CLASSES OF FINITE ORDER FORMAL SOLUTIONS OF AN ORDINARY DIFFERENTIAL EQUATION

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64 È. Â. ÃÎÐÞ ÊÈÍÀ ÅÁÛØÅÂÑÊÈÉ ÑÁÎÐÍÈÊ Òîì 17 Âûïóñê 2 ÓÄÊ 517.9 ÊËÀÑÑÛ ÔÎÐÌÀËÜÍÛÕ ÐÅØÅÍÈÉ ÊÎÍÅ ÍÎÃÎ ÏÎÐßÄÊÀ ÎÁÛÊÍÎÂÅÍÍÎÃÎ ÄÈÔÔÅÐÅÍÖÈÀËÜÍÎÃÎ ÓÐÀÂÍÅÍÈß È. Â. Ãîðþ êèíà (ã. Ìîñêâà) Àííîòàöèÿ  ýòîé ðàáîòå âûäåëÿþòñÿ îáùèå êëàññû ôîðìàëüíûõ ðåøåíèé êîíå íîãî ïîðÿäêà àëãåáðàè åñêîãî (ïîëèíîìèàëüíîãî) îáûêíîâåííîãî äèôôåðåíöèàëüíîãî óðàâíåíèÿ (ÎÄÓ), êîòîðûå ìîãóò áûòü âû èñëåíû ñ ïîìîùüþ ìåòîäîâ ïëîñêîé ñòåïåííîé ãåîìåòðèè, îñíîâàííûõ íà îïðåäåëåíèè âåäóùèõ ëåíîâ óðàâíåíèÿ ïî ìíîãîóãîëüíèêó ÍüþòîíàÁðþíî, êîòîðûé ÿâëÿåòñÿ ìíîãîóãîëüíûì ìíîæåñòâîì íà ïëîñêîñòè. Êðîìå òîãî, â ýòîé ðàáîòå äîêàçûâàåòñÿ òåîðåìà î òîì, òî åñëè ôîðìàëüíîå ðåøåíèå âûäåëåííîãî êëàññà ñóùåñòâóåò, òî ïåðâîå ïðèáëèæåíèå (óêîðî åíèå) ýòîãî ðåøåíèÿ ÿâëÿåòñÿ (ôîðìàëüíûì) ðåøåíèåì ïåðâîãî ïðèáëèæåíèÿ èñõîäíîãî óðàâíåíèÿ (óêîðî åííîãî óðàâíåíèÿ). Âû èñëÿåìûå ñ ïîìîùüþ ýòèõ ìåòîäîâ ôîðìàëüíûå ðÿäû îòíîñÿòñÿ ê åùå áîëåå îáùèì êëàññàì ôîðìàëüíûõ ðÿäîâ, íàçûâàåìûõ â èíîñòðàííîé ëèòåðàòóðå grid-based series è transseries. Òàêèå ðÿäû ÿâëÿþòñÿ îòíîñèòåëüíî íîâûìè îáúåêòàìè è, íåñìîòðÿ íà áîëüøîå èñëî ðàáîò, ïîêà ñëàáî èçó åíû. Îíè äîñòàòî íî àñòî âñòðå àþòñÿ ñðåäè ôîðìàëüíûõ ðåøåíèé äèôôåðåíöèàëüíûõ óðàâíåíèé, â òîì èñëå, âàæíûõ â ôèçèêå. Äðóãèõ îáùèõ ìåòîäîâ âû èñëåíèÿ òàêèõ ðÿäîâ ïîêà íå ñóùåñòâóåò. Ïîýòîìó òàê âàæíî âûäåëèòü êëàññû ôîðìàëüíûõ ðÿäîâ, êîòîðûå ìîæíî âû èñëèòü àëãîðèòìè åñêè ìåòîäàìè ïëîñêîé ñòåïåííîé ãåîìåòðèè. Êëþ åâûå ñëîâà: àëãåáðàè åñêîå ÎÄÓ, ôîðìàëüíîå ðåøåíèå, âû èñëåíèå ôîðìàëüíîãî ðåøåíèÿ, êëàññèôèêàöèÿ ôîðìàëüíûõ ðåøåíèé, transseries. Áèáëèîãðàôèÿ: 22 íàçâàíèÿ. CLASSES OF FINITE ORDER FORMAL SOLUTIONS OF AN ORDINARY DIFFERENTIAL EQUATION I. V. Goryuchkina (Moscow) Abstract In this paper we select general classes of nite order formal solutions of an algebraic (polynomial) ordinary dierential equation (ODE), that can be calculated by the methods of the planar power geometry based on determining leading terms of the equation by the Newton Bruno polygon. Beside that in this paper we prove the theorem that if a formal solution of the