Integral Gausa takozh vidomij yak integral Ejlera Puassona ce integral funkciyi Gausa e x2 nad usiyeyu oblastyu dijsnih chisel Nazvanij na chest nimeckogo matematika Karla Fridriha Gausa i maye viglyadGrafik f x e x2 i plosha mizh funkciyeyu ta x vissyu sho dorivnyuye p e x 2 d x p displaystyle int infty infty e x 2 dx sqrt pi Abraham de Muavr pershim vidkriv cej tip integraliv u 1733 r Todi yak Gauss opublikuvav tochnij integral u 1809 r 1 Integral maye shirokij spektr zastosuvan Napriklad za dopomogoyu prostoyi zamini zminnih vikoristovuyetsya dlya obchislennya normalizuyuchoyi konstanti normalnogo rozpodilu Cej zhe integral z fiksovanimi mezhami tisno pov yazanij yak z funkciyeyu pomilok tak i z kumulyativnoyu funkciyeyu normalnogo rozpodilu U fizici cej tip integrala chasto z yavlyayetsya napriklad v kvantovij mehanici pri znahodzhenni gustini jmovirnosti osnovnogo stanu garmonichnogo oscilyatora Cej integral takozh vikoristovuyetsya v oznachenni integrala vzdovzh trayektoriyi dlya znahodzhennya funkciyi poshirennya garmonichnogo oscilyatora a takozh u statistichnij mehanici dlya znahodzhennya funkciyi rozbittya Hocha funkciyu pomilok ne mozhna predstaviti elementarni funkciyi yak ce mozhna dovesti za dopomogoyu algoritmu Risha 2 vse zh integral Gaussa mozhna obchisliti analitichnimi metodami analizu funkcij bagatoh zminnih Tobto neviznachenij integral e x 2 d x displaystyle int e x 2 dx ne integruyetsya v elementarnih funkciyah ale viznachenij integral e x 2 d x displaystyle int infty infty e x 2 dx mozhna obchisliti Viznachenij integral dovilnoyi funkciyi Gaussa dorivnyuye e a x b 2 d x p a displaystyle int infty infty e a x b 2 dx sqrt frac pi a Zmist 1 Obchislennya 1 1 V polyarnih koordinatah 1 1 1 Povne dovedennya 1 2 U dekartovih koordinatah 2 Zv yazok z gamma funkciyeyu 3 Uzagalnennya 3 1 Integral funkciyi Gaussa 3 2 n mirne ta funkcionalne uzagalnennya 3 3 n mirnij z linijnim chlenom 3 4 Integrali podibnoyi formi 3 5 Polinomi vishih poryadiv 4 Div takozh 5 PrimitkiObchislennya RedaguvatiV polyarnih koordinatah Redaguvati Standartnij sposib obchislennya integrala Gaussa sho buv zaproponovanij Puassonom bazuyetsya na vikoristanni nastupnoyi vlastivosti 3 bazuyetsya na vikoristanni nastupnoyi vlastivosti e x 2 d x 2 e x 2 d x e y 2 d y e x 2 y 2 d x d y displaystyle left int infty infty e x 2 dx right 2 int infty infty e x 2 dx int infty infty e y 2 dy int infty infty int infty infty e x 2 y 2 dx dy nbsp Rozglyanemo funkciyu e x 2 y 2 e r 2 displaystyle e x 2 y 2 e r 2 nbsp u prostori R 2 displaystyle mathbb R 2 nbsp ta obchislimo integral vid neyi dvoma sposobami Z odniyeyi storoni yak podvijnij integral v dekatrovij sistemi koordinat vin dorivnyuye kvadrat integrala Gaussa e x 2 d x 2 displaystyle left int e x 2 dx right 2 nbsp Z inshoyi storoni za dopomogoyu metodu znahodzhennya ob yemu tila obertannya vipadok podvijnogo integrala u polyarnih koordinatah cej integral dorivnyuye p displaystyle pi nbsp Z cih dvoh obchislen otrimayemo shukane znachennya dlya integrala Gaussa hocha pri comu slid podbati pro vidpovidni nevlasni integrali R 2 e x 2 y 2 d x d y 0 2 p 0 e r 2 r d r d 8 2 p 0 r e r 2 d r 2 p 0 1 2 e s d s s r 2 p 0 e s d s p e 0 e p displaystyle begin aligned iint mathbf R 2 e x 2 y 2 dx dy amp