Диссертация (1136178), страница 18
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, n − |m| − 1,√a−11x1 ∈ [−+,22|m|√òîa−11−].22|m|Èç ôîðìóë (1.321) (1.324) âûòåêàåòËåììà 1.42.Ïðè |m| → ∞ ñïðàâåäëèâû ðàâåíñòâàsA(x1 ) =×√√(x1 − ( a + 1)/2)(x1 − ( a − 1)/2)1√√{1 −×4|m|(x1 + ( a − 1)/2)(x1 + ( a + 1)/2)111√√√+++x1 − ( a + 1)/2 x1 − ( a − 1)/2 x1 + ( a − 1)/211√+O},+|m|2x1 + ( a + 1)/21461B(x1 ) = p 2{−6x21 + a − 1−√√24 (x1 − ( a − 1)2 /4)(x1 − ( a + 1)2 /4)√11 √−[2 5 − a (k + ) + 3 a+|m|2√√4 a(a + 11)x21 − a(a − 1)21√√+]+O}.|m|216(x21 − ( a − 1)2 /4)(x21 − ( a + 1)2 /4)Âû÷èñëèì âõîäÿùèå â (1.325) ôóíêöèè. Èìååì:√√1(x1 − ( a + 1)/2)(x1 − ( a − 1)/2)√√ln A(x1 ) = ln−2(x1 + ( a − 1)/2)(x1 + ( a + 1)/2)x1 (4x21 − a − 1)1√√+O−,|m|24|m|(x21 − ( a − 1)2 /4)(x21 − ( a + 1)2 /4)pln(B(x1 ) + B 2 (x1 ) − 1) =!p22a − 1 − 6x1 + 2x1 5x1 + 5 − ap−= ln√√4 (x21 − ( a − 1)2 /4)(x21 − ( a + 1)2 /4)(1.326)√√11p5−a(k+a+{2)+322|m|x1 5x21 + 5 − a√a(4(a + 11)x21 − (a − 1)2 )1√√+, (1.327)}+O|m|216(x21 − ( a − 1)2 /4)(x21 − ( a + 1)2 /4)√ p√√12 4 (x21 − ( a − 1)2 /4)(x21 − ( a + 1)2 /4)pp=+44B 2 (x1 ) − 1x21 (5x21 + 5 − a)1+O,(1.328)|m|−1 dA(x1 )B(x1 )p=2A(x1 ) dx1B 2 (x1 ) − 1√a(4x21 − a + 1)(−6x21 + a − 1)p=−+√√16x1 5x21 + 5 − a (x21 − ( a − 1)2 /4)(x21 − ( a + 1)2 /4)1+O.(1.329)|m|−147Îòìåòèì, ÷òî ôîðìóëû (1.325) äèñêðåòíîãî ìåòîäà ÂÊÁ íå ïðèìåíèìû âáëèçè òî÷êè ïîâîðîòà, ãäå ÷èñëîj1 = j − (n − |m| − 1)/2(1.330)ðàâíî íóëþ, à òàêæå âáëèçè ãðàíè÷íûõ çíà÷åíèéj =0èj = n−|m|−1.
Äëÿ òàêèõ j òðåáóåòñÿ ñòðîèòü äîïîëíèòåëüíûå ðàçëîæåíèÿ.1/3Ïîñòðîèì àñèìïòîòèêó ζj ïðè 0 ≤ j . |m|. Èç ôîðìóëû(1.305) íàõîäèìkζ0 =ζ1 =c− (−1)√(2 + 5 − a)k+1/2√ !|m|√( a + 1) a,2(1.331)√√|m|( a − 1)(5 − a) a − 11 √+− (k + ) 5 − a ζ0 .442Ïðîèçâåäåì çàìåíó|m|j$j .ζj =j!Òîãäà êîýôôèöèåíòû$jïðè1 ≤ j . |m|1/3áóäóò óäîâëåòâîðÿòüâûòåêàþùåìó èç (1.320) ðåêóððåíòíîìó ñîîòíîøåíèþ 2 √ √4 a( a − 1)(j − 1)j(−+O)$j−1 +|m||m|2√√+(( a − 1)(5 − a) + Ojj)$j + (−4 + O)$j+1 = 0.|m||m|Ïîñêîëüêó$j+1 √√( a − 1)(5 − a)j=+O$j ,4|m|òî ñïðàâåäëèâàËåììà 1.43.Ïðè 1 ≤ j . |m|1/3 èìååò ìåñòî àñèìïòîòèêà|m|jζj =j! √ 2 √ j( a − 1)(5 − a)jζ0 (1 + O).4|m|(1.332)148 ñèëó ôîðìóëû Ñòèðëèíãà ïðè1ζj = √2πjjïîðÿäêà|m|1/3√√ j|m|e( a − 1)(5 − a)1).ζ0 (1 + O4j|m|1/3Ïîäñòàâèì â (1.325) ðàçëîæåíèÿ (1.326) (1.329).
