Ne plutati z Funkciyeyu Gausa Funkciya pomilok abo Funkciya pomilok Gausa 1 ce neelementarna funkciya sho vikoristovuyetsya v teoriyi jmovirnosti statistici matematichnij fizici i viznachayetsya yakGrafik funkciyi pomilok erf x 2 p 0 x e t 2 d t displaystyle operatorname erf x frac 2 sqrt pi int limits 0 x e t 2 mathrm d t U bagatoh iz cih zastosuvan argumentom funkciyi ye dijsne chislo Yaksho argument funkciyi ye dijsnim to znachennya funkciyi takozh ye dijsnim U statistici dlya nevid yemnih znachen x funkciya pomilok maye take traktuvannya dlya vipadkovoyi velichini Y yaka maye normalnij rozpodil iz matematichnim spodivannyam 0 ta dispersiyeyu 1 2 erf x ce jmovirnist togo sho Y potraplyaye v interval x x Dopovnyuvalna funkciya pomilok sho poznachayetsya erfc x displaystyle operatorname erfc x inodi zastosovuyetsya poznachennya Erf x displaystyle operatorname Erf x viznachayetsya cherez funkciyu pomilok erfc x 1 erf x 2 p x e t 2 d t displaystyle operatorname erfc x 1 operatorname erf x frac 2 sqrt pi int limits x infty e t 2 mathrm d t Uyavna funkciya pomilok sho poznachayetsya w x displaystyle w x takozh viznachayetsya cherez funkciyu pomilok w x e x 2 erfc i x displaystyle w x e x 2 operatorname erfc ix Zmist 1 Nazva 2 Zastosuvannya 3 Vlastivosti 4 Asimptotichnij rozklad 5 Rozvinennya v lancyugovij drib 6 Integral funkciyi pomilok z funkciyeyu rozpodilu Gausa 7 Faktorialnij ryad 8 Sporidneni funkciyi 8 1 Uzagalneni funkciyi pomilok 8 2 Iterovani integrali dodatkovoyi funkciyi pomilok 9 Primitki 10 LiteraturaNazva RedaguvatiNazva funkciya pomilok ta yiyi abreviatura erf zaproponovani Dzhejmsom Glejsherom en v 1871 r cherez yiyi zv yazok z teoriyeyu jmovirnosti i osoblivo teoriyeyu pomilok 2 Dopovnyuvalni funkciyi pomilok takozh obgovoryuvalosya Glejsherom togo zh roku v okremij publikaciyi 3 Dlya zakonu ob yekta pomilok shilnist yakogo maye viglyad f x c p 1 2 e c x 2 displaystyle f x left frac c pi right tfrac 1 2 rm e cx 2 nbsp normalnij rozpodil Glejsher obchislyuvav jmovirnist pomilki sho lezhit mizh p displaystyle p nbsp i q displaystyle q nbsp yak c p 1 2 p q e c x 2 d x 1 2 erf q c erf p c displaystyle left frac c pi right tfrac 1 2 int p q rm e cx 2 rm d x tfrac 1 2 left operatorname erf left q sqrt c right operatorname erf left p sqrt c right right nbsp Zastosuvannya RedaguvatiYaksho rezultati seriyi vimiryuvan opisuyutsya normalnim rozpodilom iz serednokvadratichnim vidhilennyam s displaystyle sigma nbsp ta matematichnim spodivannyam 0 to erf a s 2 displaystyle operatorname erf left frac a sigma sqrt 2 right nbsp ce jmovirnist togo sho pohibka odnogo vimiryuvannya lezhit mizh a ta a pri dodatnomu a Ce korisno napriklad pri viznachenni koeficiyenta bitovih pomilok cifrovoyi sistemi zv yazku Funkciyu pomilok ta dopovnyuvalnu funkciyu pomilok zastosovuyut napriklad u rozv yazkah rivnyannya teploprovidnosti yaksho granichni