiroko vikoristovuyutsya u dijsnomu i kompleksnomu analizi yak ryadi Tejlora funkcij a takozh v kombinatorici teoriyi jmovirnostej ta in Zmist 1 Operaciyi zi stepenevimi ryadami 1 1 Dodavannya i vidnimannya 1 2 Mnozhennya i dilennya 2 Zbizhnist stepenevih ryadiv 2 1 Oznaki zbizhnosti 3 Pohidna i integral 4 Stepenevi ryadi bagatoh zminnih 5 Div takozh 6 Literatura 7 PosilannyaOperaciyi zi stepenevimi ryadami RedaguvatiDodavannya i vidnimannya Redaguvati Dlya stepenevih ryadiv f i g navkolo tochki c mozhna viznachiti yih sumu i riznicyu Yaksho f x n 0 a n x c n displaystyle f x sum n 0 infty a n x c n nbsp g x n 0 b n x c n displaystyle g x sum n 0 infty b n x c n nbsp todi f x g x n 0 a n b n x c n displaystyle f x pm g x sum n 0 infty a n pm b n x c n nbsp Mnozhennya i dilennya Redaguvati Dlya mnozhennya i dilennya oderzhuyutsya formuli f x g x n 0 a n x c n n 0 b n x c n displaystyle f x g x left sum n 0 infty a n x c n right left sum n 0 infty b n x c n right nbsp i 0 j 0 a i b j x c i j displaystyle sum i 0 infty sum j 0 infty a i b j x c i j nbsp n 0 i 0 n a i b n i x c n displaystyle sum n 0 infty left sum i 0 n a i b n i right x c n nbsp Poslidovnist m n i 0 n a i b n i displaystyle m n sum i 0 n a i b n i nbsp nazivayetsya konvolyuciyeyu poslidovnostej a n displaystyle a n nbsp i b n displaystyle b n nbsp Dlya dilennya vikonuyetsya f x g x n 0 a n x c n n 0 b n x c n n 0 d n x c n displaystyle f x over g x sum n 0 infty a n x c n over sum n 0 infty b n x c n sum n 0 infty d n x c n nbsp f x n 0 b n x c n n 0 d n x c n displaystyle f x left sum n 0 infty b n x c n right left sum n 0 infty d n x c n right nbsp i znachennya znahodyatsya z formul konvolyuciyi Zbizhnist stepenevih ryadiv RedaguvatiStepenevij ryad nazivayetsya zbizhnim v tochci x0 yaksho zbizhnim ye vidpovidnij chislovij ryad n 0 a n x 0 n a 0 a 1 x 0 a 2 x 0 2 a 3 x 0 3 displaystyle sum n 0 infty a n x 0 n a 0 a 1 x 0 a 2 x 0 2 a 3 x 0 3 ldots nbsp Stepenevij ryad ye zbizhnim v deyakij oblasti yaksho vin ye zbizhnim v kozhnij tochci ciyeyi oblasti Oznaki zbizhnosti Redaguvati Dlya stepenevih ryadiv ye dekilka teorem sho opisuyut umovi i harakter yih zbizhnosti Persha teorema Abelya Nehaj ryad S a n x n displaystyle Sigma a n x n nbsp ye zbizhnim v tochci x 0 displaystyle x 0 nbsp Todi cej ryad ye absolyutno zbizhnim v krugu x lt x 0 displaystyle x lt x 0 nbsp rivnomirno po x displaystyle x nbsp na bud yakij kompaktnij pidmnozhini cogo kruga Navpaki yaksho stepenevij ryad ye rozbizhnim pri x x 0 displaystyle x x 0 nbsp vin ye rozbizhnim pri vsih x displaystyle x nbsp takih sho x gt x 0 displaystyle x gt x 0 nbsp Z pershoyi teoremi Abelya takozh viplivaye sho isnuye takij radius kruga R displaystyle R nbsp mozhlivo nulovij abo neskinchennij sho pri x lt R displaystyle x lt R nbsp ryad ye absolyutno zbizhnim i zbizhnist ye rivnomirnoyu po x displaystyle x nbsp na kompaktnih pidmnozhinah kruga x lt R displaystyle x lt R nbsp a pri x gt R displaystyle x gt R nbsp ryad ye rozbizhnim Ce znachennya R displaystyle R nbsp nazivayetsya radiusom zbizhnosti ryadu a krug x lt R displaystyle x lt R nbsp krugom zbizhnosti Formula Koshi Adamara Znachennya radiusu zbizhnosti stepenevogo ryadu mozhe buti obchisleno za formuloyu 1 R lim n a n 1 n displaystyle 1 over R varlimsup limits n rightarrow infty a n 1 n nbsp Nehaj F x displaystyle F x nbsp i G x displaystyle G x nbsp dva stepenevi ryadi z radiusami zbizhnosti R F displaystyle R F nbsp i R G displaystyle R