int 0 2 pi int 0 infty e r 2 r dr d theta 6pt amp 2 pi int 0 infty re r 2 dr 6pt amp 2 pi int infty 0 tfrac 1 2 e s ds amp amp s r 2 6pt amp pi int infty 0 e s ds 6pt amp pi e 0 e infty 6pt amp pi end aligned nbsp de mnozhnik r yakobian yakij z yavlyayetsya pri perehodi do polyarnih koordinat u podvijnomu integrali r dr d8 standartna mira na ploshini zapisana v polyarnih koordinatah i integruvannya vklyuchaye zaminu s r2 a tomu ds 2r dr Takim chinom e x 2 d x 2 p displaystyle left int infty infty e x 2 dx right 2 pi nbsp Todi e x 2 d x p displaystyle int infty infty e x 2 dx sqrt pi nbsp Povne dovedennya Redaguvati Dlya obgruntuvannya nevlasnih podvijnih integraliv ta pririvnyuvannya dvoh spivvidnoshen rozpochnemo z aproksimuyuchoyi funkciyi I a a a e x 2 d x displaystyle I a int a a e x 2 dx nbsp Yakbi integral e x 2 d x displaystyle int infty infty e x 2 dx nbsp buv bi absolyutno zbizhnim to mi b otrimali golovne znachennya integrala za Koshi tobto granicya lim a I a displaystyle lim a to infty I a nbsp spivpadala b z integralom e x 2 d x displaystyle int infty infty e x 2 dx nbsp Shob pobachiti ce vrahuyemo sho e x 2 d x lt 1 x e x 2 d x 1 1 e x 2 d x 1 x e x 2 d x lt displaystyle int infty infty e x 2 dx lt int infty 1 xe x 2 dx int 1 1 e x 2 dx int 1 infty xe x 2 dx lt infty nbsp Takim chinom dlya obchislennya integrala e x 2 d x displaystyle int infty infty e x 2 dx nbsp potribno znajti granicyu lim a I a displaystyle lim a to infty I a nbsp Pidnosyachi I a displaystyle I a nbsp do kvadratu otrimayemo I 2 a a a e x 2 d x a a e y 2 d y a a a a e y 2 d y e x 2 d x a a a a e x 2 y 2 d y d x displaystyle begin aligned I 2 a amp left int a a e x 2 dx right left int a a e y 2 dy right 6pt amp int a a left int a a e y 2 dy right e x 2 dx 6pt amp int a a int a a e x 2 y 2 dy dx end aligned nbsp Vikoristovuyuchi teoremu Fubini vishevkazanij podvijnij integral mozhna rozglyadati yak poverhnevij integral a a a a e x 2 y 2 d x y displaystyle iint a a times a a e x 2 y 2 d x y nbsp vzyatij nad kvadratom z vershinami a a a a a a a a na ploshini xy Oskilki dlya vsih dijsnih chisel eksponencialna funkciya bilsha za 0 to zvidsi viplivaye sho integral vzyatij nad vpisanim krugom povinen buti menshim za I a 2 displaystyle I a 2 nbsp i analogichno integral vzyatij nad opisanim krugom povinen buti bilshim nizh I a 2 displaystyle I a 2 nbsp Integrali nad dvoma krugami legko obchisliti perejshovshi vid dekartovoyi sistemi koordinat do polyarnoyi sistemi koordinat x r cos 8 y r sin 8 displaystyle begin aligned x amp r cos theta y amp r sin theta end aligned nbsp J r 8 x r x 8 y r y 8 cos 8 r sin 8 sin 8 r cos 8 displaystyle mathbf J r theta begin bmatrix dfrac partial x partial r amp dfrac partial x partial theta 1em dfrac partial y partial r amp dfrac partial y partial theta end bmatrix begin bmatrix cos theta amp r sin theta sin theta amp r cos theta end bmatrix nbsp d x y J r 8 d r 8 r d r 8 displaystyle d x y J r theta d r theta r d r theta nbsp 0 2 p 0 a r e r 2 d r d 8 lt I 2 a lt 0 2 p 0 a 2 r e r 2 d r d 8 displaystyle int 0 2 pi int 0 a re r 2 dr d theta lt I 2 a lt int 0 2 pi int 0 a sqrt 2 re r 2 dr d theta nbsp Div Polyarni koordinati z dekartovih koordinat shodo vidpovidnih peretvoren Pislya integruvannya otrimuyemo p 1 e