Peretvorennya Lapla sa integralne peretvorennya sho pov yazuye funkciyu F s displaystyle F s kompleksnoyi zminnoyi zobrazhennya z funkciyeyu f x displaystyle f x dijsnoyi zminnoyi original Za jogo dopomogoyu doslidzhuyut vlastivosti dinamichnih sistem i rozv yazuyutsya diferencialni i integralni rivnyannya Odniyeyu z osoblivostej peretvorennya Laplasa yaki zumovili jogo shiroke poshirennya v naukovih i inzhenernih rozrahunkah ye te sho bagatom spivvidnoshennyam i operaciyam nad originalam vidpovidayut prostishi spivvidnoshennya mizh yihnimi zobrazhennyami Tak zgortka dvoh funkcij zvoditsya v prostori zobrazhen do operaciyi mnozhennya a linijni diferencialni rivnyannya stayut algebrayichnimi Zmist 1 Oznachennya 1 1 Pryame peretvorennya Laplasa 1 2 Obernene peretvorennya Laplasa 1 3 Dvostoronnye peretvorennya Laplasa 1 4 Diskretne peretvorennya Laplasa 2 Vlastivosti 3 Pryame i obernene peretvorennya Laplasa deyakih funkcij 4 Zastosuvannya peretvorennya Laplasa 5 Literatura 6 Internet resursi 7 Div takozhOznachennya RedaguvatiPryame peretvorennya Laplasa Redaguvati Peretvorennyam Laplasa funkciyi dijsnoyi zminnoyi f x displaystyle f x nbsp nazivayetsya funkciya F s displaystyle F s nbsp kompleksnoyi zminnoyi s s i w displaystyle s sigma i omega nbsp taka sho F s L f t 0 e s t f t d t displaystyle F s mathcal L left f t right int limits 0 infty limits e st f t dt nbsp Prava chastina cogo virazu nazivayetsya integralom Laplasa Obernene peretvorennya Laplasa Redaguvati Obernenim peretvorennyam Laplasa funkciyi kompleksnoyi zminnoyi F s displaystyle F s nbsp nazivayetsya funkciya f x displaystyle f x nbsp dijsnoyi zminnoyi taka sho f x L 1 F s 1 2 p i s 1 i s 1 i e s x F s d s displaystyle f x mathcal L 1 F s frac 1 2 pi i int limits sigma 1 i cdot infty sigma 1 i cdot infty limits e sx F s ds nbsp de s 1 displaystyle sigma 1 nbsp deyake dijsne chislo Prava chastina cogo virazu nazivayetsya integralom Bromvicha Dvostoronnye peretvorennya Laplasa Redaguvati Dokladnishe Dvostoronnye peretvorennya LaplasaDvostoronnye peretvorennya Laplasa viznachayetsya takim chinom F s L f x e s x f x d x displaystyle F s mathcal L left f x right int limits infty infty limits e sx f x dx nbsp Diskretne peretvorennya Laplasa Redaguvati Rozriznyayut D displaystyle D nbsp peretvorennya i Z displaystyle Z nbsp peretvorennya D displaystyle D nbsp peretvorennyaNehaj x d t n 0 x n T d t n T displaystyle x d left t right sum limits n 0 infty x left nT right cdot delta left t nT right nbsp diskretna funkciya tobto znachennya ciyeyi funkciyi viznacheni tilki v diskretni momenti chasu n T displaystyle nT nbsp de n displaystyle n nbsp cile chislo a T displaystyle T nbsp period diskretizaciyi Todi zastosovuyuchi peretvorennya Laplasa oderzhuyemo D x d t n 0 x n T e s n T displaystyle mathcal D left x d left t right right sum limits n 0 infty x left nT right cdot e snT nbsp Z displaystyle Z nbsp peretvorennyaYaksho vikoristati navedenu zaminu zminnih z e s T displaystyle z e sT nbsp oderzhimo Z peretvorennya Z x d t n 0 x n T z n displaystyle mathcal Z left x d left t right right sum limits n 0 infty x left nT right cdot z n nbsp Vlastivosti RedaguvatiAbsolyutna zbizhnistYaksho integral Laplasa ye absolyutno zbizhnim pri s s 0 displaystyle sigma sigma 0 nbsp tobto isnuye granicya lim b 0 b f x e s 0 x d x 0 f x e s 0 x d x displaystyle lim b to infty int limits 0 b limits f x e sigma 0 x dx int limits 0 infty limits f x e sigma 0 x dx nbsp to vin ye zbizhnim absolyutno i