Диссертация (1136178), страница 44
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Ïóñòü ξ = (x− xe− ) 3 Ω− /h2/3 , e− = ε 3 Ω− /h2/3 .2/3+2/19Òîãäà√ (x0 − x3Ω− 0e)−edx +√ ln |x − x |k− Hh2/3xe− +h2/3 / 3 Ω−ZΩ−xe− +ε02/32/3e− |×k− ) ln |x − xe− | = 2h2/3 Ω− ln |x − x+2h2/3 Ω− D−1 (eZ ∞h iek1−00e√ 0 − θ(ξ − 1)H(ξ ) dξ 0 +× A−1 (k− ) +2 0ξZ e−p2/3 2/3 e02/3 3+Ω− h k−ln |ξ − ξ | + ln |h / Ω− | H(ξ 0 ) dξ 0 =1√2/32/3= 2h2/3 Ω− ln |x − xe− |[A−1 (ek− ) + e−ek− ] + h2/3 Ω− ek− ×Z e− ξ 0 1002/3+2/19×ln 1 − H(ξ ) dξ + O hln ln |x − xe− | .ξh1Òàê êàê ïðèe−Z1h → 0, x > xe− + εZ e− ξ 0 ξ 0 dξ 01002/19ln 1 − √ 0 = O hln 1 − H(ξ ) dξ −ln ,ξξ ξh0òî (4.316) äîêàçàíî. Ðàâåíñòâî (4.317) ïðîâåðÿåòñÿ àíàëîãè÷íî.
Ëåììà äîêàçàíà.x→xe±x ∈ (ex − + ε, xe+ − ε)Ïîñêîëüêó ïðèòî ïðè8/9hôóíêöèÿS 0 (x, h) ∼pU 0 (x± )(x − xe± ),k 2/3h8/9h8/9ρ 0 2/3 = O+O.(S )(x − xe− )1/3(ex+ − x)1/3(4.318)Èòàê, â ñèëó (4.316) (4.318) óðàâíåíèå (4.290) ïðèíèìàåò âèä−(S 0 )2 + U (x)−1/3−hZxe+ −εxe− +εln |x − x0 |k dx02/3− 2h2/3 Ω− A−1 (ek− ) ln |x − xe− |−00S (x , h)382e− dξ 0 iξ 0 h2/3√−h√ 0 −3(x−xe)Ωξ−−0h √2/3 2/32/3 2/3 ee−2h Ω+ A−1 (k+ ) ln |x − xe+ | − h Ω+ k+ 2 e+ ln |x − xe+ |+2/32/3k−Ω− eZh √2 e− ln |x − xe− | +e+Z+0ln 1 −ln 1 − dξ 0 iξ 0 h2/3√ √ 0 = R∗ ,3(ex+ − x) Ω+ ξ(4.319)ãäå∗2/3+2/19R =O h1e− || + | ln |x − xe+ || +ln 1 + | ln |x − xhh8/9h8/9+O+O.(x − xe− )1/3(ex+ − x)1/3(4.320)e± , ek± îïðåäåëåíû ñîîòíîøåíèÿìè (4.257), (4.245), (4.246), ε =h26/57 , x ∈ (ex− + ε, xe+ − ε).ÇäåñüÒàêèì îáðàçîì, ïîëó÷åíà çàäà÷à äëÿ ãëàâíîãî ïðèáëèæåíèÿ êôàçå.
