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+ dn λ−n + Rn (λ), Rn (λ) = o(λ−n) , λ −→ +∞,g(λ)èëè,÷òî ðàâíñèëüíî,a0 + a1 λ−1 + ... + an λ−n + o(λ−n ) = (d0 + d1 λ−1 + ... + dn λ−n + Rn (λ))(b0 + b1 λ−1 + ... + bn λ−n + o(λ−n )).Ðàñêðûâàÿ ñêîáêè èó÷èòûâàÿ ñîîòíîøåíèÿ,îïðåäåëÿþùèådn ,ïîëó÷èìo(λ−n) = Rn (λ)(b0 + b1 λ−1 + ... + bn λ−n) + o(λ−n ) + o(λ−n )(d0 + d1 λ−1 + ... + dn λ−n ) =b0 Rn (λ) + o(1)Rn (λ) + o(λ−n ).Ò.ê.b06= 0,òîïîñëåäíåå ðàâåíñòâî ìîæåò âûïîëíÿòüñÿ ëèøü â ñëó÷àåÇàìå÷àíèå.ÝòîRn (λ) = o(λ−n ), ÷òî è òðåáîâàëîñü.λ −→ +∞.
Ðàññìîòðèìóòâåðæäåíèå ñïðàâåäëâî è äëÿ ñòåïåííûõ ðàçëîæåíèé ïðèàíàëèòè÷åñêèå ñâîéñòâà ñòåïåííûõ ðàçëîæåðèé.Òåîðåìà (îá èíòåãðèðîâàíèè ñòåïåííîãî ðàçëîæíèÿ) Åñëè ôóíêöèÿf (λ)îïðåäåëåíà è íåïðåðûâíà ïðèλ ≥ λ0 ≥ 0∞Pf (λ) 'èan λ−n , λ −→ +∞,n=0òîF (λ) =∞R∞P(f (t) − a0 − a1 t−1 )dt 'bn λn , λ −→ +∞,k=1λbn =Èíûìè ñëîâàìè,ðàçëîæåíèå ôóíêöèèan+1n , n = 1, 2, ....F (λ) ïîëó÷àåòñÿ ôîðìàëüíûì èíòåãðèðîâàíèåì ðàçëîæåíèÿ ôóíêöèèf (λ) − a0 − a1 λ−1 .Äîêàçàòåëüñòâî. Èíòåãðàë, îïðåäåëÿþùèé ôóíêöèþ F (λ),ñõîäèòñÿ ïðè âñåõ λ ≥ λ0 ,ïîñêîëüêó f (λ)−a0 − a1 λ−1 = O(λ−2 ), λ −→ +∞.Äàëåå,ïðè n=2,3,... èìååìnn−1R∞R∞ PR∞P am+1 −mak t−k dt + Rn (t)dt =+ ρn−1 (λ),(f (t) − a0 − a1 t−1 )dt =m λm=1λλ k=2λR∞ρn−1 (λ) = R(t)dt.λÒ.ê.ρn−1 (λ)ñòðåìèòñÿ ê 0 ïðèλ −→ +∞,òî ìîæíî âîñïîëüçîâàòüñÿ ïðàâèëîì Ëîïèòàëÿ è ïîëó÷èòü,÷òîρn−1 (λ) = o(λ−(n−1) )78,÷òî è äîêàçûâàåò óòâåðæäåíèå.Äèôôåðåíöèðîâàòü àñèìïòîòè÷åñêèå ðàçëîæåíèÿ, âîîáùå ãîâîðÿ, íåëüçÿ.Òàê, ôóíêöèÿf (λ) = e−λ sin(eλ ) = O(λ−∞ )λ −→ +∞,âñåò.å.èìååò ñòåïåííîå ðàçëîæåíèå ïðè,êîýôôèöèåíòû êîòîðîãî ðàâíû 0, íî ó åå ïðîèçâîäíîéòàêîãî ðàçëîæåíèÿ íåò.