umovi zadayutsya funkciyeyu Gevisajda Funkciyu pomilok ta yiyi nablizhennya mozhna vikoristovuvati dlya ocinki rezultativ yaki mayut misce z velikoyu jmovirnistyu en abo z nizkoyu jmovirnistyu Nehaj zadana vipadkova velichina X Norm m s displaystyle X sim operatorname Norm mu sigma nbsp i konstanta L lt m displaystyle L lt mu nbsp todi Pr X L 1 2 1 2 erf L m 2 s A exp B L m s 2 displaystyle Pr X leq L frac 1 2 frac 1 2 operatorname erf left frac L mu sqrt 2 sigma right approx A exp left B left frac L mu sigma right 2 right nbsp de A i B deyaki chislovi konstanti Yaksho L dostatno daleka vid matematichnogo spodivannya tobto m L s ln k displaystyle mu L geq sigma sqrt ln k nbsp todi Pr X L A exp B ln k A k B displaystyle Pr X leq L leq A exp B ln k frac A k B nbsp i jmovirnist pryamuye do 0 yaksho k displaystyle k to infty nbsp Vlastivosti RedaguvatiPidintegralni funkciyi v kompleksnij ploshini nbsp Integral exp z 2 displaystyle exp z 2 nbsp nbsp erf z displaystyle operatorname erf z nbsp Funkciya pomilok neparna erf x erf x displaystyle operatorname erf x operatorname erf x nbsp Dlya bud yakogo kompleksnogo x displaystyle x nbsp vikonuyetsyaerf x erf x displaystyle operatorname erf bar x overline operatorname erf x nbsp de riska poznachaye kompleksne spryazhennya chisla x displaystyle x nbsp Pidintegralni funkciyi f exp z 2 displaystyle f exp z 2 nbsp ta f erf z displaystyle f operatorname erf z nbsp zobrazheno v kompleksnij ploshini z na risunkah Riven Im f 0 displaystyle operatorname Im f 0 nbsp pokazano tovstoyu zelenoyu liniyeyu Vid yemni cili znachennya Im f displaystyle operatorname Im f nbsp pokazano tovstimi chervonimi liniyami Dodatni cili znachennya Im f displaystyle operatorname Im f nbsp pokazano tovstimi sinimi liniyami Promizhni rivni Im f c o n s t displaystyle operatorname Im f rm const nbsp pokazano tonkimi zelenimi liniyami Promizhni rivni Re f c o n s t displaystyle operatorname Re f rm const nbsp pokazano tonkimi chervonimi liniyami dlya vid yemnih znachen i tonkimi sinimi liniyami dlya dodatnih znachen Funkciya pomilok ne mozhe buti predstavlena cherez elementarni funkciyi ale rozkladayuchi integrovanij viraz v ryad Tejlora i integruyuchi pochlenno mi mozhemo oderzhati yiyi podannya u viglyadi ryadu erf x 2 p n 0 1 n x 2 n 1 n 2 n 1 2 p x x 3 3 x 5 10 x 7 42 x 9 216 displaystyle operatorname erf x frac 2 sqrt pi sum n 0 infty frac 1 n x 2n 1 n 2n 1 frac 2 sqrt pi left x frac x 3 3 frac x 5 10 frac x 7 42 frac x 9 216 cdots right nbsp Cya rivnist vikonuyetsya i ryad shoditsya yak dlya bud yakogo dijsnogo x displaystyle x nbsp tak i na vsij kompleksnij ploshini Poslidovnist znamennikiv utvoryuye poslidovnist A007680 z Onlajn enciklopediyi poslidovnostej cilih chisel OEIS Dlya iterativnogo obchislennya elementiv ryadu korisno predstaviti jogo v alternativnomu viglyadi erf x 2 p n 0 x i 1 n 2 i 1 x 2 i 2 i 1 2 p n 0 