Îíà ñîñòîèò èç óðàâíåíèé (4.312), (4.313), (4.315), (4.319), ãäåΩ±çàäàþòñÿ ôîðìóëàìè (4.310).2.8.Íàõîæäåíèå ãëàâíîãî ïðèáëèæåíèÿ ê ôàçåÏîñòðîèì àñèìïòîòè÷åñêîå ðåøåíèå çàäà÷è äëÿ ãëàâíîãî ïðèáëèæåíèÿ ê ôàçå. Òî÷íîñòü, ñ êîòîðîé ñòðîèòñÿ ðåøåíèå, ñîîòâåòñòâóåò òî÷íîñòè, ñ êîòîðîé ïîëó÷åíû óðàâíåíèÿ (4.312), (4.313),(4.315), (4.319).Íî ïðåæäå ââåäåì ðÿä ôóíêöèé, íåîáõîäèìûõ äëÿ ïîñòðîåíèÿ ðàçëîæåíèé, è èçó÷èì ñâîéñòâà ýòèõ ôóíêöèé. Ñðåäè íèõ åñòüíåãëàäêèå, ñ âåñüìà íåñòàíäàðòíûìè àñèìïòîòèêàìè, íàõîæäåíèåêîòîðûõ òðåáóåò äîñòàòî÷íî êðîïîòëèâûõ îöåíîê.Ïóñòün √x − xdef−θ(x− − x)+W (x, h) = U (x) − h1/3 k0 E(x) + 2πh1/3 k0 pU 0 (x− )√o πn U 00 (x )x − x+−1/6+p(x− − x)3/2 θ(x− − x)+θ(x − x+ ) + h k003/206(U(x))−U (x+ )−383oU 00 (x+ )(x − x+ )3/2θ(x − x+ ) ,+(−U 0 (x+ ))3/2ãäåU (x), E(x), θ(x)çàäàíû ôîðìóëàìè (0.55), (0.56), (4.105),x+Zk0 = 2x−z− , z+ , z− ∼ x− , z+ ∼ x+Íàéäåì ðàçëîæåíèÿ äëÿ E(x)àËåììà 4.44.(4.321)dx −1p,U (x)(4.322)òå òî÷êè, â êîòîðûõèW (z± , h) = 0.W (x, h).Èìåþò ìåñòî ðàâåíñòâà√2π x− − xE(x) = E(x− ) + pθ(x− − x) − (x − x− )Φ− +U 0 (x− )πU 00 (x− )3/22+(x−x)θ(x−x)+O(x−x), x → x− , (4.323)−−−6(U 0 (x− ))3/2√2π x − x+E(x) = E(x+ ) + pθ(x − x+ ) − (x+ − x)Φ+ +−U 0 (x+ )+πU 00 (x+ )3/22(x−x)θ(x−x)+O(x−x),+++6(−U 0 (x+ ))3/2x → x+ ,(4.324)ãäådefZ∞Φ− =x−defZx+Φ+ =−∞ θ(x − x0 )11+p−pdx0 ,0(x − x− )U (x0 )U 0 (x− )(x0 − x− ) θ(x0 − x )11−p−pdx0 .0000(x+ − x )U (x )U (x+ )(x − x+ )Ñëåäîâàòåëüíî,W (x, h) = U 0 (x− )(x − z− ) + O h1/3 (x − z− ) + O (x − z− )2 , x → z− ,(4.325)W (x, h) = −U 0 (x+ )(z+ −x)+O h1/3 (z+ −x) +O (z+ −x)2 , x → z+ ,(4.326)z− = x− + h1/3 x−,1 + O(h2/3 ),h → 0,(4.327)384z+ = x+ − h1/3 x+,1 + O(h2/3 ),Çäåñüh → 0.defx−,1 = k0 E(x− )/U 0 (x− ),(4.328)(4.329)defx+,1 = −k0 E(x+ )/U 0 (x+ ).(4.330)Áîëåå òî÷íî,W (x, h) = W 0 (z− , h)(x − z− ) + W 00 (z− , h)(x − z− )2 /2++O (x − z− )3 ,x → z− ,(4.331)W (x, h) = −W 0 (z+ , h)(z+ − x) + W 00 (z+ , h)(z+ − x)2 /2++O (z+ − x)3 , x → z+ ,(4.332)ãäåW 0 (z− , h) = U 0 (x− ) + h1/3 k0 Φ− + h1/3 x−,1 U 00 (x− ) + O(h2/3 ),W 00 (z− , h) = U 00 (x− ) + O(h1/3 ),h → 0,W 0 (z+ , h) = U 0 (x+ ) − h1/3 k0 Φ+ − h1/3 x+,1 U 00 (x+ ) + O(h2/3 ),W 00 (z+ , h) = U 00 (x+ ) + O(h1/3 ),h → 0,h → 0,h → 0.Äîêàçàòåëüñòâî.