Îäíàêî ñïðàâåäëèâî ñëåäóþùåå óòâåðæäåíèå.Òåîðåìà (î äèôôåðåíöèðîâàíèè ñòåïåííîãî ðàçëîæíèÿ). Åñëè ôóíêöèÿ f (λ) íåïðåðûâíî äèôôåðåíöèðóåìàïðèλ ≥ λ0 ≥ 0è∞Pf (λ) 'an λ−n ,n=0∞Pdf (λ)'cn λ−n , λ −→ +∞,dλn=0òîc0 = c1 = 0, cn = −(n − 1)an−1 , n = 2, 3, ...,ò.å.àñèìïòîòè÷åñêèé ðÿä ïðîèçâîäíîé ïîëó÷àåòñÿ ôîðìàëüíûì äèôôåðåíöèðîâàíèåì àñèìïòîòè÷åñêîãîðÿäà ôóíêöèè.Äîêàçàòåëüñòâî.Äîêàæåì,÷òî c0 = c1 = 0.
Ïóñòü λ1 > λ0 . Èìååìf (λ) = f (λ1 ) +RλRλ df (t)dt = f (λ1 ) + (c0 + c1 t−1 + O(t−2 )dt =dtλλ11Rλc0 λ + c1 ln λ +O(t−2)dt + f (λ1 ) − c0 λ1 − c1 ln λ1 = a0 + a1 λ−1 + O(λ−2 ).λ1λ ≥ λ1 . Ðàçäåëèâ îáå ÷àñòè ýòîãî ðàâåíñòâàc0 = 0. Àíàëîãè÷íî, äåëÿ íà ln λ, ïîëó÷èì c1 = 0.Èíòåãðàë â ïîñëåäíåì ðàâåíñòâå îãðàíè÷åí ïðèïåðåéäÿ ê ïðåäåëó ïðèλ −→ +∞,ïîëó÷èìíàλèÒàêèìîáðàçîì,df (λ)= O(λ−2 ).dλf (λ) −→ a0 ïðè λ −→ +∞, è ïðèìåíÿÿ ïðåäûäóùþþ òåîðåìó,ïîëó÷àåì òðåáóåìîå óòâåðæäåíèå.λ −→ 0ñïðàâåäëèâû ñëåäóþùèå àíàëîãè äîêàçàííûõ òåîðåì.ôóíêöèÿ f (λ) íåïðåðûâíà íà èíòåðâàëå (0, a) èÓ÷èòûâàÿ,÷òîÄëÿ ñòåïåííûõ ðàçëîæåíé ïðè1.Åñëè∞Pf (λ) 'an λn , λ −→ 0,n=0òîRλf (t)dt 'n=102.Åñëè ôóíêöèÿf (λ)∞Pan−1 nn λ , λ −→ 0.íåïðåûâíî äèôôåðåíöèðóåìà íà èíòåðâàëåf (λ) '∞Pan λn ,n=0òîcn = (n + 1)an+1 , n = 0, 1, ...(0, a)è∞Pdf (λ)'cn λn , λ −→ 0,dλn=0.3 Àñèìïòîòèêà èíòåãðàëîâ Ëàïëàñà 1.Èíòåãðàëàìè Ëàïëàñà íàçûâàþòñÿ èíòåãðàëû âèäàF (λ) =Rωf (x)eλS(x) dx,aãäåλ−âåùåñòâåííûé ïàðàìåòð,S(x)−âåùåñòâåííîçíà÷íàÿ ôóíêöèÿ.
Íàñ áóäåò èíòåðåñîâàòü ãëàâíûé÷ëåí àèìïòîòèêè òàêèõ èíòåãðàëîâ ïðèλ −→ +∞.Èñïîëüçóåìûé ìåòîä îñíîâûâàåòñÿ íà ñëåäóþùèõx0 , òî ôóíêöèÿ eλS(x) ) òàêæåèìååò â ýòîé òî÷êå ìàêñèìóì,òåì áîëåå âûðàæåííûé,÷åì áîëüøå çíà÷åíèå λ. Ïîýòîìó ìîæíî îæèäàòü,÷òîïîâåäåíèå F (λ) îïðåäåëÿåòñÿ â îñíîâíîì èíòåãðàëîì ïî ìàëîé îêðåñòíîñòè òî÷êè x0 .
