Диссертация (Редукции бездисперсионных интегрируемых иерархий и уравнение Левнера), страница 10

Èñïîëüçóÿ (AI.3), (AI.5), ëåãêî äîêàçàòü ñëåäóþùèå ñâîéñòâà ôóíêöèè ζa (x):ζa (x + 1) = ζa (x) ,è(AI.11)ζa (x + τ ) = ζa (x) − 2πiζ1 (x + τ2 ) = ζ4 (x) − πi ,ζ4 (x + τ2 ) = ζ1 (x) − πi .(AI.12)Äëÿ âû÷èñëåíèé íàì òàêæå ïîíàäîáèòñÿ ζ2 (0) = 0, ζ2 ( τ2 ) = −πi.Î÷åâèäíî, ζa - íå÷åòíûå ôóíêöèè.  ÷àñòíîñòè, ζ1 (x, τ ) =ïðè x → 0 è ζa (0, τ ) = 0 ïðè a = 2, 3, 4.Ââåäåì ôóíêöèþS(x) = log66θ1 (x, τ ).θ4 (x, τ )1x+ O(x)(AI.13)Îáîçíà÷èì ∂x S(x) = S 0 (x), ∂x2 S(x) = S 00 (x), ∂τ S(x) = Ṡ(x).

Ìîæíîäîêàçàòü ñëåäóþùèå ôîðìóëû (çäåñü è íèæå τ 0 ≡ τ2 ):S 0 (x) = πθ42 (0, τ )θ2 (x, τ )θ3 (x, τ )=θ1 (x, τ )θ4 (x, τ )= πθ3 (0, τ 0 )θ4 (0, τ 0 )S 00 (x) = −π 2 θ22 (0, τ )θ32 (0, τ )θ43 (0, τ )(AI.14)0θ2 (x, τ ),θ1 (x, τ 0 )θ4 (2x, τ )2θ1 (x, τ )θ42 (x, τ )0=0(AI.15)θ3 (x, τ )θ4 (x, τ ),θ12 (x, τ 0 )π2 402πiṠ(x) = S (x)ζ2 (x, τ ) + θ4 (0, τ ),(AI.16)22πiṠ 0 (x) = S 00 (x)ζ2 (x, τ ) − S 0 (x)℘2 (x, τ ).(AI.17)Èç (AI.14) è (AI.15) âèäíî, ÷òî S 0 (x + 1) = S 0 (x), S 0 (x + τ 0 ) = −S 0 (x),S 00 (x + 1) = S 00 (x), S 00 (x + τ 0 ) = −S 00 (x).= −π 2 θ3 (0, τ 0 )θ4 (0, τ 0 )θ22 (0, τ 0 )Çàìåòèì, ÷òîS 0 (x)S 0 (x + 21 ) = −π 2 θ44 (0, τ ).(AI.18)Ïðè x → 0, èìååì:1+ O(x),xÎòìåòèì òàêæå òîæäåñòâàS 00 (x) = −S 0 (x) =1+ O(1).x21 0℘1 (x, τ 0 ),20S (x) = 2ζ1 (x, τ ) − ζ1 (x, τ 0 ).Îòñþäà ñðàçó ñëåäóåò, ÷òîS 0 (x)S 00 (x) =2ζ2 (x, τ ) − ζ2 (x, τ 0 ) = S 0 (x + 21 ),2℘2 (x, τ ) − ℘2 (x, τ 0 ) = −S 00 (x + 21 ).Òàêæå íàì ïîíàäîáèòñÿ ñòàíäàðòíîå ñîîòíîøåíèå(θ10 (0, τ ))2 θ1 (x − y, τ )θ1 (x + y, τ )℘2 (x, τ ) − ℘2 (y, τ ) =,θ22 (x, τ )θ22 (y, τ )÷àñòíûì ñëó÷àåì êîòîðîãî ÿâëÿåòñÿ℘1 (x, τ 0 ) − ℘2 (0, τ 0 ) = (S 0 (x))2 .67(AI.19)(AI.20)(AI.21)(AI.22)(AI.23)(AI.24)7.2Ïðèëîæåíèå II.