 ñèëó (4.151), à òàêæå ñëåäóþùèõ èíòåãðàëîâ [21]Z∞1 dτ √ = 0,τ τ0Z ξ 1 p1 √ln 1 − τ dτ = −2 ξ + O √ ,τξ0Z ξ 1 p1 √2πln 1 + τ dτ = 2 ξ −+O √ ,τ3ξ0ln 1 −(4.333)ξ → +∞,ξ → +∞,(4.334)(4.335)èìååì:Zx+E(x) = E(x− ) +x−x − x− 1−ln 1 − 0 p 0x − x−U (x− )(x0 − x− )385√Z x+ x − x− 1U 00 (x− ) x0 − x− 0−−dx +ln 1 − 0 p0)x−x4(U 0 (x− ))3/2U(x−x−√1U 00 (x− ) x0 − x− 0−pdx =+4(U 0 (x− ))3/2U 0 (x− )(x0 − x− )pZ|x − x− | n ∞ sgn(x − x− ) dτ= E(x− ) + pln 1 −√ +ττU 0 (x− )0Z ∞osgn(x − x− )3/2+dτ + O(|x − x− | ) −τ 3/2(x+ −x− )/|x−x− |Z (x+ −x− )/|x−x− | sgn(x − x− ) √U 00 (x− )3/2−|x − x− |ln 1 − τ dτ −τ4(U 0 (x− ))3/20Z x+1 11p−(x − x− )−p+0U (x0 )U 0 (x− )(x0 − x− )x− x − x−√U 00 (x− ) x0 − x− 0+dx + O((x − x− )2 ) = E(x− )+03/24(U (x− ))√2πx− − xθ(x− − x) − (x − x− )Φ− ++pU 0 (x− )πU 00 (x− )+(x− − x)3/2 θ(x− − x) + O((x − x− )2 ),6(U 0 (x− ))3/2x → x− .Ðàâåíñòâî (4.323) äîêàçàíî.Äàëåå, ïîäñòàâëÿÿ (4.323) â (4.321) è ðàçëàãàÿ ôóíêöèþUïîôîðìóëå Òåéëîðà, èìååìW = U 0 (x− )(x − x− ) + U 00 (x− )(x − x− )2 /2 + O((x − x− )3 )−−h1/3 x−,1 U 0 (x− ) − h1/3 k0 (x − x− )Φ− + O(h1/3 (x − x− )2 ).(4.336)Èç (4.336) âûòåêàþò (4.325), (4.327), (4.331).
Ñîîòíîøåíèÿ (4.324),(4.326), (4.328), (4.332) ïðîâåðÿþòñÿ àíàëîãè÷íî. Ëåììà äîêàçàíà.Ëåììà 4.45.Zz+z−dxÑïðàâåäëèâî ðàâåíñòâîp=W (x, h)Zx+x−h1/3 k0 P1p++ O(h3/8 ),2U (x)dxh → 0.(4.337)386Çäåñü êîíñòàíòà P1 îïðåäåëåíà ôîðìóëîé∞n θ(x − x )θ(x − x)θ(x − x− )E(x− )−+P1 =E(x)−−3/23/2−∞U (x)U 0 (x− )(x − x− )3/2Zoθ(x+ − x)E(x+ )−dx,3/203/2− U (x+ )(x+ − x)ãäå θ(x) èìååò âèä(4.338)(4.105).Äîêàçàòåëüñòâî.
Èñïîëüçóÿ (4.325), (4.327), ïîëó÷àåìz− +h1/4Zpz−=p=pZ1U 0 (x− )z−1W (x, h)z− +h1/4z− +h1/4Zdx−x−dx√−x − z−Zdxp=U (x)z− +h1/4x−dx √+O(h3/8 ) =x − x−[−h5/24 x−,1 +h7/24 x2−,1 /4]+O(h3/8 ),U 0 (x− )h → 0.(4.339)Àíàëîãè÷íî, â ñèëó (4.326), (4.328)Zz+z+ −h1/4ZdxpW (x, h)x+−z+ −h1/4dxp=U (x)1=p[−h5/24 x+,1 + h7/24 x2+,1 /4] + O(h3/8 ),0−U (x+ )h → 0.(4.340)Íàêîíåö, èñïîëüçóÿ (4.329), (4.330) è ôîðìóëó Òåéëîðà, èìååìZz+ −h1/4z− +h1/411p−pdx =W (x, h)U (x)Zz+ −h1/4z− +h1/4n h1/3 k E(x)0+2(U (x))3/23 h2/3 k02 (E(x))2 oh1/3 k0 P1 h1/3 h x−,13/8p+dx + O(h ) =+×8 (U (x))5/222U 0 (x− )Z ∞Z z+ −h1/4idxx+,1dx×+p+3/2(x+ − x)3/2−U 0 (x+ ) −∞z− +h1/4 (x − x− )3873 2/3 h x2−,1p+ h8U 0 (x− )Zz+ −h1/4×−∞Z∞z− +h1/4x2+,1dx+p×(x − x− )5/2−U 0 (x+ )h xidxh1/3 k0 P1−,15/243/8p+h++O(h)=2(x+ − x)5/2U 0 (x− )ix2+,1h7/24 h x2−,1p+p−+p+ O(h3/8 ),0004−U (x+ )U (x− )−U (x+ )x+,1ih → 0.(4.341)Îñòàåòñÿ ñëîæèòü (4.339) (4.341).