x 2 n 1 i 1 n x 2 i displaystyle operatorname erf x frac 2 sqrt pi sum n 0 infty left x prod i 1 n frac 2i 1 x 2 i 2i 1 right frac 2 sqrt pi sum n 0 infty frac x 2n 1 prod i 1 n frac x 2 i nbsp oskilki 2 i 1 x 2 i 2 i 1 displaystyle frac 2i 1 x 2 i 2i 1 nbsp spivmnozhnik sho peretvoryuye i displaystyle i nbsp j chlen ryadu v i 1 displaystyle i 1 nbsp j vvazhayuchi pershim chlenom x displaystyle x nbsp Uyavna funkciya pomilok maye duzhe shozhij ryad Maklorena a sameerfi z 2 p n 0 z 2 n 1 n 2 n 1 2 p z z 3 3 z 5 10 z 7 42 z 9 216 displaystyle operatorname erfi z frac 2 sqrt pi sum n 0 infty frac z 2n 1 n 2n 1 frac 2 sqrt pi left z frac z 3 3 frac z 5 10 frac z 7 42 frac z 9 216 cdots right nbsp dlya bud yakogo kompleksnogo chisla z Funkciya pomilok na neskinchennosti rivna odinici prote ce spravedlivo tilki pri nablizhenni do neskinchennosti po dijsnij osi oskilki Pri rozglyadi funkciyi pomilok v kompleksnij ploshini tochka z displaystyle z infty nbsp bude dlya neyi istotno osoblivoyu Pohidna funkciyi pomilok vivoditsya bezposeredno z viznachennya funkciyi d d z erf z 2 p e z 2 displaystyle frac rm d rm d z operatorname erf z frac 2 sqrt pi rm e z 2 nbsp Zvidsi pohidna uyavnoyi funkciyi pomilok d d z erfi z 2 p e z 2 displaystyle frac rm d rm d z operatorname erfi z frac 2 sqrt pi rm e z 2 nbsp Pervisnoyu funkciyi pomilok yaku mozhna otrimati za dopomogoyu integruvannya chastinami ye z erf z e z 2 p displaystyle z operatorname erf z frac rm e z 2 sqrt pi nbsp Pervisnoyu uyavnoyi funkciyi pomilok yaku takozh mozhna otrimati integruvannyam chastinami ye z erfi z e z 2 p displaystyle z operatorname erfi z frac rm e z 2 sqrt pi nbsp Pohidni vishih poryadkiv zadayutsya formulami erf k z 2 1 k 1 p H k 1 z e z 2 2 p d k 1 d z k 1 e z 2 k 1 2 displaystyle operatorname erf k z frac 2 1 k 1 sqrt pi H k 1 z rm e z 2 frac 2 sqrt pi frac rm d k 1 rm d z k 1 left rm e z 2 right qquad k 1 2 dots nbsp de H displaystyle mathit H nbsp ce polinomi Ermita 4 Rozklad 5 yakij zbigayetsya shvidshe dlya vsih dijsnih znachen x displaystyle x nbsp nizh ryad Tejlora otrimuyetsya za dopomogoyu teoremi Gansa Genriha Byurmana en 6 erf x 2 p sgn x 1 e x 2 1 1 12 1 e x 2 7 480 1 e x 2 2 5 896 1 e x 2 3 787 276480 1 e x 2 4 2 p sgn x 1 e x 2 p 2 k 1 c k e k x 2 displaystyle begin aligned operatorname erf x amp frac 2 sqrt pi operatorname sgn x sqrt 1 rm e x 2 left 1 frac 1 12 left 1 rm e x 2 right frac 7 480 left 1 rm e x 2 right 2 frac 5 896 left 1 rm e x 2 right 3 frac 787 276480 left 1 rm e x 2 right 4 cdots right 10pt amp frac 2 sqrt pi operatorname sgn x sqrt 1 rm e x 2 left frac sqrt pi 2 sum k 1 infty c k rm e kx 2 right end aligned nbsp sgn displaystyle operatorname sgn nbsp ce signum funkciya Zberigayuchi lishe pershi dva koeficiyenti ta vibirayuchi c 1 31 200 displaystyle c 1 frac 31 200 nbsp ta c 2 341 8000 displaystyle c 2 frac 341 8000 nbsp otrimane