G nbsp Todi R F G min R F R G displaystyle R F G geq min R F R G nbsp R F G min R F R G displaystyle R F cdot G geq min R F R G nbsp R F R F displaystyle R F R F nbsp Yaksho u ryadu G x displaystyle G x nbsp vilnij chlen nulovij todi R F G R F R F 1 R G displaystyle R F circ G geq R F over R F 1 R G nbsp Pitannya pro zbizhnist ryadu v tochkah mezhi x R displaystyle x R nbsp kruga zbizhnosti potrebuye dodatkovogo analizu Oznaka D Alambera Yaksho pri n gt N displaystyle n gt N nbsp i a gt 1 displaystyle alpha gt 1 nbsp vikonano nerivnist a n a n 1 R 1 a n displaystyle left a n over a n 1 right geq R left 1 alpha over n right nbsp todi stepenevij ryad S a n x n displaystyle Sigma a n x n nbsp ye absolyutno zbizhnim v usih tochkah kola x R displaystyle x R nbsp i zbizhnist ye rivnomirnoyu po x displaystyle x nbsp Oznaka Dirihle Yaksho vsi koeficiyenti stepenevogo ryadu S a n x n displaystyle Sigma a n x n nbsp dodatni i poslidovnist a n displaystyle a n nbsp monotonno zbigayetsya do nulya todi cej ryad ye zbizhnim v usih tochkah kola x 1 displaystyle x 1 nbsp okrim mozhlivo tochki x 1 displaystyle x 1 nbsp Druga teorema Abelya Nehaj stepenevij ryad ye zbizhnim v tochci x x 0 displaystyle x x 0 nbsp Todi vin ye rivnomirno zbizhnim po x displaystyle x nbsp na vidrizku sho spoluchaye tochki 0 i x 0 displaystyle x 0 nbsp Pohidna i integral RedaguvatiYaksho deyaka funkciya rivna sumi stepenevogo ryadu v deyakij oblasti to yiyi pohidnu i integral mozhna viznachiti pochlenno prodiferenciyuvavshi i prointegruvavshi dodanki stepenevogo ryadu f x n 1 a n n x c n 1 n 0 a n 1 n 1 x c n displaystyle f prime x sum n 1 infty a n n left x c right n 1 sum n 0 infty a n 1 left n 1 right left x c right n nbsp dd f x d x n 0 a n x c n 1 n 1 k n 1 a n 1 x c n n k displaystyle int f x dx sum n 0 infty frac a n left x c right n 1 n 1 k sum n 1 infty frac a n 1 left x c right n n k nbsp dd Radiusi zbizhnosti oboh cih ryadiv dorivnyuyut radiusu pochatkovogo ryadu Stepenevi ryadi bagatoh zminnih RedaguvatiStepenevij ryad vid n zminnih ryad vidu F X 1 X 2 X n k 1 k 2 k n 0 a k 1 k 2 k n X 1 k 1 X 2 k 2 X n k n displaystyle F X 1 X 2 dots X n sum limits k 1 k 2 dots k n 0 infty a k 1 k 2 dots k n X 1 k 1 X 2 k 2 ldots X n k n nbsp abo v multiindeksnih poznachennyah F X a a a X a displaystyle F X sum limits alpha a alpha X alpha nbsp de X displaystyle X nbsp ce vektor X X 1 X 2 X n displaystyle X X 1 X 2 dots X n nbsp a displaystyle alpha nbsp multiindeks a k 1 k 2 k n displaystyle alpha k 1 k 2 dots k n nbsp X a displaystyle X alpha nbsp odnochlen X a X 1 k 1 X 2 k 2 X n k n displaystyle X alpha X 1 k 1 X 2 k 2 ldots X n k n nbsp Div takozh RedaguvatiRyad matematika Formalnij stepenevij ryad Ryad Tejlora Teorema Abelya Ryad PyuyizoLiteratura RedaguvatiFihtengolc G M Kurs differencialnogo i integralnogo ischisleniya Moskva Nauka 1964 T 2 800 s ros Grishenko A O Nagnibida M I Nastasiv P P Teoriya funkcij kompleksnoyi zminnoyi K Visha shkola 1994 375 st Shabat B V Vvedenie v kompleksnyj analiz M Nauka 1969 Zill Dennis G Shanahan Patrick D A first course in complex analysis with applications Jones and Bartlett Publishers Inc ISBN 0763714372Posilannya RedaguvatiStepenevi ryadi Teorema Abelya Radius zbizhnosti Oblast zbizhnosti stepenevogo ryadu Visha matematika v prikladah i zadachah Klepko V Yu Golec V L 2 ge vidannya K Centr uchbovoyi literaturi 2009 S 520 594 s Otrimano z https uk wikipedia org w index php title Stepenevij ryad amp oldid 40291129