Ëåììà äîêàçàíà.ÏóñòüZdefk1 = 2(1 − h1/3 [b− + b+ ])z+z−ãäåb− , b+dx−1pW (x, h),(4.342)çàäàíû ôîðìóëàìè (0.58). Òîãäà â ñèëó (4.337)k1 = k0 − h1/3 k03 P1 /4 − h1/3 k0 (b− + b+ ) + O(h3/8 ),Ëåììà 4.46.Zz+k1z−Zz+k1z−h → 0.(4.343)Ïðè h → 0 ñïðàâåäëèâû ðàâåíñòâà1ln |z− − x0 | 01/3pdx = k0 E(x− ) + O h ln ,hW (x0 , h)(4.344)ln |z+ − x0 | 011/3pdx = k0 E(x+ ) + O h ln .hW (x0 , h)(4.345)Çäåñü E(x), k0 , k1 îïðåäåëåíû ôîðìóëàìè(0.56), (4.322), (4.342).Äîêàçàòåëüñòâî. Èñïîëüçóÿ (4.343), èìååìZz+k1z−= k0hZz+z−ln |z− − x0 | 0pdx − k0 E(x− ) =W (x0 , h)ln |z− − x0 | 0pdx −W (x0 , h)Zx+x−ln |x− − x0 | 0 ipdx + O(h1/3 ).U (x0 )(4.346)388Òàê êàê√τ −1/2 ln τ dτ = 2 ξ(ln ξ − 2),Rξ0z− +h2/9Zz−ln |x0 − z− | 0pdx −W (x0 , h)Zòîz− +h2/9x−ln |x0 − x− | 0pdx =U (x0 )z− +h2/9ln |x0 − z− | dx0p−001/3002U(x)(x−z)+O(h(x−z))+O((x−z))z−−−−−Z z− +h2/9ln |x0 − x− | dx0p=−U 0 (x− )(x0 − x− ) + O((x0 − x− )2 )x−h2/9 x−,112/91/3= −pln h + O h ln , h → 0,(4.347)hU 0 (x− )Z=à òàêæåZz+z+ −h2/9ln |x0 − z− | 0pdx −W (x0 , h)Zx+z+ −h2/9ln |x0 − x− | 0pdx =U (x0 )h2/9 x+,1= −pln |x+ − x− | + O(h1/3 ),0−U (x+ )h → 0.(4.348)Êðîìå òîãî, ïîëó÷àåìZz+ −h2/9z− +h2/9Zz+ −h2/9dx0= −(z− − x− )×(ln |z− − x0 | − ln |x− − x0 |) pU (x0 )dx0p+ O h2/300(x − x− ) U (x )×z− +h2/9Íàêîíåö, òàêZz+ −h2/9z− +h2/9dx0p=(x0 − x− )2 U (x0 )2h2/9 x−,1+ O(h1/3 ), h → 0.= −p0U (x− )R ∞ −3/2√êàêτlnτdτ=2(lnξ+2)/ξ,ξz+ −h2/9z− +h2/9ZZz− +h2/9òî11 0ln |z− − x | p−pdx =W (x0 , h)U (x0 )(z+ −z− )/2=(4.349)0E(x− )h1/3 k0 nln |x − z− |+2(U 0 (x− ))3/2 (x0 − x− )3/20389Z z+ −h2/9 no1+O √ 0dx0 +ln |x+ − z− |×x − x−(z+ −z− )/2 h1/3 oh1/3 k0 E(x+ )101/3=×+O √dx + O h lnh2(−U 0 (x+ ))3/2 (x+ − x0 )3/2x+ − x0h2/9 x−,1 ln h2/9 2h2/9 x−,1 h2/9 x+,1 ln |x+ − x− |11/3p= p+p++ O h ln .hU 0 (x− )U 0 (x− )−U 0 (x+ )(4.350)Ðàâåíñòâî (4.344) âûòåêàåò èç (4.346) (4.350).