Äîêàçàòåëüñòâî óðàâíåíèÿ (110)Äîêàæåì ôîðìóëó (110) ïðîèçâîäíîé ïî τ îò ôóíêöèè S(u|τ ) = logπ2 42πi ∂τ S(u|τ ) = ∂u S(u|τ ) ζ2 (u|τ ) +θ (0|τ ) .2 4θ1 (u|τ ):θ4 (u|τ )(AII.1)Ïîõîæàÿ ôîðìóëà áûëà äîêàçàíà â [69] â êîíòåêñòå ñâÿçè ÏåíëåâåÊàëîäæåðî.Ìû íà÷íåì ñ òàêîãî ïðåäñòàâëåíèÿ S 0 (u):S 0 (u) = ζ1 (u) − ζ4 (u) = πθ42 (0)θ2 (u) θ3 (u),θ1 (u) θ4 (u)(AII.2)êîòîðîå ëåãêî äîêàçûâàåòñÿ ñ ïîìîùüþ óðàâíåíèÿ (AI.8) ñðàâíåíèåìàíàëèòè÷åñêèõ ñâîéñòâ îáåèõ ñòðîí ðàâåíñòâà. Î÷åâèäíî, ÷òî θ1 è θ4äîëæíû áûòü â çíàìåíàòåëå, ÷òîáû îáåñïå÷èòü ñîâïàäàþùèå ó îáåèõñòîðîí ïîëþñû. Òåïåðü ïðîâåðèì, ÷òî íóëè ïðàâîé ÷àñòè îáíóëÿþò ëåâóþ, ò.å ïðè ïîäñòàíîâêå ω1 = 12 è ω2 = 1+τ2 , íàïðèìåð. ×òîáû íàéòè10 10çíà÷åíèå S ( 2 ), âû÷èñëèì S (u + 2 ), çàòåì ïîëîæèì u = 0,111S u+= ζ1 u +− ζ4 u += ζ2 (u) − ζ3 (u)2220Ïðè ïîäñòàíîâêå u = 0 ýòî âûðàæåíèå ðàâíî íóëþ, ñëåäîâàòåëüíî, íóëüïðàâîé ÷àñòè ÿâëÿåòÿ íóëåì ëåâîé.Ïðîâåðèì âòîðîé íóëü ïðàâîé ÷àñòè:S01+τu+21+τ1+τ= ζ1 u +− ζ4 u +=22B 0 (u)θ3 (u) + B(u)θ30 (u) B 0 (u)θ2 (u) + B(u)θ20 (u)=−= ζ3 (u) − ζ2 (u).B(u)θ3 (u)B(u)θ2 (u)Ïîëîæèâ u = 0, ïîëó÷àåì íóëü, ò.å.