nablizhennya pokazuye svoyu najbilshu vidnosnu pohibku pri x 1 379 6 displaystyle x pm 1 3796 nbsp yaka mensha nizh 3 612 7 10 3 displaystyle 3 6127 cdot 10 3 nbsp erf x 2 p sgn x 1 e x 2 p 2 31 200 e x 2 341 8000 e 2 x 2 displaystyle operatorname erf x approx frac 2 sqrt pi operatorname sgn x sqrt 1 rm e x 2 left frac sqrt pi 2 frac 31 200 rm e x 2 frac 341 8000 rm e 2x 2 right nbsp Obernena funkciya pomilok ye ryadomerf 1 x k 0 c k 2 k 1 p 2 x 2 k 1 displaystyle operatorname erf 1 x sum k 0 infty frac c k 2k 1 left frac sqrt pi 2 x right 2k 1 nbsp de c0 1 i c k m 0 k 1 c m c k 1 m m 1 2 m 1 1 1 7 6 127 90 displaystyle c k sum m 0 k 1 frac c m c k 1 m m 1 2m 1 left 1 1 frac 7 6 frac 127 90 ldots right nbsp Tomu ryad mozhna podati v nastupnomu viglyadi pomitimo sho drobi skorocheni erf 1 x 1 2 p x p x 3 12 7 p 2 x 5 480 127 p 3 x 7 40320 4369 p 4 x 9 5806080 34807 p 5 x 11 182476800 displaystyle operatorname erf 1 x frac 1 2 sqrt pi left x frac pi x 3 12 frac 7 pi 2 x 5 480 frac 127 pi 3 x 7 40320 frac 4369 pi 4 x 9 5806080 frac 34807 pi 5 x 11 182476800 dots right nbsp 1 Poslidovnosti chiselnikiv i znamennikiv pislya skorochennya A092676 i A132467 u OEIS poslidovnist chiselnikiv do skorochennya A002067 u OEIS Pri z lt 1 maye misce spivvidnoshennya erf erf 1 z z displaystyle operatorname erf left operatorname erf 1 z right z nbsp Obernena dopovnyuvalna funkciya pomilok viznachayetsya yak erfc 1 1 z erf 1 z displaystyle operatorname erfc 1 1 z operatorname erf 1 z nbsp Dlya dijsnogo x isnuye yedine dijsne chislo erfi 1 x displaystyle operatorname erfi 1 x nbsp sho erfi erfi 1 x x displaystyle operatorname erfi left operatorname erfi 1 x right x nbsp Obernena uyavna funkciya pomilok viznachayetsya yak erfi 1 x displaystyle operatorname erfi 1 x nbsp 7 Dlya bud yakogo dijsnogo x mozhna vikoristovuvati metod Nyutona dlya obchislennya erfi 1 x displaystyle operatorname erfi 1 x nbsp a pri 1 x 1 displaystyle 1 leq x leq 1 nbsp nastupnij ryad Makrolena ye zbizhnim erfi 1 z k 0 1 k c k 2 k 1 p 2 z 2 k 1 displaystyle operatorname erfi 1 z sum k 0 infty frac 1 k c k 2k 1 left frac sqrt pi 2 z right 2k 1 nbsp de koeficiyenti ck viznacheni vishe nbsp Dopovnyuvalna funkciya pomilokAsimptotichnij rozklad RedaguvatiKorisnim asimptotichnim rozkladom dopovnyuvalnoyi funkciyi pomilok a otzhe i funkciyi pomilok dlya velikih dijsnih x displaystyle x nbsp ye erfc x e x 2 x p 1 n 1 1 n 1 3 5 2 n 1 2 x 2 n e x 2 x p n 0 1 n 2 n 1 2 x 2 n displaystyle operatorname erfc x frac rm e x 2 x sqrt pi left 1 sum n 1 infty 1 n frac 1 cdot 3 cdot 5 cdots 2n 1 2x 2 n right frac rm e x 2 x sqrt pi sum n 0 infty 1 n frac 2n 1 2x 2 n nbsp de 2n 1 podvijnij faktorial chisla 2n 1 yakij ye dobutkom usih neparnih chisel do 2n 1 vklyuchno Cej ryad rozbigayetsya dlya bud yakogo skinchennogo x displaystyle x nbsp i jogo zmist yak asimptotichnogo rozkladu polyagaye v tomu sho dlya