selected class exists then the rst approximation (the truncation) of this solution is a (formal) solution of the rst approximation of the initial equation (that is called the truncated equation). Calculated formal solutions by means of these methods relate to much more general classes of the formal solutions that are called grid-based series and transseries in the foreign papers. Grid-based series and transseries are fairly new objects and in spite of the large number of publications they are slightly studied. Such series appear among formal solutions of dierential equations including equations that are important in physics. Other general methods of the calculation of such series

ÊËÀÑÑÛ ÔÎÐÌÀËÜÍÛÕ ÐÅØÅÍÈÉ ÊÎÍÅ ÍÎÃÎ ÏÎÐßÄÊÀ... 65 do not exist yet. Therefore it is important to select the classes of formal solutions that can be calculated algorithmically by the methods of the planar power geometry. Keywords: algebraic ODE, formal solution, calculation of formal solution, classication of formal solutions, transseries. Bibliography: 22 titles. Îáîçíà åíèÿ D îòêðûòîå ñâÿçíîå ïîäìíîæåñòâî ìíîæåñòâà êîìïëåêñíûõ èñåë, çàìûêàíèå êîòîðîãî ñîäåðæèò íóëü; D îòêðûòîå ñâÿçíîå ïîäìíîæåñòâî ìíîæåñòâà êîìïëåêñíûõ èñåë, çàìûêàíèå êîòîðîãî ñîäåðæèò áåñêîíå íîñòü; R ìíîæåñòâî âåùåñòâåííûõ èñåë, ïîïîëíåííîå + è ; R è C ìíîæåñòâà âåùåñòâåííûõ è êîìïëåêñíûõ èñåë áåç íóëÿ ñîîòâåòñòâåííî; Z + è R + ìíîæåñòâà íåîòðèöàòåëüíûõ öåëûõ è âåùåñòâåííûõ èñåë ñîîòâåòñòâåííî; O(D) ìíîæåñòâî ãîëîìîðôíûõ â îáëàñòè D ôóíêöèé; C[x] ìíîæåñòâî ìíîãî ëåíîâ ñ êîìïëåêñíûìè êîýôôèöèåíòàìè; C[x, 1/x] ìíîæåñòâî ìíîãî ëåíîâ Ëîðàíà ñ êîìïëåêñíûìè êîýôôèöèåíòàìè; C[[x]] ìíîæåñòâî ôîðìàëüíûõ ðÿäîâ Òåéëîðà ñ êîìïëåêñíûìè êîýôôèöèåíòàìè â îêðåñòíîñòè òî êè x = 0; C(x) ìíîæåñòâî ðàöèîíàëüíûõ ôóíêöèé ñ êîìïëåêñíûìè êîýôôèöèåíòàìè; C((x)) ìíîæåñòâî ôîðìàëüíûõ ðÿäîâ Ëîðàíà ñ êîíå íîé ãëàâíîé àñòüþ è êîìïëåêñíûìè êîýôôèöèåíòàìè â îêðåñòíîñòè òî êè x = 0; i ìíèìàÿ åäèíèöà; O (ϕ(x)) ôóíêöèÿ, ïî ìîäóëþ íå ïðåâîñõîäÿùàÿ ïðîèçâåäåíèÿ ïîëîæèòåëüíîé êîíñòàíòû è ìîäóëÿ ôóíêöèè ϕ(x), íî íå ÿâëÿþùàÿñÿ áåñêîíå íî ìàëîé âåëè èíîé ïî ñðàâíåíèþ ñ ϕ(x) âî âñåõ ïðåäåëüíûõ òî êàõ ðàññìàòðèâàåìîé îáëàñòè (â îòëè èå îò ñòàíäàðòíîãî O (ϕ(x)), êîòîðîå äîïóñêàåò o (ϕ(x))). Ïîäðîáíåå îá îáîçíà åíèÿõ O è o ñì. â ãë. 1 [1]. 