rivnomirno dlya s s 0 displaystyle sigma geqslant sigma 0 nbsp i F s displaystyle F s nbsp analitichna funkciya pri s s 0 displaystyle sigma geqslant sigma 0 nbsp s R e s displaystyle sigma operatorname mathrm Re s nbsp dijsna chastina kompleksnoyi zminnoyi s displaystyle s nbsp Tochna nizhnya gran s a displaystyle sigma a nbsp mnozhini chisel s displaystyle sigma nbsp pri yakih cya umova vikonuyetsya nazivayetsya abscisoyu absolyutnoyi zbizhnosti peretvorennya Laplasa dlya funkciyi f x displaystyle f x nbsp Dostatni umovi isnuvannya pryamogo peretvorennya LaplasaPeretvorennya Laplasa L f x displaystyle mathcal L f x nbsp isnuye v sensi absolyutnoyi zbizhnosti v nastupnih vipadkah Vipadok s 0 displaystyle sigma geqslant 0 nbsp peretvorennya Laplasa isnuye yaksho isnuye integral 0 f x d x displaystyle int limits 0 infty limits f x dx nbsp Vipadok s gt s a displaystyle sigma gt sigma a nbsp peretvorennya Laplasa isnuye yaksho integral 0 x 1 f x d x displaystyle int limits 0 x 1 limits f x dx nbsp isnuye dlya kozhnogo skinchennogo x 1 gt 0 displaystyle x 1 gt 0 nbsp i f x K e s a x displaystyle f x leqslant Ke sigma a x nbsp dlya x gt x 2 0 displaystyle x gt x 2 geqslant 0 nbsp Vipadok s gt 0 displaystyle sigma gt 0 nbsp abo s gt s a displaystyle sigma gt sigma a nbsp yaka iz granic bilsha peretvorennya Laplasa isnuye yaksho isnuye peretvorennya Laplasa dlya funkciyi f x displaystyle f x nbsp pohidna do f x displaystyle f x nbsp dlya s gt s a displaystyle sigma gt sigma a nbsp Dostatani umovi isnuvannya obernenogo peretvorennya Laplasa1 Yaksho F s displaystyle F s nbsp analitichna funkciya dlya s s a displaystyle sigma geqslant sigma a nbsp i maye poryadok menshe 1 to obernene peretvorennya dlya neyi isnuye i ye neperervnim dlya vsih znachen argumenta prichomu L 1 F s 0 displaystyle mathcal L 1 F s 0 nbsp dlya t 0 displaystyle t leqslant 0 nbsp 2 Nehaj F s f F 1 s F 2 s F n s displaystyle F s varphi F 1 s F 2 s dots F n s nbsp tak shob f z 1 z 2 z n displaystyle varphi z 1 z 2 dots z n nbsp analitichna vidnosno kozhnogo z k displaystyle z k nbsp i rivna nulyu dlya z 1 z 2 z n 0 displaystyle z 1 z 2 dots z n 0 nbsp i F k s L f k x s gt s a k k 1 2 n displaystyle F k s mathcal L f k x sigma gt sigma ak colon k 1 2 dots n nbsp todi obernene peretvorennya isnuye i vidpovidne pryame peretvorennya maye abscisu absolyutnoyi zbizhnosti Teorema pro zgortkuPeretvorennyam Laplasa zgortki dvoh originaliv ye dobutok zobrazhen cih originaliv L f x g x L f x L g x displaystyle mathcal L f x g x mathcal L f x cdot mathcal L g x nbsp Mnozhennya zobrazhenf x g 0 0 x f x t g t d t L 1 s F s G s displaystyle f x g 0 int limits 0 x limits f x tau g tau d tau mathcal L 1 sF s G s nbsp Liva chastina cogo virazu nazivayetsya integralom Dyuamelya Diferenciyuvannya i integruvannya originaluDlya peretvorennya Laplasa vid pohidnoyi funkciyi vikonuyetsya rivnist L f x s F s f 0 displaystyle mathcal L f x s cdot F s f 0 nbsp Dlya pohidnoyi n displaystyle n nbsp go poryadku L f n x s n F s s n 1 f 0 f n 1 0 displaystyle mathcal L left f n x right s n cdot F s s n 1 f 0 dots f n 1 0 nbsp Peretvorennya Laplasa vid integrala funkciyi dorivnyuye L 0 x f t d t F s s displaystyle mathcal L left int limits 0 x limits f t dt right frac F s s nbsp Difrenciyuvannya ta integruvannya zobrazhennyaObernene peretvorennya Laplasa vid pohidnoyi funkciyi dorivnyuye L 1 F s x f x displaystyle mathcal L 1 F s xf x nbsp Obernene peretvorennya Laplasa vid pohidnoyi funkciyi dorivnyuye L 