Òîãäà ïðè(0)1/3ðÿäêà |m|ïðàâàÿ ÷àñòü (1.332) ñîãëàñóåòñÿ ñ c1 ζj,− . Çäåñüj ïîc1 êîíñòàíòà.  ðåçóëüòàòå ïîëó÷àåìÏðè |m|1/3 . j , j1 . −|m|3/5 ñïðàâåäëèâà àñèìïòî-Ëåììà 1.44.(0)òèêà ζj = c1 ζj,− , ãäå(0)ζj,−√ p√√2 4 (x21 − ( a − 1)2 /4)(x21 − ( a + 1)2 /4)p×=4x21 (5x21 + 5 − a)Zj1 /|m|× exp{|m|Zj1 /|m|f0 (x1 )dx1 +x01f1 (x1 )dx1 + Ox011},|m|1/5(1.333)ôóíêöèèf0 (x1 ) = − ln!pa − 1 − 6x21 + 2x1 5x21 + 5 − a√√,4(x1 − ( a + 1)/2)(x1 − ( a − 1)/2)x1 (4x21 − a − 1)√√+4(x21 − ( a − 1)2 /4)(x21 − ( a + 1)2 /4)√√11p+{25−a(k+)+3a+22x1 5x21 + 5 − a√a(4(a + 11)x21 − (a − 1)2 )√√}++16(x21 − ( a − 1)2 /4)(x21 − ( a + 1)2 /4)√a(4x21 − a + 1)(−6x21 + a − 1)p,+√√16x1 5x21 + 5 − a (x21 − ( a − 1)2 /4)(x21 − ( a + 1)2 /4)(1.334)f1 (x1 ) = −(1.335)à x01 êîíñòàíòà.Îòìåòèì, ÷òî (1.333) ýêñïîíåíöèàëüíî óáûâàåò ïðè ïðèáëèæåíèè ê "òî÷êå ïîâîðîòà"j1= 0.149Äàëåå ïîñòðîèì àñèìïòîòèêó êîýôôèöèåíòîâïîâîðîòà"ïðè|j1 | .