1. Ââåäåíèå Ïðåäëîæåííûå â ðàáîòå [2] ìåòîäû ïëîñêîé ñòåïåííîé ãåîìåòðèè ïîçâîëÿþò äëÿ àíàëèòè åñêèõ îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé âû èñëÿòü èõ ôîðìàëüíûå ðåøåíèÿ (èëè ôîðìàëüíûå àñèìïòîòèêè ðåøåíèé), èìåþùèå âèä ôóíêöèîíàëüíûõ ðÿäîâ, ñîäåðæàùèõ ñòåïåííûå ôóíêöèè ñ êîìïëåêñíûìè ïîêàçàòåëÿìè ñòåïåíè, êðàòíûå è íåêðàòíûå ëîãàðèôìû. Íàïîìíèì, òî óðàâíåíèå âèäà f(x, y 0,..., y n ) = 0, y j = x j y (j), (1) ãäå f(x, y 0,..., y n ) ýòî àíàëèòè åñêàÿ ôóíêöèÿ â íåêîòîðîé îáëàñòè U C n+2, íàçûâàåòñÿ (êîìïëåêñíûì) àíàëèòè åñêèì îáûêíîâåííûì äèôôåðåíöèàëüíûì óðàâíåíèåì. àñòíûì ñëó- àåì àíàëèòè åñêèõ îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé ÿâëÿþòñÿ àëãåáðàè åñêèå îáûêíîâåííûå äèôôåðåíöèàëüíûå óðàâíåíèÿ. (Êîìïëåêñíîå) àëãåáðàè åñêîå îáûêíîâåííîå äèôôåðåíöèàëüíîå óðàâíåíèå ýòî óðàâíåíèå âèäà (1), ãäå f(x, y 0,..., y n ) C[x, y 0,..., y n ], ò. å. f(x, y 0,..., y n ) ýòî ïîëèíîì ñâîèõ àðãóìåíòîâ c ïîñòîÿííûìè êîìïëåêñíûìè êîýôôèöèåíòàìè. Äàëåå â ðàáîòå áóäóò ðàññìàòðèâàòüñÿ òîëüêî àëãåáðàè åñêèå îáûêíîâåííûå äèôôåðåíöèàëüíûå óðàâíåíèÿ è èõ ôîðìàëüíûå ðåøåíèÿ. Îòìåòèì, òî ïîä âû èñëåíèåì ôîðìàëüíîãî ðåøåíèÿ, èìåþùåãî âèä ôóíêöèîíàëüíîãî ðÿäà, îáû íî ïîíèìàåòñÿ âû èñëåíèå êîýôôèöèåíòîâ íà àëüíîé àñòè ðÿäà, îïèñàíèå åãî ñòðóêòóðû (ñèñòåìû ôóíêöèé), à òàêæå

66 È. Â. ÃÎÐÞ ÊÈÍÀ îïðåäåëåíèå êîëè åñòâà ïðîèçâîëüíûõ ïîñòîÿííûõ è ëåíîâ, â êîòîðûõ ýòè ïðîèçâîëüíûå ïîñòîÿííûå ñîäåðæàòñÿ. Íàïîìíèì, òî ôîðìàëüíûé ôóíêöèîíàëüíûé ðÿä ñ óïîðÿäî åííûìè ëåíàìè ÿâëÿåòñÿ ôîðìàëüíûì ðåøåíèåì îáûêíîâåííîãî äèôôåðåíöèàëüíîãî óðàâíåíèÿ, åñëè ïîñëå ïîäñòàíîâêè åãî â óðàâíåíèå, èñïîëüçóÿ ôîðìàëüíûå ïðàâèëà óìíîæåíèÿ, ñëîæåíèÿ è äèôôåðåíöèðîâàíèÿ ðÿäîâ, ïîëó àåòñÿ ôîðìàëüíûé ðÿä ñ íóëåâûìè êîýôôèöèåíòàìè, ê êîòîðîìó ïðèìåíèìû òå æå ñàìûå ïðàâèëà óïîðÿäî èâàíèÿ, òî è äëÿ èñõîäíîãî ðÿäà. Â ïðîöåññå ðàçâèòèÿ ìåòîäîâ ïëîñêîé ñòåïåíè ãåîìåòðèè ïîÿâèëàñü ñâÿçàííàÿ ñ âû èñëåíèÿìè êëàññèôèêàöèÿ ôîðìàëüíûõ ðåøåíèé. Òàê, äëÿ øåñòîãî óðàâíåíèÿ Ïåíëåâå ñ ïîìîùüþ ìåòîäîâ ïëîñêîé ñòåïåííîé ãåîìåòðèè [3] áûëè íàéäåíû (â óêàçàííîì â ïðåäûäóùåì àáçàöå