è ýòîò íóëü ïðàâîé ÷àñòè ÿâëÿåòñÿíóëåì ëåâîé.Ïîëó÷èëè, ÷òî íóëè è ïîëþñû ñîâïàäàþò. Êîíñòàíòó íàõîäèì, ïîäñòàâëÿÿ u = 0πθ2 (0) θ3 (0) θ42 (0)θ2 (0) θ3 (0)S (0) = ζ1 (0) == πθ42 (0).θ1 (0) θ4 (0)θ1 (0) θ4 (0)068Åùå íàì ïîíàäîáèòñÿ ÷àñòíûé ñëó÷àé ýòîãî ðàâåíñòâà, äëÿ ïîëó÷åíèÿêîòîðîãî íàì íóæíî ïîäñòàâèòü u + 12 â âûðàæåíèå äëÿ S 0 (u),11ζ1 u +− ζ4 u += ζ2 (u) − ζ3 (u),22è âçÿòü ïðîèçâîäíóþ ïî u0(ζ2 (u) − ζ3 (u)) =θ2 (u)0 θ3 (u)0−θ2 (u)θ3 (u)0=2θ3 (u)00θ3 (u)0+−=θ3 (u)θ3 (u)2 2θ2 (u)00 θ3 (u)00θ3 (u)0θ2 (u)0−+=−.θ2 (u)θ3 (u)θ3 (u)θ2 (u)θ2 (u)00−=θ2 (u)θ2 (u)0θ2 (u)2Óñòðåìëÿÿ u ê íóëþ, ïîëó÷àåìθ2 (0)00 θ3 (0)00−.(ζ2 (0) − ζ3 (0)) =θ2 (0)θ3 (0)0Òåïåðü ïðîäåëàåì òî æå ñàìîå ñ ïðàâîé ñòîðîíîé (AII.2). Ïîäñòàâëÿåì−πθ42 (0)θ2 (u + 12 ) θ3 (u + 21 )θ1 (u)θ4 (u)2,=−πθ(0)4θ2 (u)θ3 (u)θ1 (u + 12 ) θ4 (u + 21 )áåðåì ïðîèçâîäíóþπθ42 (0)θ1 (u)0 θ4 (u)0 θ2 (u)0 θ3 (u)0+−−θ1 (u)θ4 (u)θ2 (u)θ3 (u)θ1 (u) θ4 (u).θ2 (u) θ3 (u)Òåïåðü, ïîëîæèâ u = 0, ïîëó÷àåì, ÷òî â ñêîáêå âûæèâàåò òîëüêî ïåðâûé÷ëåí−πθ42 (0)Äîêàçàëè:πθ2 (0)θ3 (0)θ4 (0)θ4 (0)θ10 (0) θ4 (0)= −πθ42 (0)= −π 2 θ44 (0).θ2 (0) θ3 (0)θ2 (0) θ3 (0)θ300 (0) θ200 (0)−= π 2 θ44 (0).θ3 (0) θ2 (0)(AII.3)Èç (79) ìû âèäèì, ÷òî Ṡ(u + 1) = Ṡ(u), Ṡ(u + τ ) = Ṡ(u) − S 0 (u),ñëåäîâàòåëüíî, îáå ñòîðîíû óðàâíåíèÿ (AII.1) ïåðèîäè÷íû îòíîñèòåëüíî69ñäâèãîâ u → u + 1 è ïîëó÷àþò àääèòèâíûé âêëàä −2πiS 0 (u) ïðè ñäâèãåu → u + τ (ñì.

(AI.11)). Òàêèì îáðàçîì, ôóíêöèÿg(u) := 4πi Ṡ(u) − 2S 0 (u)ζ2 (u) − π 2 θ44 (0)(AII.4)äâàæäû ïåðèîäè÷åñêàÿ ñ ïåðèîäàìè 1, τ . Èñïîëüçóÿ óðàâíåíèåòåïëîïðîâîäíîñòè (AI.9), èìååì:θ100 (u) θ400 (u)θ3 (u) θ20 (u)2g(u) =−− 2πθ4 (0)− π 2 θ44 (0).θ1 (u) θ4 (u)θ1 (u) θ4 (u)Äëÿ òîãî, ÷òîáû äîêàçàòü, ÷òî g(u) ≡ 0, äîñòàòî÷íî ïîêàçàòü, ÷òî îíàðåãóëÿðíà â u = 0, u = τ2 (íóëè çíàìåíàòåëåé) è g(u0 ) = 0 â êàêîé-òîòî÷êå u0 (óäîáíî âûáðàòü u0 = 1+τ2 ).