|m|3/5 .Ïðè òàêèõj1ζjâáëèçè "òî÷êèòðåõ÷ëåííîå ðåêóððåíòíîåñîîòíîøåíèå (1.320) ïðèáëèæåííî çàìåíÿåòñÿ ñîîòíîøåíèåì√√√4 a4 5−a14 aj1)ζj−1 + (−2 ++(k + )+(1 −|m|(a − 1)|m|(a − 1) |m|(a − 1)2√20j124 aj1+ 2)ζj + (1 +)ζj+1 ≈ 0,|m| (a − 1)|m|(a − 1)â êîòîðîì ÷èñëàìåíójèj1ñâÿçàíû ðàâåíñòâîì (1.330). Ïðîèçâåäåì çà-√2 a j12ζj = exp −ϑj .(a − 1)|m|Òîãäà êîýôôèöèåíòûϑjáóäóò ïðèáëèæåííî óäîâëåòâîðÿòü ñëåäóþ-ùåìó òðåõ÷ëåííîìó ðåêóððåíòíîìó ñîîòíîøåíèþ√4 5−a14(5 − a)j12ϑj−1 + (−2 +(k + ) −)ϑj + ϑj+1 ≈ 0,|m|(a − 1)2(a − 1)2 |m|2êîòîðîå ÿâëÿåòñÿ ðàçíîñòíûì àíàëîãîì óðàâíåíèÿ Âåáåðà. Äåéñòâèòåëüíî, åñëèϑ(τ ) ∈ C 4 ,òî ïðèh→0ϑ(τ − h) − 2ϑ(τ ) + ϑ(τ + h)= ϑ00 (τ ) + O(h2 ).2hÑëåäîâàòåëüíî, ãëàâíûé ÷ëåí àñèìïòîòèêè(0)ϑj = ϑãäåϑ(τ )(0)ϑjïðåäñòàâèì â âèäå!√42 5 − a j1√p,a − 1 |m| ðåøåíèå óðàâíåíèÿ Âåáåðàd2 ϑ(τ )τ21+ (− + k + )ϑ(τ ) = 0.2dτ42(1.336)150Ðåøåíèÿ (1.336) âûðàæàþòñÿ ÷åðåç ôóíêöèè ïàðàáîëè÷åñêîãîöèëèíäðà:ϑ = c2 Dk (τ ) + c3 D−k−1 (iτ ),(1.337)c2 , c3 êîíñòàíòû.
Ñ ïîìîùüþ ôîðìóë (1.110), (1.111) ñîãëàñóåì(0)3/5ðàçëîæåíèÿ ïðè j1 ïîðÿäêà −|m|. Ïîñêîëüêó ζj,− ýêñïîíåíöèàëüíîóáûâàåò, òî â ðàâåíñòâå (1.337) êîíñòàíòà c3 = 0. ÑïðàâåäëèâàãäåËåììà 1.45.Èìååò ìåñòî àñèìïòîòèêà!√42 5 − a j1√p(1 + O(|m|−1/5 )).a − 1 |m|√2 a j12Dkζj = c2 exp −(a − 1)|m|(1.338)Çäåñü |j1 | . |m|3/5 , |m| → ∞.Äàëåå ïðè(0)c4 ζj,− ,ïîðÿäêà|m|3/5ïðàâàÿ ÷àñòü (1.338) ñîãëàñóåòñÿ ñ(0)ζj,− îïðåäåëÿåòñÿ ôîðìóëîé3/5(1.333). Ýòà ôîðìóëà çàäàåò àñèìïòîòèêó ζj ïðè |m|. j1 , |m|1/3 .j2 , ãäåj2 = n − |m| − j.(1.339)ãäåc4j1 íåêîòîðàÿ êîíñòàíòà, àÍàêîíåö, àíàëîãè÷íî (1.332) ïîêàçûâàåòñÿ, ÷òî ïðè|m|j2 −1ζj =(j2 − 1)!Çäåñü ÷èñëàj2è1 ≤ j2 .