Àíàëîãè÷íî ïðîâåðÿåòñÿ (4.345). Ëåììà äîêàçàíà.ÎïðåäåëèìdefΩ(x) =∞n θ(x0 − x )θ(x − x0 )−+ln |x − x |E(x0 )−03/2(U (x ))−∞Z0oθ(x0 − x− )E(x− )θ(x+ − x0 )E(x+ )− 0−dx0 .3/203/203/203/2(U (x− )) (x − x− )(−U (x+ )) (x+ − x )Ëåììà 4.47.(4.351)Ïðè x ∈ (z− + ε, z+ − ε), ãäå ε = h26/57 , ñïðàâåäëèâîðàâåíñòâîZz+z−√ln |x − x0 |h1/3 k02π x− − x0pdx − E(x) =Ω(x) − pθ(x− − x)−2W (x0 , h)U 0 (x− )√2π x − x+−pθ(x−x+ )+O h3/8 (1+| ln |x−z− ||+| ln |x−z+ ||) , h → 0.−U 0 (x+ )(4.352)Äîêàçàòåëüñòâî. Èìååìz− +h1/4Zx−Zx+z+ −h1/4ln |x − x0 | 01pdx = pU (x0 )U 0 (x− )Zz− +h1/4x−ln |x − x0 | 0√ 0dx +x − x−+O h3/8 (1 + | ln |x − z− ||) ,Z x+ln |x − x0 | 01ln |x − x0 | 0p√dx = pdx +U (x0 )−U 0 (x+ ) z+ −h1/4 x+ − x0+O h3/8 (1 + | ln |x − z+ ||) ,390z− +h1/4Zz−Zz+z+ −h1/4z− +h1/4ln |x − x0 | 0√ 0dx +x − z−z−+O h3/8 (1 + | ln |x − z− ||) ,Z z+1ln |x − x0 | 0ln |x − x0 |0p√dx = pdx +W (x0 , h)−U 0 (x+ ) z+ −h1/4 z+ − x0+O h3/8 (1 + | ln |x − z+ ||) .1ln |x − x0 |pdx0 = pW (x0 , h)U 0 (x− )ZÄàëåå, ïîëó÷àåìZz+ −h1/4z− +h1/4ln |x − x | p0z+ −h1/41W (x0 , h)1−pdx0 =0U (x )n h1/3 k E(x0 ) 3 h2/3 k 2 (E(x0 ))2 o00=ln |x − x |+dx0 +03/205/28 (U (x ))2(U (x ))z− +h1/4Z x+1/3hk0+O h3/8 (1 + | ln |x − z− || + | ln |x − z+ ||) =ln |x − x0 |×2x−on E(x0 )E(x+ )E(x− )−××−(U (x0 ))3/2 (U 0 (x− ))3/2 (x0 − x− )3/2 (−U 0 (x+ ))3/2 (x+ − x0 )3/2Z z+ −h1/4n1/3hkE(x− )000×dx ++ln |x − x |0 (x ))3/2 (x0 − x )3/221/4(U−−z− +hoE(x+ )+dx0 +03/203/2(−U (x+ )) (x+ − x )Z1/4n3 2/3 2 z+ −h(E(x− ))20+ h k0ln |x − x |+8(U 0 (x− ))5/2 (x0 − x− )5/2z− +h1/4o(E(x+ ))2dx0 ++05/205/2(−U (x+ )) (x+ − x )+O h3/8 (1 + | ln |x − z− || + | ln |x − z+ ||) .Z0Èòàê,Zz+ −h1/4z− +h1/411 0 h1/3 k0ln |x − x | p−pdx =Ω(x)+2W (x0 , h)U (x0 )0391Π+ (x, h)Π− (x, h)+p+ O h3/8 (1 + | ln |x − z− || + | ln |x − z+ ||) ,+pU 0 (x− )−U 0 (x+ )(4.353)ãäåΩ(x)çàäàåòñÿ ôîðìóëîé (4.351), àh1/4ZΠ− (x, h) =0dtln |(x − z− ) − t| √ −th1/3 x−,1+2Zz− +h1/4 −x−Z0dtln |(x − x− ) − t| √ +t∞ln |(x − x− ) − t|z− +h1/4 −x−3+ h2/3 x2−,18Zdt+t3/2∞ln |(x − x− ) − t|z− +h1/4 −x−dt,t5/2Z x+ −z+ +h1/4dtdtln |(x+ − x) − t| √ +Π+ (x, h) =ln |(z+ − x) − t| √ −tt00Zdth1/3 x+,1 ∞ln |(x+ − x) − t| 3/2 ++2tx+ −z+ +h1/4Z ∞dt3 2/3 2ln |(x+ − x) − t| 5/2 .+ h x+,18tx+ −z+ +h1/4h1/4ZÓïðîñòèì âûðàæåíèå äëÿ∞ZΠ− (x, h) =0Π− (x, h).Ïî ôîðìóëå Òåéëîðàdt{ln |(x − z− ) − t| − ln |(x − x− ) − t|} √ +t+O(h3/8 (1 + | ln |x − z− ||)).Âû÷èñëèì ïîëó÷èâøèåñÿ èíòåãðàëû.
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