Ðåãóëÿðíîñòü â u = 0 î÷åâèäíà, òàê000êàê θ1 (u) è θ2 (u) èìåþò ïðîñòûå íóëè â u = 0, êîòîðûå ñîêðàùàþò íóëèâ çíàìåíàòåëå. Ðåãóëÿðíîñòü â u = τ2 ìåíåå î÷åâèäíà, ïîýòîìó ðàñïèøåìâñå ïîäðîáíî. Ïîäñòàâèì u = τ2 è âîñïîëüçóåìñÿ (AI.6):θ100τ= i(B 00 (u)θ4 (u) + 2B 0 (u)θ40 (u) + B(u)θ400 (u))u+2θ100 (u + τ2 ) iB(u)(−π 2 θ4 (u) − 2iπθ40 (u) + θ400 (u))θ400 (u)2==−π−2iπζ(u)+.4θ1 (u + τ2 )iB(u)θ4 (u)θ4 (u)Àíàëîãè÷íî ïîëó÷àåìθ400 (u + τ2 )θ100 (u)2.=−π−2iπζ(u)+1θ4 (u + τ2 )θ4 (u)Òåì æå ñïîñîáîìθ20τ= B(u)[−iπθ3 (u) + θ30 (u)],u+2òîãäà−2πθ42 (0) B 2 (u)θ (u)θ (u)(−iπ)θ3 (u + τ2 ) θ20 (u + τ2 )232−ττ = −2πθ4 (0)θ1 (u + 2 ) θ4 (u + 2 )i2 B 2 (u)θ1 (u)θ4 (u)B 2 (u)θ2 (u)θ30 (u) θ2 (u)θ30 (u)θ2 (u)θ3 (u)2 22− 2 2= −2iπ θ4 (0)+ 2πθ4 (0).i B (u)θ1 (u)θ4 (u)θ1 (u)θ4 (u)θ1 (u)θ4 (u)Ïîëó÷àåì, ÷òî70τθ400 (u)θ100 (u)22g u+= −π − 2iπζ4 (u) ++ π + 2iπζ1 (u) −−2θ4 (u)θ4 (u)θ2 (u)θ3 (u)−2iπ 2 θ42 (0)θ1 (u)θ4 (u)+θ2 (u)θ30 (u)22πθ4 (0)θ1 (u)θ4 (u)− π 2 θ42 (0) =θ400 (u) θ100 (u)θ2 (u)θ3 (u)θ2 (u)θ30 (u) 2 22 22=−+2iπ (ζ1 (u) − ζ4 (u))−2iπ θ4 (0)+2πθ4 (0)−π θ4 (0) =θ4 (u) θ1 (u)θ1 (u)θ4 (u)θ1 (u)θ4 (u)θ2 (u)θ3 (u)θ2 (u)θ3 (u)θ2 (u)θ30 (u) 2 2θ400 (u) θ100 (u)2 22 22−+2iπ θ4 (0)−2iπ θ4 (0)+2πθ4 (0)−π θ4 (0) ==θ4 (u) θ1 (u)θ1 (u)θ4 (u)θ1 (u)θ4 (u)θ1 (u)θ4 (u)θ400 (u) θ100 (u)θ2 (u)θ30 (u)2=−+ 2πθ4 (0)− π 2 θ42 (0).θ4 (u) θ1 (u)θ1 (u)θ4 (u)Òàêèì îáðàçîì, ìû ïîëó÷èëè, ÷òî íóëè â çíàìåíàòåëå ó âòîðîãî èòðåòüåãî ÷ëåíîâ ñîêðàùàþòñÿ ñ íóëÿìè èõ ÷èñëèòåëåé.

Ïîëó÷èëè, ÷òîg( τ2 ) ðåãóëÿðíà.Íàêîíåö, ïîêàæåì, ÷òî g( 1+τ2 ) ðàâíÿåòñÿ íóëþ. Äëÿ ýòîãî ðàññìîòðèì:θ100 (u + 1+τθ300 (u)22 )= −π − 2iπζ3 (u) +,θ(u)θ1 (u + 1+τ)32θ400 (u + τ2 )θ200 (u)2= −π − 2iπζ2 (u) +θ4 (u + τ2 )θ2 (u)Ïîäñòàâèì èõ â âûðàæåíèå äëÿ g (AII.4):θ100 (u + 1+τθ400 (u + 1+τ1+τ2 42 )2 )g u+=−1+τ1+τ − π θ4 (0) =2θ1 (u + 2 ) θ4 (u + 2 )= 2iπ(ζ2 (u) − ζ3 (u)) +θ300 (u) θ200 (u)−− π 2 θ44 (0).θ3 (u) θ2 (u)Òåïåðü ïîëîæèì u = 0, ïîëó÷èìg1+τ2= 2iπ(ζ2 (0) − ζ3 (0)) + π 2 θ44 (0) − π 2 θ44 (0) = 0. ðåçóëüòàòå ìû ïîëó÷èëè, ÷òî g(u) ≡ 0, à ýòî, â ñâîþ î÷åðåäü, ýêâèâàëåíòíî (AII.1).717.3Ïðèëîæåíèå III. Äîêàçàòåëüñòâî îñíîâíîãî òîæäåñòâàÇäåñü ìû äîêàæåì îñíîâíîå òîæäåñòâî, êîòîðîå ïîçâîëÿåò âûâåñòè ýëëèïòè÷åñêîå óðàâíåíèå Ëåâíåðà èç (111).