a 2 lt I 2 a lt p 1 e 2 a 2 displaystyle pi 1 e a 2 lt I 2 a lt pi 1 e 2a 2 nbsp Za teoremoyu pro dvoh policejskih otrimayemo znachennya integrala Gaussa e x 2 d x p displaystyle int infty infty e x 2 dx sqrt pi nbsp U dekartovih koordinatah Redaguvati Insha tehnika yaka shodit do Laplasa 1812 3 polyagaye v nastupnomu Poklademo y x s d y x d s displaystyle begin aligned y amp xs dy amp x ds end aligned nbsp Oskilki granici vidnosno s priy zalezhat vid znaku x to dlya sproshennya obchislen vikoristovuyemo toj fakt sho e x2 ye parnoyu funkciyeyu i otzhe integral na mnozhini vsih dijsnih chisel udvichi bilshij nizh integral vid nulya do neskinchennosti e x 2 d x 2 0 e x 2 d x displaystyle int infty infty e x 2 dx 2 int 0 infty e x 2 dx nbsp Takim chinom pri x 0 dlya zminnih y and s mayemo odnakovi granici Tobto I 2 4 0 0 e x 2 y 2 d y d x 4 0 0 e x 2 y 2 d y d x 4 0 0 e x 2 1 s 2 x d s d x 4 0 0 e x 2 1 s 2 x d x d s 4 0 1 2 1 s 2 e x 2 1 s 2 x 0 x d s 4 1 2 0 d s 1 s 2 2 arctan s 0 p displaystyle begin aligned I 2 amp 4 int 0 infty int 0 infty e x 2 y 2 dy dx 6pt amp 4 int 0 infty left int 0 infty e x 2 y 2 dy right dx 6pt amp 4 int 0 infty left int 0 infty e x 2 1 s 2 x ds right dx 6pt amp 4 int 0 infty left int 0 infty e x 2 1 s 2 x dx right ds 6pt amp 4 int 0 infty left frac 1 2 1 s 2 e x 2 1 s 2 right x 0 x infty ds 6pt amp 4 left frac 1 2 int 0 infty frac ds 1 s 2 right 6pt amp 2 Big arctan s Big 0 infty 6pt amp pi end aligned nbsp Otzhe yak i ochikuvalosya I p displaystyle I sqrt pi nbsp Zv yazok z gamma funkciyeyu RedaguvatiPidintegralna funkciya ce parna funkciya tomu e x 2 d x 2 0 e x 2 d x displaystyle int infty infty e x 2 dx 2 int 0 infty e x 2 dx nbsp Takim chinom cej integral pislya zamini zminnoyi x t displaystyle x sqrt t nbsp peretvoryuyetsya na integral Ejlera 2 0 e x 2 d x 2 0 1 2 e t t 1 2 d t G 1 2 p displaystyle 2 int 0 infty e x 2 dx 2 int 0 infty frac 1 2 e t t frac 1 2 dt Gamma left frac 1 2 right sqrt pi nbsp de G z 0 t z 1 e t d t displaystyle Gamma z int 0 infty t z 1 e t dt nbsp gamma funkciya Ce pokazuye chomu faktorial napivcilogo chisla ye racionalnim domnozhinim na p displaystyle sqrt pi nbsp U zagalnomu vipadku 0 x n e a x b d x G n 1 b b a n 1 b displaystyle int 0 infty x n e ax b dx frac Gamma left n 1 b right ba n 1 b nbsp yakij mozhna otrimati vikonavshi zaminu t a x b displaystyle t ax b nbsp v pidintegralnij funkciyi gamma funkciyi G z a z b 0 x b z 1 e a x b d x displaystyle Gamma z a z b int 0 infty x bz 1 rm e ax b rm d x nbsp Uzagalnennya RedaguvatiIntegral funkciyi Gaussa Redaguvati Dokladnishe Integral funkciyi GaussaRezultatom obchislennya integrala dovilnoyi funkciyi Gaussa ye e a x b 2 d x p a displaystyle int infty infty e a x b 2 dx sqrt frac pi a nbsp Alternativnoyu formoyu ye e a x 2 b x c d x p a e b 2 4 a c displaystyle int infty infty e ax 2 bx c dx sqrt frac pi a e frac b 2 4a c nbsp Cya forma korisna dlya obchislennya matematichnih spodivan deyakih neperervnih rozpodiliv pov yazanih iz normalnim rozpodilom napriklad dlya lognormalnogo rozpodilu n mirne ta funkcionalne uzagalnennya Redaguvati Dokladnishe bagatovimirnij normalnij rozpodilPripustimo A simetrichna dodatno viznachena otzhe maye obernenu matricya n n yaka ye obernenoyu do kovariacijnoyi matrici Tomu