|m|1/3 √ 2 √ j −1( a − 1)(5 − a) 2j2ζn−|m|−1 (1 + O).4|m|jñâÿçàíû ðàâåíñòâîì (1.339).×òîáû ïîëó÷èòü ôîðìóëó ñâÿçè ìåæäóζ0èζn−|m|−1îñòàåò-ñÿ âûðàçèòü äðóã ÷åðåç äðóãà âõîäÿùèå â ðàçëîæåíèÿ êîíñòàíòû.Ñâÿçü ìåæäó íèìè íàõîäèòñÿ â ïðîöåññå ñîãëàñîâàíèÿ àñèìïòîòèê. ðåçóëüòàòå, â ôîðìóëå ñâÿçè â êà÷åñòâå ìíîæèòåëÿ âîçíèêàåò ýêñïîíåíòà, ïîêàçàòåëü êîòîðîé ñîäåðæèò èíòåãðàëû îò ôóíêöèéf0 , f1 .Íî ïîñêîëüêó ôóíêöèè (1.334), (1.335) íå÷åòíûå, à ïðåäåëû èíòåãðèðîâàíèÿ ñèììåòðè÷íû îòíîñèòåëüíî íóëÿ, òî ýòè èíòåãðàëû ðàâíûíóëþ. Òàê êàê ãëàâíûå ÷ëåíû àñèìïòîòèêèëè÷àþòñÿ ëèøü íà ìíîæèòåëü(−1)k ,Dk (τ )òî ïîëó÷àåìïðèτ → ±∞îò-151Òåîðåìà 1.7.Èìååò ìåñòî ôîðìóëà ñâÿçè êîýôôèöèåíòîâζn−|m|−1 = (−1)k ζ0 (1 + O(|m|−1/5 )),|m| → ∞.(1.340)Ïîñêîëüêó â ñèëó (1.282)KBΦW(z) = (−1)k ζ0 z n−|m|−1 (1 + O−ãäå| z | |m|,à êîýôôèöèåíòζ01|m|KBΦW(z)−è ïðè+O|m|),zèìååò âèä (1.331), òî èç ðà-âåíñòâ (1.316), (1.319), (1.340) âûòåêàåò ìàëîñòüñN (z) ïî ñðàâíåíèþ| z | |m|.3.10.Àñèìïòîòèêà íîðìûÏóñòü ìíîãî÷ëåíΦ(z)Φ(z) çàäàí ôîðìóëîé (0.12), ãäå ôóíêöèÿ p(u) àñèìïòîòè÷åñêîå ðåøåíèå ìíîãîòî÷å÷íîé ñïåêòðàëüíîé çàäà÷è.P[m, n].Ïðåäâàðèòåëüíî íàéäåì àñèìïòîòèêó %(r) ïðè |m| → ∞.
Ôóíêöèÿ %(r) óäîâëåòâîðÿåò çàäà÷å (1.221), (1.222), à èñêîìàÿ àñèìïòîòèW KBêà ñîâïàäàåò ñ ÂÊÁ-ïðèáëèæåíèåì %(r) äëÿ ðåøåíèÿ ýòîé çàäà÷è. Îòìåòèì, ÷òî ïðè a > 1 è r ≥ 0 òî÷êè ïîâîðîòà ó óðàâíåíèÿÂû÷èñëèì àñèìïòîòèêó íîðìûΦ(z)â ïðîñòðàíñòâå(1.221) îòñóòñòâóþò. ÑïðàâåäëèâàËåììà 1.46.Ïóñòü âûïîëíåíû óñëîâèÿ(1.230), (1.231).Òîãäà ïðèr |m| èìååò ìåñòî ðàâåíñòâî%(r) = %W KB (r)(1 + O(r1) + O()),|m||m|ãäåp∗c(Λ1 (r) + r + 1)|m|W KBp%(r) = p,√4Λ1 (r)( Λ1 (r) + a(r + 1))n+1/2Λ1 (r) = r2 + (4a − 2)r + 1,(1.341)152à êîíñòàíòàp√ √ √|m|1+a a( a − 1)c∗ =2π(1 +√√a)a!|m|2.Êðîìå òîãî, ïðè r |m| ñïðàâåäëèâà àñèìïòîòèêà%(r) =n(n − |m|) −n+|m|−1|m|r(1 + O()).2π|m|r(1.342)Äîêàçàòåëüñòâî.