Ïîëîæèìϕ(x1 , x2 ) := −ζ1 (x1 ) − ζ4 (x1 ) + ζ1 (x2 ) + ζ4 (x2 ) + 2ζ2 (x1 − x2 ).Ðàññìîòðèì âûðàæåíèåS 0 (x1 − x2 )ϕ(x1 , x2 ) + π 2 θ44 (0) = S 0 (x1 )S 0 (x2 ).(AIII.1)×òîáû äîêàçàòü åãî, çàìåòèì, ÷òî ϕ(x1 , x2 ) äîïóñêàåò ñëåäóþùåå ïðåäñòàâëåíèå:θ1 (x1 −x2 ) θ4 (x1 −x2 ) θ2 (x1 +x2 ).θ1 (x1 )θ4 (x1 )θ1 (x2 )θ4 (x2 )θ2 (x1 −x2 )(AIII.2)Èñïîëüçóåì ñòàíäàðòíûé ìåòîä äîêàçàòåëüñòâà â òåîðèè ýëëèïòè÷åñêèõôóíêöèé.

Äëÿ êðàòêîñòè áóäåì èñïîëüçîâàòü îáîçíà÷åíèå θa (0) = θa .Ìû äîëæíû ïðîâåðèòü:a) îáå ñòîðîíû äâàæäû ïåðèîäè÷íû êàê ôóíêöèè x1 ñ ïåðèîäàìè 1 è τϕ(x1 , x2 ) = πθ2 (0)θ3 (0)θ42 (0)ϕ(x1 + 1, x2 ) = πθ2 θ3 θ42θ1 (x1 + 1−x2 ) θ4 (x1 + 1−x2 ) θ2 (x1 + 1+x2 ).θ1 (x1 + 1)θ4 (x1 + 1)θ1 (x2 )θ4 (x2 )θ2 (x1 + 1−x2 )Òàê êàê θ1,2 (u + 1) = −θ1,2 (u) è θ3,4 (u + 1) = θ3,4 (u), ñðàçó ïîëó÷àåì, ÷òîϕ(x1 + 1, x2 ) = ϕ(x1 , x2 ).Àíàëîãè÷íî ñ τϕ(x1 + τ, x2 ) = πθ2 θ3 θ42= πθ2 θ3 θ42θ1 (x1 + τ −x2 ) θ4 (x1 + τ −x2 ) θ2 (x1 + τ +x2 )=θ1 (x1 + τ )θ4 (x1 + τ )θ1 (x2 )θ4 (x2 )θ2 (x1 + τ −x2 )−A(x1 )θ1 (x1 −x2 ) (−A(x1 ))θ4 (x1 −x2 ) A(x1 )θ2 (x1 +x2 ).−A(x1 )θ1 (x1 )(−A(x1 ))θ4 (x1 )θ1 (x2 )θ4 (x2 )A(x1 )θ2 (x1 −x2 )ϕ(x1 + τ, x2 ) = ϕ(x1 , x2 ).Äëÿ êðàòêîñòè ìû èñïîëüçîâàëè çäåñü îáîçíà÷åíèå A(u) = e−πi(2x1 +τ ) .á) îáå ñòîðîíû èìåþò îäèíàêîâûå íóëè è ïîëþñû.72Ïðîâåðèì íóëè.Íóëü x1 − x2 = 0, ò.å.

x1 = x2 = xϕ(x, x) = −ζ1 (x) − ζ4 (x) + ζ1 (x) + ζ4 (x) + 2ζ2 (0) = 0.Íóëü x1 − x2 = τ2 , ò.å. x1 = x + τ2 , ïîëó÷èì:τ τττ − ζ4 x ++ ζ1 (x) + ζ4 (x) + 2ζ2=ϕ x + , x = −ζ1 x +2222= −ζ4 (x) + iπ − ζ1 (x) + iπ + ζ1 (x) + ζ4 (x) − 2iπ = 0.Íóëü x1 + x2 = 21 , ïîäñòàâèì x1 = −x +12111ϕ −x + , x = −ζ1 −x +− ζ4 −x ++ ζ1 (x) + ζ4 (x)+2221= −ζ2 (−x) − ζ3 (−x) + ζ1 (x) + ζ4 (x)++2ζ2 −2x +2+2ζ1 (−2x) = ζ2 (x) + ζ3 (x) + ζ1 (x) + ζ4 (x) − 2ζ1 (2x).Çäåñü íàì íàäî âîñïîëüçîâàòüñÿ èçâåñòíûì òîæäåñòâîìθ1 (2u)θ2 (0)θ3 (0)θ4 (0) = 2θ1 (u)θ1 (u)θ1 (u)θ1 (u).Èç íåãî âûâîäèì:10θ10 (2x)2 (2θ1 (x)θ1 (x)θ1 (x)θ1 (x))==θ1 (2x)2θ1 (x)θ1 (x)θ1 (x)θ1 (x)1= [ζ1 (x) + ζ2 (x) + ζ3 (x) + ζ4 (x)].2Òåïåðü ìû âèäèì, ÷òî âûðàæåíèå îáðàùàåòñÿ â íóëü:ζ2 (x) + ζ3 (x) + ζ1 (x) + ζ4 (x) − 2ζ1 (2x) =1= ζ2 (x) + ζ3 (x) + ζ1 (x) + ζ4 (x) − 2 [ζ1 (x) + ζ2 (x) + ζ1 (3) + ζ4 (x)] = 0.2Ñëåäîâàòåëüíî, íóëè ñîâïàäàþò.