1 s F s d s f x x displaystyle mathcal L 1 left int limits s infty limits F s ds right frac f x x nbsp Zapiznennya originaliv i zobrazhen Granichni teoremiZapiznennya zobrazhen L e a x f x F s a displaystyle mathcal L left e ax f x right F s a nbsp L 1 F s a e a x f x displaystyle mathcal L 1 left F s a right e ax f x nbsp Zapiznennya originaliv L f t a u t a e a s F s displaystyle mathcal L left f t a u t a right e as F s nbsp L 1 e a s F s f x a u x a displaystyle mathcal L 1 left e as F s right f x a u x a nbsp de u x displaystyle u x nbsp Funkciya Gevisajda Inshi vlastivostiLinijnist L a f x b g x a F s b G s displaystyle mathcal L left af x bg x right aF s bG s nbsp Mnozhennya na chislo L f a x 1 a F s a displaystyle mathcal L left f ax right frac 1 a F left frac s a right nbsp Pryame i obernene peretvorennya Laplasa deyakih funkcij Redaguvati Funkciya Chasova oblast x t L 1 X s displaystyle x t mathcal L 1 left X s right nbsp Chastotna oblast X s L x t displaystyle X s mathcal L left x t right nbsp Oblast zbizhnosti1 idealne zapiznennya d t t displaystyle delta t tau nbsp e t s displaystyle e tau s nbsp 1a odinichnij impuls d t displaystyle delta t nbsp 1 displaystyle 1 nbsp s displaystyle forall s nbsp 2 zapiznennya n go poryadku z chastotnim zsuvom t t n n e a t t u t t displaystyle frac t tau n n e alpha t tau cdot u t tau nbsp e t s s a n 1 displaystyle frac e tau s s alpha n 1 nbsp s gt 0 displaystyle s gt 0 nbsp 2a stepeneva n go poryadku t n n u t displaystyle t n over n cdot u t nbsp 1 s n 1 displaystyle 1 over s n 1 nbsp s gt 0 displaystyle s gt 0 nbsp 2a 1 stepeneva q go poryadku t q G q 1 u t displaystyle t q over Gamma q 1 cdot u t nbsp 1 s q 1 displaystyle 1 over s q 1 nbsp s gt 0 displaystyle s gt 0 nbsp 2a 2 odinichna funkciya u t displaystyle u t nbsp 1 s displaystyle 1 over s nbsp s gt 0 displaystyle s gt 0 nbsp 2b odinichna funkciya z zapiznennyam u t t displaystyle u t tau nbsp e t s s displaystyle e tau s over s nbsp s gt 0 displaystyle s gt 0 nbsp 2c shodinka shvidkosti t u t displaystyle t cdot u t nbsp 1 s 2 displaystyle frac 1 s 2 nbsp s gt 0 displaystyle s gt 0 nbsp 2d n go poryadku z chastotnim zsuvom t n n e a t u t displaystyle frac t n n e alpha t cdot u t nbsp 1 s a n 1 displaystyle frac 1 s alpha n 1 nbsp s gt a displaystyle s gt alpha nbsp 2d 1 eksponencijne zatuhannya e a t u t displaystyle e alpha t cdot u t nbsp 1 s a displaystyle 1 over s alpha nbsp s gt a displaystyle s gt alpha nbsp 3 eksponencijne nablizhennya 1 e a t u t displaystyle 1 e alpha t cdot u t nbsp a s s a displaystyle frac alpha s s alpha nbsp s gt 0 displaystyle s gt 0 nbsp 4 sinus sin w t u t displaystyle sin omega t cdot u t nbsp w s 2 w 2 displaystyle omega over s 2 omega 2 nbsp s gt 0 displaystyle s gt 0 nbsp 5 kosinus cos w t u t displaystyle cos omega t cdot u t nbsp s s 2 w 2 displaystyle s over s 2 omega 2 nbsp s gt 0 displaystyle s gt 0 nbsp 6 giperbolichnij sinus sinh a t u t displaystyle sinh alpha t cdot u t nbsp a s 2 a 2 displaystyle alpha over s 2 alpha 2 nbsp s gt a displaystyle s gt alpha nbsp 7 giperbolichnij kosinus cosh a t u t displaystyle cosh alpha t cdot u t nbsp s s 2 a 2 displaystyle s over s 2 alpha 2 nbsp s gt a displaystyle s gt alpha nbsp 8 eksponencijno zatuhayuchij sinus e a t sin w t u t displaystyle e alpha t sin omega t cdot u t nbsp w s a 2 w 2 displaystyle omega over s alpha 2 omega 2 nbsp s gt a displaystyle s gt alpha nbsp 9 eksponencijno zatuhayuchij kosinus e a t cos w t u t displaystyle