Åñëè x(0 ) âíóòðèñîîáðàæåíèÿõ.Åñëè ôóíêöèÿS(x)äîñòèãàåò ñâîåãî ìàêñèìóìà â òî÷êåïðîìåæóòêà èíòåãðèðîâàíèÿ,òî,êàê ìû â äàëüíåéøåì óâèäèì,èíòåãðàë ïî îêðåñòíîñòè ìîæíî çàìåíòüèíòåãðàëîì ÷àñòíîãî âèäà,äëÿ êîòîðîãî ãëàâíûé ÷ëåí àñèìïòîòèêè íàõîäèòñÿ ñðàâíèòåëüíî ëåãêî. Ïðåæäå÷åì êîíêðåòèçèðîâàòü ýòè ñîîáðàæåíèÿ,ðàññìîòðèì äâà ïðèìåðà:79F1 (λ) =R∞e−λx dt, F2 (x) =0R∞2e−λx , λ > 0.0Ëåãêî âèäåòü,÷òîF1 (λ) = λ−1 , F2 (λ) = O( √1 ), λ −→ +∞λ.Ðàçëè÷èå â ïîâåäåíèè èíòåãðàëîâ, îêàçûâàåòñÿ,îáúÿñíÿåòñÿ òåì, ÷òî ïåðâàÿ ïðèçâîäíàÿ ôóíêöèèS(x)âòî÷êå ìàêñèìóìà â îäíîì ñëó÷àå îòëè÷íà îò íóëÿ, à â äðóãîì ðàâíà íóëþ.  ýòîì ïàðàãðàôå ðàññìàòðèâàåòñÿñëó÷àé îòëè÷íîé îò íóëÿ ïðîèçâîäíîé.Ñíà÷àëà äîêàæåì ïðîñòîå,íî âàæíîå äëÿ äàëüíåéøåãî óòâåðæäåíèå.Ëåììà îá ýêñïîíåíöèàëüíîé îöåíêå.
Ïðåäïîëîæèì,÷òî ïðè íåêîòîðîì ïîëîæèòåëüíîì λ0 èíòåãðëF (λ)ñõîäèòñÿ àáñîëþòíî èS(x) ≤ c, c = constíà ïðîìåæóòêå(a, ω).Òîãäàñóùåñòâóåò êîíñòàíòàM,òàêàÿ,÷òî|F (λ)| ≤ M ecλ , λ > λ0 .Äîêàçàòåëüñòâî. Ïóñòü λ > λ0 , òîãäà|F (λ)| ≤Rω|f (x)|eλS(x) dx =aRω|f (x)|eλ0 S(x) e(λ−λ0 )S(x) dx ≤ e(λ−λ0 )caM = e−λ0 cRωRω|f (x)|eλ0 S(x) dx = M eλc ,a|f (x)|,a÷òî è òðåáîâàëîñü.Íàïîìíèì,÷òî òî÷êà îáëàñòè îïðåäåëåíèÿ äèôôåðåíöèðóåìîé ôóíêöèè íçûâàåòñÿâýòîé òî÷êå ïåðâàÿ ïðîèçâîäíàÿ ôóíêöèè ðàâíâ íóëþ.Ñòàöèîíàðíàÿ òî÷êà íàçûâàåòñÿñòàöèîíàðíîé,åñëèíåâûðîæäåííîé,åñëèâ ýòîé òî÷êå âòîðàÿ ïðîèçâîäíàÿ îòëè÷íà îò íóëÿ.Òåïåðü ìû ïîëó÷èì ãëàâíûé ÷ëåí àñèïòîòèêè èíòåãðàëà Ëàïëàñà â ïðåäïîëîæåíèè,÷òî îáëàñòüþèíòåãðèðîâàíèÿ ÿâëÿåòñÿ êîíå÷íûé îòðåçîêìàêñèìóì äîñòèãàåòñÿ âa[a, b]è òî÷êà ìàêñèìóìàS(x)íå ÿâëÿåòñÿ ñòàöèîêàðíîé,ò.å.b.èëè âÒåîðåìà (òî÷êà ìàêñèìóìà íå ÿâëÿåòñÿ ñòàöèîíàðíîé). Ïðåäïîëîæèì, ÷òî ôóíêöèè f (x), S(x)áåñêîíå÷íî äèôôåðåíöèðóåìû íà îòðåçêå[a, b],ôóíêöèÿ S(x) äîñòèãàåò ìàêñèìóìà â åäèíñòâåííîé òî÷êå,êîòîðàÿíå ÿâëÿåòñÿ ñòàöèîíàðíîé.