Òî, ÷òî ïîëþñû ñîâïàäàþò âèäíî íåâîîðóæåííûì ãëàçîì, è îíè îòëè÷àþòñÿ íà x1 -íåçàâèñèìûé, ðàâíûé 1 ôàêòîð.Òåïåðü ïîäñòàâèì ÿâíûé âèä S 0 (x) èç (AII.2) â ëåâóþ ñòîðîíó(AIII.1) è äîêàæåì åãî ïðÿìûì âû÷èñëåíèåì.S 0 (x1 − x2 )ϕ(x1 , x2 ) + π 2 θ44 (0) =73θ2 (x1 − x2 ) θ3 (x1 − x2 )θ1 (x1 −x2 ) θ4 (x1 −x2 ) θ2 (x1 +x2 )πθ2 (0)θ3 (0)θ42 (0)+θ1 (x1 − x2 ) θ4 (x1 − x2 )θ1 (x1 )θ4 (x1 )θ1 (x2 )θ4 (x2 )θ2 (x1 −x2 )θ3 (x1 − x2 )θ2 (x1 +x2 )+ π 2 θ44 (0) =+π 2 θ44 (0) = π 2 θ2 (0)θ3 (0)θ44 (0)θ1 (x1 )θ4 (x1 )θ1 (x2 )θ4 (x2 )θ2 (0)θ3 (0)θ3 (x1 − x2 )θ2 (x1 +x2 )2 4= π θ4 (0)+1 =θ1 (x1 )θ4 (x1 )θ1 (x2 )θ4 (x2 )θ2 (x1 )θ3 (x1 ) 2 θ2 (x2 )θ3 (x2 )θ2 (x1 )θ3 (x1 )θ2 (x2 )θ3 (x2 )== πθ42 (0)πθ (0)== π 2 θ44 (0)θ1 (x1 )θ4 (x1 )θ1 (x2 )θ4 (x2 )θ1 (x1 )θ4 (x1 ) 4 θ1 (x2 )θ4 (x2 )= πθ42 (0)S 0 (x1 )S 0 (x2 ). äîêàçàòåëüñòâå ìû èñïîëüçîâàëè òîæäåñòâîθ2 (y + z)θ3 (y − z)θ2 θ3 = θ2 (y)θ3 (y)θ2 (z)θ3 (z) − θ1 (y)θ4 (y)θ1 (z)θ4 (z).Äîêàçàòü åãî, åñòåcòâåííî, ìîæíî áûëî áû è àíàëèòè÷åñêèì ìåòîäîì.7.4Ïðèëîæåíèå IV.

