trigonometricheskiy ryad. vichislenie koeffitsientov ryada fure. ryad fure dlya funktsii s periodom 2 1. primenenie priblizitelnogo rascheta ryadov.
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![a funktsiy {1,cos(x),sin(x),cos(2x),sin(2x),…,cos(nx),sin(nx),…} nazivaetsya ortogonalnoy na otrezke [−π,π], esli integral ot proizvedeniya lyubix dvux razlichnix funktsiy etoy sistemi raven nulyu. naprimer: ∫−ππcos(nx)sin(mx)dx=0 pri lyubix n,m∈n∫−ππcos(nx)cos(mx)dx=0 pri n=m∫−ππsin(nx)sin(mx)dx=0 pri n=m imenno svoystvo ortogonalnosti ispolzuetsya dlya "otdeleniya" koeffitsientov. usloviya dirixle (dirichlet conditions) dlya togo chtobi funktsiya f(x) mogla bit razlojena v sxodyashiysya ryad fure, ona doljna udovletvoryat tak nazivaemim usloviyam dirixle na intervale razlojeniya: 1. periodichnost: funktsiya doljna bit periodicheskoy s periodom t (ili rassmatrivatsya na otrezke dlini t). 2. ogranichennost i konechnoe chislo razrivov: funktsiya doljna bit ogranichennoy i imet na dannom periode ne bolee chem konechnoe chislo tochek razriva pervogo roda (kogda sushestvuyut konechnie odnostoronnie predeli). 3. konechnoe chislo ekstremumov: funktsiya doljna imet na dannom periode ne bolee chem konechnoe chislo maksimumov i minimumov. sxodimost ryada fure: · v tochkax neprerivnosti: esli x — tochka, v kotoroy funktsiya f(x) neprerivna i udovletvoryaet usloviyam dirixle, to ryad fure sxoditsya k samoy …](https://cdn.faylhub.uz/previews/pages/p2_201042.webp "trigonometricheskiy ryad. vichislenie koeffitsientov ryada fure. ryad fure dlya funktsii s periodom 2 1. primenenie priblizitelnogo rascheta ryadov. - Page 2")
![m ryadom fure: f(x)=2a0+k=1∑∞(akcos(kx)+bksin(kx)) vichislenie koeffitsienta a0 dlya naxojdeniya a0, prointegriruem obe chasti ravenstva po intervalu [−π,π]: $$\int_{-\pi}^{\pi} f(x) dx = \int_{-\pi}^{\pi} \frac{a_0}{2} dx + \sum_{k=1}^{\infty} \left[ a_k \int_{-\pi}^{\pi} \cos(kx) dx + b_k \int_{-\pi}^{\pi} \sin(kx) dx \right]$$v silu ortogonalnosti, integrali ot $\cos(kx)$ i $\sin(kx)$ na etom simmetrichnom intervale ravni nulyu:$$\int_{-\pi}^{\pi} \cos(kx) dx = 0 \text{ i } \int_{-\pi}^{\pi} \sin(kx) dx = 0 \text{ pri } k \ge 1$$ ostaetsya tolko pervoe slagaemoe: ∫−ππf(x)dx=2a0⋅2π=a0π otsyuda poluchaem formulu dlya a0: a0=π1∫−ππf(x)dx (koeffitsient a0 proportsionalen srednemu znacheniyu funktsii na periode). vichislenie koeffitsientov an (dlya n≥1) chtobi nayti an, umnojim obe chasti isxodnogo razlojeniya na cos(nx) i prointegriruem: ∫−ππf(x)cos(nx)dx=2a0∫−ππcos(nx)dx+k=1∑∞[ak∫−ππcos(kx)cos(nx)dx+bk∫−ππsin(kx)cos(nx)dx] v silu ortogonalnosti, vse integrali, krome togo, gde k=n, ravni nulyu. edinstvenniy nenulevoy integral: ∫−ππcos2(nx)dx=π. sledovatelno: ∫−ππf(x)cos(nx)dx=anπ otsyuda formula dlya an: an=π1∫−ππf(x)cos(nx)dx(n=1,2,3,…) vichislenie koeffitsientov bn analogichno, umnojim na sin(nx) i prointegriruem. nenulevim ostanetsya tolko integral s bn: ∫−ππsin2(nx)dx=π. bn=π1∫−ππf(x)sin(nx)dx(n=1,2,3,…) ryad fure dlya funktsii s proizvolnim …](https://cdn.faylhub.uz/previews/pages/p3_201042.webp "trigonometricheskiy ryad. vichislenie koeffitsientov ryada fure. ryad fure dlya funktsii s periodom 2 1. primenenie priblizitelnogo rascheta ryadov. - Page 3")