Òîãäà äëÿ èíòåãðàëà ËàïëàñàF (λ) =Rbf (x)eλS(x) dxañïðàâåäëèâî ñëåäóþùåå.1. Åñëè ìàêñèìóìS(x)a,òîäîñòèãàåòñÿ âλS(a)F (λ) = − e 0 (f (a) + O(λ−1 )), λ −→ +∞;λS (a)b,òî2. Åñëè ìàêñèìóì äîñòèãàåòñÿ âλS(b)F (λ) = e 0 (f (b) + O(λ−1 )), λ −→ +∞.λS (b)Äîêàçàòåëüñòâî.Îãðàíè÷èìñÿ ïåðâûì ñëó÷àåì, âòîðîé ðàññìàòðèâàåòñÿ àíàëîãè÷íî.Èç óñëîâèéòåîðåìû ñëåäóåò, ÷òî íàéäóòñÿ ïîëîæèòåëüíûåx ∈ [a, a + ε].ÏðåäñòàâèìF (λ)α, ε, δ ,òàêèå,÷òî S 0 (x) ≤ −αèS(x) − S(a) ≤ −δïðèâ âèäåF (λ) = eλS(a)Rbf (x)eλ(S(x)−S(a)) dx = eλS(a) J.aÈíòåãðàëJïðåäñòàâèì êàê ñóììóJ1 + J2èíòåãðàëîâ ïî îòðåçêàì[a, a + ε], [a + ε, b].Ïîëåììå îáýêñïîíåíöèàëüíîé îöåíêå|J2 | ≤ M e−λδ ,ò.å.J2 = O(λ−∞ ).ÂJ1 =Îöåíèìa+εRèíòåãðàëåJ1ïðîèçâåäåì èíòåãðèðîâàíèå ïî ÷àñòÿì:aa+εR f (x) 0 λ(S(x)−S(a))f (x)f (a + ε) λ(S(a+ε)−S(a))f (a)d(eλ(S(x)−S(a)) ) =e−−1edx =000λS (x)λS (a + ε)λS (a) λ a S 0 (x)f (a)− 0+ A1 + A2 .λS (a)A1èA2 .ÏîñêîëüêóS(a + ε) − S(a) < 0,òîA1 = O(λ−∞ ).80Äàëåå, ïî ôîðìóëå ËàãðàíæàS(x) − S(a) = S 0 (ξ)(x − a) ≤ −α(x − a), x ∈ [a, a + ε], ξ ∈ (a, b),è ò.ê.