Ðàññìîòðèì çàäàííóþ ôîðìóëîé (1.220) ôóíêöèþ%(r).Âîñïîëüçîâàâøèñü èíòåãðàëüíûì ïðåäñòàâëåíåì äëÿ ãè-ïåðãåîìåòðè÷åñêîé ôóíêöèè [5], èìååì:n(n − |m|)(|m| + 1)n Γ(2n + 2) r|m|%(r) =E(r).2π(n + 1)n+1 Γ(n + |m| + 1)Γ(n − |m| + 1)(1.343)ÇäåñüZ1E(r) =tn+|m| (1 − t)n−|m|dt =(1 − t(1 − r))n+1Z10e|m|Φ(t)dt,1 − t(1 − r)(1.344)0ãäå√√√Φ(t) = ( a + 1) ln t + ( a − 1) ln (1 − t) − a ln (1 − t(1 − r)).ÀñèìïòîòèêàE(r)ïðè|m| → ∞ìîæåò áûòü íàéäåíà ñ ïîìî-ùüþ ìåòîäà Ëàïëàñà [87]. Óðàâíåíèå√Φ0 (t) =√√a(1 − r)t2 + (−2 a − 1 + r)t + a + 1=0t(1 − t)(1 − t(1 − r))èìååò êîðåíüp√2 a + 1 − r − Λ1 (r)√t∗ =∈ (0, 1).2 a(1 − r)153Ïîñêîëüêó√√2ta(1−r)−2a−1+r∗=Φ00 (t∗ ) =t∗ (1 − t∗ )(1 − t∗ (1 − r))p=−p√Λ1 (r)( Λ1 (r) + a(r + 1))4a3/2p< 0,(a − 1)( Λ1 (r) + r − 1)2(1.345)t = t∗ ôóíêöèÿ Φ(t) äîñòèãàåò ìàêñèìàëüíîãî çíà÷åíèÿ, ïðèýòîò ìàêñèìóì åäèíñòâåííûé íà [0, 1].òî ïðè÷åìÄàëåå âîñïîëüçóåìñÿ ðàâåíñòâàìèpp√√(2 a + 1 − r − Λ1 (r))(−2 ar + r − 1 − Λ1 (r))√ √=4 a( a − 1)(r − 1)pΛ1 (r) + r + 1√=,2( a − 1)p(1 − r)2 (r − 1 + Λ1 (r))pp=√√(2 a + 1 − r − Λ1 (r))(−2 ar + r − 1 + Λ1 (r))p√Λ1 (r) + a(r + 1)=,2(a − 1)ñîãëàñíî êîòîðûìr|m| e|m|Φ(t∗ ) =!|m|pΛ1 (r) + r + 1√2( a − 1)2(a − 1)p√Λ1 (r) + a(r + 1)√×(2 a)−n .!n×(1.346)Íàêîíåö, ïîäñòàâëÿÿ (1.345), (1.346) â ôîðìóëó ìåòîäà Ëàïëàñà, àòàêæå ïðèìåíÿÿ ôîðìóëó Ñòèðëèíãà, èìååì:|m|%(r) ="n(n − |m|)Γ(n + 1) r2πΓ(|m| + 1)Γ(n − |m| + 1)√#s−2πe|m|Φ(t∗ ) ×00|m|Φ (t∗ )2 a11r× p+ O() = %W KB (r)(1 + O() + O()).|m||m||m|Λ1 (r) + r − 1154Çäåñür |m|.Àñèìïòîòèêà (1.342) ïîëó÷àåòñÿ èç (1.344), åñëè ïðèr |m|ðàçëîæèòü ôóíêöèþE(r) =1Z1|m|−1trn+1n−|m|(1 − t)|m|) =dt 1 + O(r0Γ(|m|)Γ(n − |m| + 1)|m|=) ,1 + O(Γ(n + 1) rn+1r(1.347)è äàëåå ïîäñòàâèòü (1.347) â (1.343).