exp 1 2 i j 1 n A i j x i x j d n x exp 1 2 x T A x d n x 2 p n det A 1 det A 2 p det 2 p A 1 displaystyle int infty infty exp left frac 1 2 sum limits i j 1 n A ij x i x j right d n x int infty infty exp left frac 1 2 x T Ax right d n x sqrt frac 2 pi n det A sqrt frac 1 det A 2 pi sqrt det 2 pi A 1 nbsp de integral rozumiyetsya nad Rn Cej fakt zastosovuyetsya pri doslidzhenni bagatovimirnogo normalnogo rozpodilu Takozh x k 1 x k 2 N exp 1 2 i j 1 n A i j x i x j d n x 2 p n det A 1 2 N N s S 2 N A 1 k s 1 k s 2 A 1 k s 2 N 1 k s 2 N displaystyle int x k 1 cdots x k 2N exp left frac 1 2 sum limits i j 1 n A ij x i x j right d n x sqrt frac 2 pi n det A frac 1 2 N N sum sigma in S 2N A 1 k sigma 1 k sigma 2 cdots A 1 k sigma 2N 1 k sigma 2N nbsp de s perestanovka mnozhin 1 2N a dodatkovij koeficiyent u pravij chastini ce suma za vsima kombinatornimi parami mnozhini 1 2N z N kopij matrici A 1 Krim togo 4 f x exp 1 2 i j 1 n A i j x i x j d n x 2 p n det A exp 1 2 i j 1 n A 1 i j x i x j f x x 0 displaystyle int f vec x exp left frac 1 2 sum limits i j 1 n A ij x i x j right d n x sqrt 2 pi n over det A left exp left 1 over 2 sum limits i j 1 n A 1 ij partial over partial x i partial over partial x j right f vec x right vec x 0 nbsp dlya deyakoyi analitichnoyi funkciyi f za umovi sho vona zadovolnyaye deyakim vidpovidnim obmezhennyam shodo svogo zrostannya ta deyakim inshim tehnichnim kriteriyam Dlya deyakih funkcij vona pracyuye napriklad dlya polinomiv A dlya deyakih ni Pidnesennya do stepenya diferencialnim operatorom rozumiyetsya v sensi stepenevih ryadiv Hocha funkcionalni integrali ne mayut chitkogo oznachennya abo navit yih nemozhlivo strogo obchisliti v bilshosti vipadkiv prote mi mozhemo viznachiti funkcionalnij integral Gaussa analogichno skinchennovimirnomu vipadku Ale problema vse zh zalishayetsya oskilki 2 p displaystyle 2 pi infty nbsp ye neskinchennistyu a takozh funkcionalnij determinant bude neskinchennim u zagalnomu vipadku Prote ce mozhna obijti yaksho rozglyanuti vidnoshennya f x 1 f x 2 N exp 1 2 A x 2 N 1 x 2 N 2 f x 2 N 1 f x 2 N 2 d d x 2 N 1 d d x 2 N 2 D f exp 1 2 A x 2 N 1 x 2 N 2 f x 2 N 1 f x 2 N 2 d d x 2 N 1 d d x 2 N 2 D f 1 2 N N s S 2 N A 1 x s 1 x s 2 A 1 x s 2 N 1 x s 2 N displaystyle frac int f x 1 cdots f x 2N exp left iint frac 1 2 A x 2N 1 x 2N 2 f x 2N 1 f x 2N 2 d d x 2N 1 d d x 2N 2 right mathcal D f int exp left iint frac 1 2 A x 2N 1 x 2N 2 f x 2N 1 f x 2N 2 d d x 2N 1 d d x 2N 2 right mathcal D f frac 1 2 N N sum sigma in S 2N A 1 x sigma 1 x sigma 2 cdots A 1 x sigma 2N 1 x sigma 2N nbsp V poznachennyah DeVita ce spivvidnoshennya viglyadaye analogichno skinchennovimirnomu vipadku n mirnij z linijnim chlenom Redaguvati Yaksho A znovu ye simmetrichnoyu dodatno viznachenoyu matriceyu todi pri umovi sho vsi vektori ye vektor stovpcyami e 1 2 i j 1 n A i j x i x j i 1 n B i x i d n x e 1 2 x T A x B T x d n x 2 p n det A e 1 2 B T A 1 B displaystyle int e frac 1 2 sum limits i j 1 n A ij x i x j sum limits i 1 n B i x i d n x int e frac 1 2 vec x T mathbf A vec x vec B T vec x d n x sqrt frac 2 pi n det A e frac 1 2 vec B T mathbf A 1 vec B nbsp Integrali podibnoyi formi Redaguvati 0 x 2 n e x 2 a 2 d x p a 2 n 1 2 n 1 2 n 1 displaystyle int 0 infty x 2n e frac x 2 a 2 dx sqrt