ïîäèíòåãðàëüíàÿ ôóíêöèÿ îãðàíè÷åíà, òîa+εR −αλ(x−a)edx = C12 (1 − e−αλε ), C1 = const > 0,|A2 | ≤ C1λ aαλïîýòîìóA2 = O(λ−2 ).Òàêèì îáðàçîì,λS(a)λS(a)F (λ) = − e 0 f (a) + eλS(a) (O(λ−2 ) + O(λ−∞ )) = − e 0 (f (a) + O(λ−∞ )),λS (a)λS (a)λ −→ +∞÷òî è òðåáîâàëîñü.Çàìå÷àíèå. Òåîðìà ñïðàâåäëèâà è âñëó÷àå íåîãðàíè÷åííîãî ïðîìåæóòêà, åñëè ê íåìó ìîæíî ïðèìåíòüëåììó îá ýêñïîíåíöèàëüíîé îöåíêå.Ïðèìåð.(Èíòåãðàë âåðîÿòíîñòåé îøèáîê). Òàê íàçûâàþò èíòåãðàë2err(x) = √πZx2e−t dt0,Èçó÷èì åãî ïîâåäåíèå ïðèx −→ +∞.Èìååì:2 err(x) = √πZ∞e−t2Z∞dt −x01 − √2 Err(x)πÏðåîáðàçóåìErr(x),ïîëàãàÿ2e−t dt =.t = xu :Z∞Err(x) = x22e−xu2du = xe−x(1 + O(x−2 )) =2x212e−x (1 + O(x−2 )).2xÒàêèì îáðàçîì2e−xerr(x) = 1 − √ (1 + O(x−2 )), x −→ +∞πx.3 Àñèìïòîòèêà èíòåãðàëîâ Ëàïëàñà 2.Ïðîäîëæèèì èçó÷åíèå èíòåãðàëîâ Ëàïëàñà, ïðåäïîëàãàÿ,÷òîS(x) äîñòèãàåò ìàêñèìóìà â íåâûðîæäåííîéñòàöèîíàðíîé òî÷êå. Ðàññìîòðèì ñíà÷àëà èíòåãðàëû ñïåöèàëüíîãî âèäà.Ëåììà Âàòñîíà. ÏóñòüZdW (λ) =αf (x)xβ e−λx dx, λ > 0.0α > 0, β > −1 è ôóíêöèÿ f (x) áåñêîíå÷íî äèôôåðåíöèðóåìà íà íåêîòîðîì îòðåçêå 0, δ],òî ïðèn = 0, 1, ...Òîãäà, åñëèâñåõ81nW (λ) = k + β + 21 X k + β + 1 f (k) (0) − k + β + 1ααΓλ+ O λ−, λ −→ +∞.ααk!k=0Äîêàçàòåëüñòâî.îòðåçêó[0, δ].Ëåììà îá ýêñïîíåíöèàëüíîé îöåêå ïîçâîëÿåò îãðàíè÷èòüñÿ èíòåãðàëîìWδ (λ)ïîÈñïîëüçóÿ ôîðìóëó Òåéëîðà,ïîëó÷èìδWδ (λ) =ZnXf (k) (0)k=0k!xk+β −λxαeZδdx +0αrn (x)e−λx dx, rn (x) = O(xn+1 ), x −→ 0.0Äàëåå,Zδxn+β −λxαeZ∞dx =x0u = λxeZ∞dx −0Ïî ëåììå îá ýêñïîíåíöèàëüíîé îöåíêåαn+β −λxααxn+β e−λx dx = In1 − In2 .δIn2 = O(λ−∞ ),ïîñêîëüêó−xα ≤ −δ α < 0.ÈíòåãðàëIn,1çàìåíîéñâîäèòñÿ ê ãàììà -ôóíêöèè,÷òî äàåò1Γ k +β +1 .In1 = ααÊðîìå òîãî, δZZ∞ k+β+2α−λx rn (x)e ≤ C xn+β+1 e−λxα dx = C Γ k + β + 2 λ−αdx.αα00Ñîáèðàÿ âñå ïîëó÷åííûå îöåíêè, ïðèõîäèì ê óòâåðæäåíèþ ëåììû.Ñëåäóþùàÿ ëåììà, êîòîðàÿ ïðèâîäèòñÿ áåç äîêàçàòåëüñòâà, ïîçâîëÿåò íàéòè çàìåíó ïåðåìåííîé, ñïîìîùüþ êîòîðîé ïðîèçâîëüíûé èíòåãðàë Ëàïëàñà ïî ñóùåñòâó ñâîäèòñÿ ê èíòåãðàëó ðàññìîòðåííîãî âëåììå Âàòñîíà âèäà.Ëåììà Ìîðñà .
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