Âûâîä ýëëèïòè÷åñêîãî óðàâíåíèÿ Ëåâíåðà èç (111)Ìû äîëæíû ïîêàçàòü, ÷òî çàìåíà (112), 4πi ∂τ u = −ζ1 (u + ξ) − ζ4 (u + ξ) + ζ1 (ξ) + ζ4 (ξ),02(AIV.1)4πi ∂τ log R = (S (ξ)) ,ïåðåâîäèò (111) â òîæäåñòâî. Äåéñòâèòåëüíî, ïîñëå ïîäñòàíîâêè óðàâíåíèå (111) ïðèîáðåòàåò âèäπ 2 θ44 (0) 0π 2 θ44 (0)S (u2 ) ϕ(u2 +ξ, ξ) + 0=S (u1 ) ϕ(u1 +ξ, ξ) + 0S (u1 )S (u2 )2 4πθ(0)= (S 0 (ξ))2 S 0 (u1 −u2 ) ϕ(u1 +ξ, u2 +ξ) + 0 4.S (u1 −u2 )Îñòàåòñÿ ïðèìåíèòü òîæäåñòâî (AIII.1) äëÿ (x1 , x2 ) = (u1 +ξ, ξ),(x1 , x2 ) = (u2 +ξ, ξ) è (x1 , x2 ) = (u1 +ξ, u2 +ξ).0Äàëåå ìû äîëæíû ïðîâåðèòü, ÷òî óðàâíåíèå (115),4πi ∂τ log R = (S 0 (ξ(τ )))2 ,ÿâëÿåòñÿ ïðåäåëüíûì ñëó÷àåì (113) ïðè z → ∞. Ïðè ñòðåìëåíèè z êáåñêîíå÷íîñòè, u ñòðåìèòñÿ ê íóëþ. Òîãäà ðàçäåëèì îáå ÷àñòè íà u4πi∂τ uζ1 (u + ξ) − ζ1 (ξ) ζ4 (u + ξ) − ζ4 (ξ)=−−,uuu74Ïîäñòàâëÿÿ (72) è ñðàâíèâàÿ ëèäèðóþùèå ÷ëåíû, ïîëó÷èì:4πi ∂τ log c1 = −ζ1 0 (ξ(τ )) − ζ4 0 (ξ(τ )),ãäå ζa (u) = ∂u ζa (u).

Íàïîìíèì, ÷òî log R = log(πc1 ) + log θ2 θ30(ñì.(70), (76)), òîãäà4πi ∂τ log R = −ζ1 (ξ) − ζ4 (ξ) + 4πi ∂τ log θ2 (0)θ3 (0) .00Ïîñëåäíèé ÷ëåí ìîæíî ïðåîáðàçîâàòü ñ ïîìîùüþ óðàâíåíèÿ òåïëîïðîâîäíîñòè (AI.9) äëÿ òýòà-ôóíêöèé. À èìåííî:∂τ θa (u) ∂u2 θa (u)=.4πiθa (u)θa (u)Òàê êàê∂uïåðåïèøåì∂u θa (u)θa (u)∂u2 θa (u)=−θa (u)4πi ∂τ log θa (u) = ∂u∂u θa (u)θa (u)∂u θa (u)θa (u)+2,∂u θa (u)θa (u)2.Ó÷èòûâàÿ, ÷òî θ20 (0) = θ30 (0) = 0, èìååì:4πi ∂τ log R ==− ∂x2 logθ1 (x|τ ) θ4 (x|τ ) − ∂x2 log θ1 (x| τ2 )+x=ξ+x=ξ∂x2 logθ2 (x|τ ) θ3 (x|τ ) ∂x2 log θ1 (x| τ2 )x= 12.Ìû èñïîëüçîâàëè õîðîøî èçâåñòíûå òîæäåñòâà:2θ1 (u|τ )θ4 (u|τ ) = θ2 (0| τ2 ) θ1 (u| τ2 ),2θ2 (u|τ )θ3 (u|τ ) = θ2 (0| τ2 ) θ2 (u| τ2 ).Âèäèì, ÷òî ðàâåíñòâî, êîòîðîå ìû ñîáèðàåìñÿ äîêàçàòü, ò.å.,024πi ∂τ log R = (S (ξ)) =π 2 θ44 (0|τ )θ22 (ξ| τ2 ),θ12 (ξ| τ2 )ýêâèâàëåíòíî ðàâåíñòâó− ∂x2 log θ1 (x| τ2 )+∂x2 log θ1 (x| τ2 )75x= 21=π 2 θ44 (0|τ )θ22 (x| τ2 ).θ12 (x| τ2 )=x=0Ïîñäëåäíåå äîêàçûâàåòñÿ ñòàíäàðòíûìè ðàññóæäåíèÿìè, êîòîðûå ìûìíîãî ðàç èñïîëüçîâàëè â äàííîé ðàáîòå.