pi frac a 2n 1 2n 1 2 n 1 nbsp 0 x 2 n 1 e x 2 a 2 d x n 2 a 2 n 2 displaystyle int 0 infty x 2n 1 e frac x 2 a 2 dx frac n 2 a 2n 2 nbsp 0 x 2 n e a x 2 d x 2 n 1 a n 2 n 1 p a displaystyle int 0 infty x 2n e ax 2 dx frac 2n 1 a n 2 n 1 sqrt frac pi a nbsp 0 x 2 n 1 e a x 2 d x n 2 a n 1 displaystyle int 0 infty x 2n 1 e ax 2 dx frac n 2a n 1 nbsp 0 x n e a x 2 d x G n 1 2 2 a n 1 2 displaystyle int 0 infty x n e ax 2 dx frac Gamma frac n 1 2 2a frac n 1 2 nbsp De n displaystyle n nbsp naturalne chislo displaystyle nbsp podvijnij faktorial Prostij sposib yih otrimannya ce diferenciyuvannya pid znakom integrala x 2 n e a x 2 d x 1 n n a n e a x 2 d x 1 n n a n e a x 2 d x p 1 n n a n a 1 2 p a 2 n 1 2 a n displaystyle begin aligned int infty infty x 2n e alpha x 2 dx amp left 1 right n int infty infty frac partial n partial alpha n e alpha x 2 dx left 1 right n frac partial n partial alpha n int infty infty e alpha x 2 dx 6pt amp sqrt pi left 1 right n frac partial n partial alpha n alpha frac 1 2 sqrt frac pi alpha frac 2n 1 left 2 alpha right n end aligned nbsp Takozh mozhna vikoristovuvati integruvati chastinami ta znajti vidpovidne rekurentne spivvidnoshennya Polinomi vishih poryadiv Redaguvati Vikoristannya linijnoyi zamini bazisu pokazuye sho integral eksponenti v stepeni yakoyi odnoridnij mnogochlen vid n zminnih mozhe zalezhati tilki vid S L n displaystyle SL n nbsp invariantiv mnogochlena Odnim z takih invariantiv ye diskriminant nuli yakogo poznachayut osoblivosti integrala Prote integral takozh mozhe zalezhati vid drugih invariantiv Eksponentu v stepeni yakoyi drugi parni mnogochleni mozhna chiselno obchisliti za dopomogoyu ryadiv Voni mozhut buti interpritovani yak formalnye obchislennya koli nemaye zbizhnosti Napriklad obchislennya integrala vid eksponenti v stepeni yakoyi polinom chetvertogo poryadku maye viglyad e a x 4 b x 3 c x 2 d x f d x 1 2 e f n m p 0 n p 0 mod 2 b n n c m m d p p G 3 n 2 m p 1 4 a 3 n 2 m p 1 4 displaystyle int infty infty e ax 4 bx 3 cx 2 dx f dx frac 1 2 e f sum begin smallmatrix n m p 0 n p 0 mod 2 end smallmatrix infty frac b n n frac c m m frac d p p frac Gamma left frac 3n 2m p 1 4 right a frac 3n 2m p 1 4 nbsp Vimoga do togo shobn p 0 ta te sho ostacha vid dilennya dorivnyuye 2 mod 2 polyagaye v tomu sho integral vid do 0 daye mnozhnik 1 n p 2 a integral vid 0 do daye mnozhnik 1 2 kozhnogo razu Ci integrali z yavlyayutsya v takih oblastyah yak kvantova teoriya polya Div takozh Redaguvati nbsp Portal Mathematics nbsp Portal Physics Perelik integraliv funkcij Gaussa Zagalni integrali v teoriyi kvantovih poliv Normalnij rozpodil Tablicya integraliv eksponencialnih funkcij Funkciya pomilok Integral BerezinaPrimitki Redaguvati Stahl Saul April 2006 The Evolution of the Normal Distribution MAA org Arhiv originalu za 25 sichnya 2016 Procitovano 25 travnya 2018 Cherry G W 1985 Integration in Finite Terms with Special Functions the Error Function Journal of Symbolic Computation 1 3 283 302 doi 10 1016 S0747 7171 85 80037 7 a b The Probability Integral Arhiv originalu za 10 chervnya 2020 Procitovano 10 chervnya 2020 Reference for Multidimensional Gaussian Integral Stack Exchange 30 bereznya 2012 Otrimano z https uk wikipedia org w index php title Integral Gausa amp oldid 35810044