e alpha t cos omega t cdot u t nbsp s a s a 2 w 2 displaystyle s alpha over s alpha 2 omega 2 nbsp s gt a displaystyle s gt alpha nbsp 10 korin n go poryadku t n u t displaystyle sqrt n t cdot u t nbsp s n 1 n G 1 1 n displaystyle s n 1 n cdot Gamma left 1 frac 1 n right nbsp s gt 0 displaystyle s gt 0 nbsp 11 naturalnij logarifm ln t t 0 u t displaystyle ln left t over t 0 right cdot u t nbsp t 0 s ln t 0 s g displaystyle t 0 over s ln t 0 s gamma nbsp s gt 0 displaystyle s gt 0 nbsp 12 funkciya Besselya pershogo rodu poryadku n J n w t u t displaystyle J n omega t cdot u t nbsp w n s s 2 w 2 n s 2 w 2 displaystyle frac omega n left s sqrt s 2 omega 2 right n sqrt s 2 omega 2 nbsp s gt 0 displaystyle s gt 0 nbsp n gt 1 displaystyle n gt 1 nbsp 13 modifikovana funkciya Besselya pershogo rodu poryadku n I n w t u t displaystyle I n omega t cdot u t nbsp w n s s 2 w 2 n s 2 w 2 displaystyle frac omega n left s sqrt s 2 omega 2 right n sqrt s 2 omega 2 nbsp s gt w displaystyle s gt omega nbsp 14 funkciya Besselya drugogo rodu nulovogo poryadku Y 0 a t u t displaystyle Y 0 alpha t cdot u t nbsp 15 modifikovana funkciya Besselya drugogo rodu nulovogo poryadku K 0 a t u t displaystyle K 0 alpha t cdot u t nbsp 16 funkciya pomilok e r f t u t displaystyle mathrm erf t cdot u t nbsp e s 2 4 erfc s 2 s displaystyle e s 2 4 operatorname erfc left s 2 right over s nbsp s gt 0 displaystyle s gt 0 nbsp Primitki do tablici u t displaystyle u t nbsp Funkciya Gevisajda d t displaystyle delta t nbsp delta funkciya G z displaystyle Gamma z nbsp gamma funkciya g displaystyle gamma nbsp stala Ejlera Maskeroni t displaystyle t nbsp dijsna zminna s displaystyle s nbsp kompleksna zminna a displaystyle alpha nbsp b displaystyle beta nbsp t displaystyle tau nbsp i w displaystyle omega nbsp dijsni chisla n displaystyle n nbsp cile chislo Zastosuvannya peretvorennya Laplasa RedaguvatiPeretvorennya Laplasa shiroko vikoristovuyetsya v matematici fizici i tehnici Rozv yazok sistem diferencialnih i integralnih rivnyan za dopomogoyu peretvorennya Laplasa legko perehoditi vid skladnih ponyat matematichnogo analizu do prostishih algebrayichnih vidnoshen Rozrahunok vihidnih signaliv dinamichnih sistem v teoriyi upravlinnya i obrobci signaliv Rozrahunok elektrichnih shem za dopomogoyu operatornogo metodu Rozv yazok nestacionarnih zadach matematichnoyi fiziki Literatura RedaguvatiVan der Pol B Bremer H Operacionnoe ischislenie na osnove dvustoronnego preobrazovaniya Laplasa M IL 1952 Ditkin V A Prudnikov A P Integralnye preobrazovaniya i operacionnoe ischislenie M Fizmatgiz 1961 Ditkin V A Prudnikov A P Integralnye preobrazovaniya i operacionnoe ischislenie M Fizmatgiz 1974 542 s Karslou H Eger D Operacionnye metody v prikladnoj matematike M IL 1948 Kozhevnikov N I Krasnoshekova T I Shishkin N E Ryady i integraly Fure Teoriya polya Analiticheskie i specialnye funkcii Preobrazovaniya Laplasa M Nauka 1964 Krasnov M L Makarenko G I Operacionnoe ischislenie Ustojchivost dvizheniya M Nauka 1964 103 s Mikusinskij Ya Operatornoe ischislenie M IL 1956 Romanovskij P I Ryady Fure Teoriya polya Analiticheskie i specialnye funkcii Preobrazovaniya Laplasa M Nauka 1980 336 s Internet resursi RedaguvatiOperacijne chislennya Peretvorennya Laplasa Div takozh RedaguvatiPeretvorennya Fur ye Diferencialni rivnyannya Peretvorennya Besselya Peretvorennya Mellina Peretvorennya Karsona Hevisajda Otrimano z https uk wikipedia org w index php title Peretvorennya Laplasa amp oldid 34421339