bud yakogo N N displaystyle N in mathbb N nbsp erfc x e x 2 x p n 0 N 1 1 n 2 n 1 2 x 2 n R N x displaystyle operatorname erfc x frac rm e x 2 x sqrt pi sum n 0 N 1 1 n frac 2n 1 2x 2 n R N x nbsp de zalishok v poznachennyah O velikogo R N x O x 1 2 N e x 2 displaystyle R N x O left x 1 2N rm e x 2 right nbsp pri x displaystyle x to infty nbsp Dijsno tochne znachennya zalishku stanovit R N x 1 N p 2 1 2 N 2 N N x t 2 N e t 2 d t displaystyle R N x frac 1 N sqrt pi 2 1 2N frac 2N N int x infty t 2N rm e t 2 rm d t nbsp yake legko otrimuyetsya za dopomogoyu indukciyi z vikoristannyam formulie t 2 2 t 1 e t 2 displaystyle rm e t 2 2t 1 left rm e t 2 right nbsp ta integruvannya chastinami Dlya dosit velikih znachen x displaystyle x nbsp potribni lishe pershi kilka chleniv cogo asimptotichnogo rozkladu shob otrimati garne nablizhennya dlya funkciyi erfc x displaystyle operatorname erfc x nbsp todi yak dlya ne nadto velikih znachen x displaystyle x nbsp vishenavedenij ryad Tejlora u tochci 0 displaystyle 0 nbsp zabezpechuye bilsh shvidku zbizhnist Rozvinennya v lancyugovij drib RedaguvatiPredstavlennya dopovnyuvalnoyi funkciyi pomilok cherez lancyugovij drib maye viglyad 8 erfc z z p e z 2 1 z 2 a 1 1 a 2 z 2 a 3 1 a m m 2 displaystyle operatorname erfc z frac z sqrt pi rm e z 2 cfrac 1 z 2 cfrac a 1 1 cfrac a 2 z 2 cfrac a 3 1 dotsb qquad a m frac m 2 nbsp Integral funkciyi pomilok z funkciyeyu rozpodilu Gausa Redaguvati erf a x b 1 2 p s 2 e x m 2 2 s 2 d x erf a m b 1 2 a 2 s 2 a b m s R displaystyle int infty infty operatorname erf left ax b right frac 1 sqrt 2 pi sigma 2 rm e frac x mu 2 2 sigma 2 rm d x operatorname erf left frac a mu b sqrt 1 2a 2 sigma 2 right qquad a b mu sigma in mathbb R nbsp yakij otrimanij Ng ta Gellerom za dopomogoyu zmini zminnih formula 13 u paragrafi 4 3 9 Faktorialnij ryad RedaguvatiObernenij faktorialna ryaderfc z e z 2 p z n 0 1 n Q n z 2 1 n e z 2 p z 1 1 2 1 z 2 1 1 4 1 z 2 1 z 2 2 displaystyle begin aligned operatorname erfc z amp frac rm e z 2 sqrt pi z sum n 0 infty frac 1 n Q n z 2 1 bar n amp frac rm e z 2 sqrt pi z left 1 frac 1 2 frac 1 z 2 1 frac 1 4 frac 1 z 2 1 z 2 2 cdots right end aligned nbsp dd ye zbizhnim pri Re z 2 gt 0 displaystyle operatorname Re z 2 gt 0 nbsp Tut Q n def 1 G 1 2 0 t t 1 t n 1 t 1 2 e t d t k 0 n 1 2 k s n k displaystyle Q n stackrel text def frac 1 Gamma 1 2 int 0 infty tau tau 1 cdots tau n 1 tau 1 2 rm e tau rm d tau sum k 0 n left frac 1 2 right bar k s n k nbsp cherez z n displaystyle z bar n nbsp poznacheno zrostayuchij faktorial a s n k displaystyle s n k nbsp chislo Stirlinga pershogo rodu 10 11 Predstavlennya neskinchennoyu sumoyu sho mistit podvijnij faktorial erf z 2 p n 0 2 n 2 n 1 2 n 1 z 2 n 1 displaystyle operatorname erf z frac 2 sqrt pi sum n 0 infty frac 2 n 2n 1 2n 1 z 2n 1 nbsp Sporidneni funkciyi RedaguvatiZ tochnistyu do masshtabu i zsuvu funkciya