Ëåììà äîêàçàíà.Çàïèøåì ãëàâíûå ÷ëåíû ÂÊÁ-ïðèáëèæåíèé (1.305), (1.341) ââèäåKB|m|S(z)ΦW,−,0 (z) = c− t(z)e2%W KB (| z |2 ) = c∗ t1 (| z |2 )e|m|S1 (|z| ) .(1.348)ÇäåñüS(z) =√pp√a ln ( Λ(z) + a(z + 1)) − ln ( Λ(z) + z + 1),S1 (r) = ln (pp√√Λ1 (r) + r + 1) − a ln ( Λ1 (r) + a(r + 1)).(1.349)(1.350)Åñëè ïîäñòàâèòü ôóíêöèè (1.348) â ôîðìóëó (1.219) äëÿ ñêàëÿðíîãîïðîèçâåäåíèÿ, òî ïîëó÷èì èíòåãðàëKB 2kΦW−,0 kP[m,n]Z| c− |2 c∗ | t(z) |2 t1 (| z |2 )e|m|=Ω(z,z)dz dz×C× 1 + O(|m|−1 ) ,(1.351)Ω(z, z) = S(z) + S(z) + S1 (| z |2 ).(1.352)ãäå ôóíêöèÿÎòìåòèì, ÷òî ýòà ôóíêöèÿ íåïðåðûâíà ïðè âñåõ çíà÷åíèÿõ àðãóìåíòîâ, íî íå ÿâëÿåòñÿ äèôôåðåíöèðóåìîé íà äóãå ^ z − , z + .155Íàéäåì òî÷êó, ãäå äîñòèãàåòñÿ ãëîáàëüíûé ìàêñèìóìΩ(z, z).Òîãäà àñèìïòîòèêà èíòåãðàëà (1.351) áóäåò ðàâíà èíòåãðàëó ïî ìàëîé îêðåñòíîñòè ýòîé òî÷êè.Ïðåäâàðèòåëüíî äîêàæåì ëåììó.Ëåììà 1.47.Ôóíêöèÿ Ω(z, z) èìååò åäèíñòâåííóþ ñòàöèîíàð-íóþ òî÷êó z = z = 1.Äîêàçàòåëüñòâî.
Äèôôåðåíöèðóÿ (1.349), (1.350), íàõîäèì, ÷òîñòàöèîíàðíûå òî÷êè óäîâëåòâîðÿþò ñèñòåìå óðàâíåíèé∂Ω= 0,∂z(1.353)∂Ω= 0,∂z(1.354)ãäåp√√(2 a − 1)z 2 + 3( a − 1)z − 1 − (z − 1) Λ(z)∂Ω=−∂z2z(z 2 + 3z + 1)√ √2 a( a − 1)z,(1.355)−p√Λ1 (zz) + zz(2 a − 1) + 1p√√(2 a − 1)z 2 + 3( a − 1)z − 1 − (z − 1) Λ(z)∂Ω=−∂z2z(z 2 + 3z + 1)√ √2 a( a − 1)z−p.(1.356)√Λ1 (zz) + zz(2 a − 1) + 1Òî÷êàz =z =1óäîâëåòâîðÿåò óðàâíåíèÿì (1.353), (1.354). Ïîêà-æåì, ÷òî äðóãèõ ðåøåíèé ñèñòåìà (1.353), (1.354) íå èìååò.Ïðåæäå âñåãî çàìåòèì, ÷òî ïðèz = z = zj îñîáûå òî÷êè (1.251), çíà÷åíèÿ ïðîèçâîäíûõz j , j = 1, 2, 3∂Ω/∂z , ∂Ω/∂z , êî, ãäåòîðûå âû÷èñëÿþòñÿ â ðåçóëüòàòå ïðåäåëüíîãî ïåðåõîäà â ôîðìóëàõz = z = −1 íåíà äóãå ^ z − , z + ,(1.355), (1.356), íå ðàâíû íóëþ. Êðîìå òîãî, òî÷êàÿâëÿåòñÿ ñòàöèîíàðíîé, ïîñêîëüêóãäå ôóíêöèÿΩ(z, z)z = −1íå äèôôåðåíöèðóåìà.ëåæèò156Ïðåîáðàçóåì óðàâíåíèå (1.353) ê âèäópp√√√2 az 2 + 3 az − (z − 1) Λ(z) 2 azz − Λ1 (zz)=,zz − 1z 2 + 3z + 1è äàëåå ïîñëå çàìåíû√u = z − 1, u = z − 1ïîëó÷àåì óðàâíåíèåqa(u+1)(3uu+5u+5u+10)+(uu+u+u)u u2 + (5 − a)u + 5 − a == (u2 + 5u + 5)p(uu + u + u)2 + 4a(uu + u + u + 1).(1.357)Âîçâåäåì ïðàâóþ è ëåâóþ ÷àñòè (1.357) â êâàäðàò.
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