![vajnix instrumentov v prikladnoy matematike, injenerii i fizike. ryad fure dlya funktsii s periodom 2l=2 sluchay 2l=2 oznachaet, chto period funktsii t=2, i, sledovatelno, l=1. razlojenie provoditsya na intervale [−1,1]. podstavlyaya l=1 v obshie formuli dlya perioda 2l, poluchaem: ryad fure dlya t=2 (l=1): f(x)≈2a0+n=1∑∞(ancos(nπx)+bnsin(nπx)) koeffitsienti: a0=11∫−11f(x)dxan=∫−11f(x)cos(nπx)dx(n≥1)bn=∫−11f(x)sin(nπx)dx(n≥1) takoy ryad ispolzuetsya, naprimer, pri analize periodicheskix signalov ili kolebaniy, zadannix v edinichnom vremennom intervale. ispolzovanie ryadov fure dlya priblijennogo rascheta na praktike, osobenno v injenerii i obrabotke dannix, mi redko ispolzuem beskonechniy ryad fure. vmesto etogo ispolzuetsya chastichnaya summa ryada ili konechnoe garmonicheskoe razlojenie dlya approksimatsii funktsii. f(x)≈sn(x)=2a0+n=1∑n(ancos(nx)+bnsin(nx)) chem bolshe chlenov n mi berem, tem tochnee approksimatsiya. priblijenniy raschet koeffitsientov (garmonicheskiy analiz) v realnix zadachax (naprimer, analiz elektrokardiogrammi, zvukovix voln) funktsiya f(x) chasto zadaetsya empiricheskimi dannimi (naborom tochek) ili ne imeet prostoy analiticheskoy formuli, chto delaet pryamoe integrirovanie nevozmojnim. v etix sluchayax ispolzuetsya chislenniy (priblijenniy) raschet koeffitsientov, nazivaemiy garmonicheskim analizom. 1. chislennoe …](https://cdn.faylhub.uz/previews/pages/p4_201042.webp "trigonometricheskiy ryad. vichislenie koeffitsientov ryada fure. ryad fure dlya funktsii s periodom 2 1. primenenie priblizitelnogo rascheta ryadov. - Page 4")
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trigonometricheskiy ryad. vichislenie koeffitsientov ryada fure. ryad fure dlya funktsii s periodom 2 1. primenenie priblizitelnogo rascheta ryadov. plan 1. trigonometricheskie ryadi: osnovnie ponyatiya i usloviya predstavleniya funktsii 2. vichislenie koeffitsientov ryada fure 3. prakticheskoe primenenie ryadov fure i priblijennie vichisleniya 1. trigonometricheskie ryadi: osnovnie ponyatiya i usloviya predstavleniya funktsii obshee ponyatie trigonometricheskogo ryada trigonometricheskiy ryad — eto funktsionalniy ryad spetsialnogo vida, sostoyashiy iz sinusov i kosinusov kratnix argumentov. obshiy vid takogo ryada: f(x)≈2a0+n=1∑∞(ancos(nx)+bnsin(nx)) gde a0,an,bn — chislovie koeffitsienti ryada. trigonometricheskiy ryad imeet period t=2π, poskolku period funktsiy cos(nx) i sin(nx)...
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