pomilok zbigayetsya z funkciyeyu rozpodilu jmovirnostej normalnogo rozpodilu sho poznachayetsya F x displaystyle Phi x nbsp F x 1 2 1 erf x 2 displaystyle Phi x frac 1 2 left 1 operatorname erf frac x sqrt 2 right nbsp Zvorotna funkciya do F displaystyle Phi nbsp vidoma yak normalna kvantilna funkciya inodi poznachayetsya probit displaystyle operatorname probit nbsp i virazhayetsya cherez normalnu funkciyu pomilok yak probit p F 1 p 2 erf 1 2 p 1 displaystyle operatorname probit p Phi 1 p sqrt 2 operatorname erf 1 2p 1 nbsp Normalnij integralnij rozpodil chastishe zastosovuyetsya v teoriyi jmovirnosti i matematichnij statistici todi yak funkciya pomilok chastishe zastosovuyetsya v inshih rozdilah matematiki Funkciya pomilok ye okremim vipadkom funkciyi Mittag Leflera a takozh mozhe buti predstavlena yak virodzhena gipergeometrichna funkciya funkciya Kummera erf x 2 x p 1 F 1 1 2 3 2 x 2 displaystyle operatorname erf x frac 2x sqrt pi 1 F 1 left frac 1 2 frac 3 2 x 2 right nbsp Funkciya pomilok virazhayetsya takozh cherez integral Frenelya U terminah regulyarizovanoyi nepovnoyi gamma funkciyi P i nepovnoyi gamma funkciyi erf x sign x P 1 2 x 2 sign x p g 1 2 x 2 displaystyle operatorname erf x operatorname sign x P left frac 1 2 x 2 right operatorname sign x over sqrt pi gamma left frac 1 2 x 2 right nbsp Uzagalneni funkciyi pomilok Redaguvati nbsp Grafik uzagalnenih funkcij pomilok E n x displaystyle E n x nbsp sira liniya E 1 x 1 e x p displaystyle E 1 x 1 e x sqrt pi nbsp chervona liniya E 2 x erf x displaystyle E 2 x operatorname erf x nbsp zelena liniya E 3 x displaystyle E 3 x nbsp sinya liniya E 4 x displaystyle E 4 x nbsp zhovta liniya E 5 x displaystyle E 5 x nbsp Takozh mozhna rozglyanuti zagalnishi funkciyi E n x n p 0 x e t n d t n p p 0 1 p x n p 1 n p 1 p displaystyle E n x frac n sqrt pi int limits 0 x e t n mathrm d t frac n sqrt pi sum p 0 infty 1 p frac x np 1 np 1 p nbsp Okremimi vartimi uvagi vipadkami ye E 0 x displaystyle E 0 x nbsp pryama liniya sho prohodit cherez pochatok koordinat E 0 x x e p displaystyle E 0 x frac x e sqrt pi nbsp E 2 x displaystyle E 2 x nbsp funkciya pomilok erf x displaystyle operatorname erf x nbsp Pislya dilennya na n displaystyle n nbsp vsi E n displaystyle E n nbsp z neparnimi n displaystyle n nbsp viglyadayut shozhe ale ne identichno Vsi E n displaystyle E n nbsp z parnimi n displaystyle n nbsp tezh viglyadayut shozhe ale ne identichno pislya dilennya na n displaystyle n nbsp Vsi uzagalneni funkciyi pomilok z n gt 0 displaystyle n gt 0 nbsp viglyadayut shozhe na napivosi x gt 0 displaystyle x gt 0 nbsp Na napivosi x gt 0 displaystyle x gt 0 nbsp vsi uzagalneni funkciyi mozhut buti virazheni cherez gamma funkciyu E n x x x n 1 n G n G 1 n G 1 n x n p x gt 0 displaystyle E n x frac x left x n right 1 n Gamma n left Gamma left frac 1 n right Gamma left frac 1 n x n right right sqrt pi qquad x gt 0 nbsp Otzhe mi mozhemo viraziti funkciyu pomilok cherez gamma funkciyu erf x 1 G 1 2 x 2 p displaystyle operatorname erf x 1 frac Gamma left frac 1 2 x 2 right sqrt pi nbsp Iterovani integrali dodatkovoyi funkciyi pomilok Redaguvati Iterovani integrali dodatkovoyi funkciyi pomilok viznachayutsya yak i n erfc z z i n 1 erfc z d z displaystyle i n operatorname erfc z int limits z infty i n 1 operatorname erfc zeta mathrm d zeta nbsp Yih mozhna rozklasti v ryad i n erfc z j 0 z j 2 n j j G 1 n j 2 displaystyle i n operatorname erfc z sum j 0 infty frac z j 2 n j j Gamma left 1 frac n j 2 right nbsp zvidki viplivayut vlastivosti simetriyi i 2 m erfc z i 2 m erfc z q 0 m z 2 q 2 2 m q 1 2 q m q displaystyle i 2m operatorname erfc z i 2m operatorname erfc z sum q 0 m frac z 2q 2 2 m q 1 2q m q nbsp i i 2 m 1 erfc z i 2 m 1 erfc z q 0 m z 2 q 1 2 2 m q 1 2 q 1 m q displaystyle i 2m 1 operatorname erfc z i 2m 1 operatorname erfc z sum q 0 m frac z 2q 1 2 2 m q 1 2q 1 m q nbsp Primitki Redaguvati Modul math Specialni funkciyi ta konstanti bestprog net uk ukr 1 listopada 2019 Procitovano 7 zhovtnya 2023 Glaisher James Whitbread Lee July 1871 On a class of definite integrals London Edinburgh and Dublin Philosophical Magazine and Journal of Science 4 42 277 294 302 doi 10 1080 14786447108640568 Procitovano 6 grudnya 2017 Glaisher James Whitbread Lee September 1871 On a class of definite integrals Part II London Edinburgh and Dublin Philosophical Magazine and Journal of Science 4 42 279 421 436 doi 10 1080 14786447108640600 Procitovano 6 grudnya 2017 Weisstein Eric W Erf MathWorld Wolfram H M Schopf and P H Supancic On Burmann s Theorem and Its Application to Problems of Linear and Nonlinear Heat Transfer and Diffusion The Mathematica Journal 2014 doi 10 3888 tmj 16 11 Schopf Supancic Weisstein E W Burmann s Theorem Wolfram MathWorld A Wolfram Web Resource Bergsma Wicher 2006 On a new correlation coefficient its orthogonal decomposition and associated tests of independence arXiv math 0604627 Cuyt Annie A M Petersen Vigdis B Verdonk Brigitte Waadeland Haakon Jones William B 2008 Handbook of Continued Fractions for Special Functions Springer Verlag ISBN 978 1 4020 6948 2 Ng Edward W Geller Murray January 1969 A table of integrals of the Error functions Journal of Research of the National Bureau of Standards Section B 73B 1 1 doi 10 6028 jres 073B 001 Schlomilch Oskar Xavier 1859 Ueber facultatenreihen Zeitschrift fur Mathematik und Physik de nim 4 390 415 Procitovano 4 grudnya 2017 Eq 3 on page 283 of Nielson Niels 1906 Handbuch der Theorie der Gammafunktion nim Leipzig B G Teubner Procitovano 4 grudnya 2017 Literatura RedaguvatiMilton Abramowitz and Irene A Stegun eds Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables New York Dover 1972 Otrimano z https uk wikipedia org w index php title Funkciya pomilok amp oldid 40582207