ñìûñëå) âñå ôîðìàëüíûå ðåøåíèÿ â îêðåñòíîñòÿõ îñîáûõ è íåîñîáûõ òî åê ïðè âñåõ çíà åíèÿõ ïàðàìåòðîâ óðàâíåíèÿ, óñëîâíî ðàçäåëåííûå íà ïÿòü òèïîâ: ñòåïåííûå, ñòåïåííî-ëîãàðèôìè- åñêèå, ñëîæíûå, ýêçîòè åñêèå è ïîëóýêçîòè åñêèå. Âñå ôîðìàëüíûå ðåøåíèÿ îáûêíîâåííîãî äèôôåðåíöèàëüíîãî óðàâíåíèÿ ýòèõ ïÿòè òèïîâ â êàæäîé îñîáîé òî êå ìîæíî çàïèñàòü åäèíûì âûðàæåíèåì. Íàïðèìåð, âñå ôîðìàëüíûå ðåøåíèÿ øåñòîãî óðàâíåíèÿ Ïåíëåâå ýòèõ ïÿòè òèïîâ â îêðåñòíîñòè òî êè x = 0 ìîæíî çàïèñàòü â âèäå k=0 s c k k (x) x, (2) ãäå s k C, Re s k < Re s k+1, lim Re s k = +, à êîýôôèöèåíòû c k (x) ðàçëè àþòñÿ ñîãëàñíî k òèïàì: c 0 (x) C, îñòàëüíûå c k (x) C[x αi, x αi ], α R (ñòåïåííûå ðàçëîæåíèÿ), c 0 (x) C, îñòàëüíûå c k (x) C[ln x] (ñòåïåííî-ëîãàðèôìè åñêèå ðàçëîæåíèÿ), âñå c k (x) C((ln 1 x)) (ñëîæíûå ðàçëîæåíèÿ), c βi βi 0 (x) C[x, x ], îñòàëüíûå c βi k (x) C(x ) C((x βi )), β R (ïîëóýêçîòè åñêèå ðàçëîæåíèÿ), âñå c k (x) C(x γi ) C((x γi )), γ R (ýêçîòè åñêèå ðàçëîæåíèÿ). Íà ñàìîì äåëå, ïîêàçàòåëè ñòåïåíè s k öåëûå äëÿ âñåõ ôîðìàëüíûõ ðåøåíèé âèäà (2) øåñòîãî óðàâíåíèÿ Ïåíëåâå, êðîìå ñòåïåííûõ ðàçëîæåíèé. Åùå ðàç íàïîìíèì, òî ñîãëàñíî ââåäåííûì â íà àëå ñòàòüè îáîçíà åíèÿì C((x βi )) ýòî ìíîæåñòâî ôîðìàëüíûõ ðÿäîâ Ëîðàíà ñ êîíå íîé ãëàâíîé àñòüþ â òî êå x βi = 0. Ïðè ýòîì èñëî β îäèíàêîâîå äëÿ âñåõ c k (x), ò. å. êàæäûé êîýôôèöèåíò c k (x) ýòî ñòåïåííîé ðÿä ïî âîçðàñòàþùèì ñòåïåíÿì x βi, ãäå β ìîæåò áûòü êàê ïîëîæèòåëüíûì, òàê è îòðèöàòåëüíûì âåùåñòâåííûì èñëîì. Îòìåòèì, òî ñðåäè ôîðìàëüíûõ ðåøåíèé êîíå íîãî ïîðÿäêà îñòàëüíûõ ïÿòè óðàâíåíèé Ïåíëåâå òàêæå èìåþòñÿ òîëüêî ðàçëîæåíèÿ ïÿòè âûøåîïèñàííûõ òèïîâ (ñì., íàïðèìåð, [4] [8]). Íî, â îòëè èå îò øåñòîãî óðàâíåíèÿ Ïåíëåâå, äëÿ íèõ èìåþòñÿ åùå ôîðìàëüíûå ðåøåíèÿ â âèäå ýêñïîíåíöèàëüíûõ ðàçëîæåíèé c e φ(x) + k=2 b k(x) (c e φ(x) ) k, ãäå c C, b k (x) C((x)), φ(x) C[ln x]((x)) (ñì. [9]). Ýòè ðàçëîæåíèÿ â ýòîé ðàáîòå íå ðàññìàòðèâàþòñÿ, ïîñêîëüêó èìåþò áåñêîíå íûé ïîðÿäîê ëåíîâ. Â ýòîé ðàáîòå ðàññìàòðèâàþòñÿ âñåâîçìîæíûå òèïû ôîðìàëüíûõ ðåøåíèé êîíå íîãî ïîðÿäêà àëãåáðàè åñêîãî îáûêíîâåííîãî äèôôåðåíöèàëüíîãî óðàâíåíèÿ, êîòîðûå ìîãóò áûòü ïîëó åíû ñ ïîìîùüþ ìåòîäîâ ïëîñêîé ñòåïåííîé ãåîìåòðèè. Â ðàçäåëå 2 ñòðîèòñÿ êëàññ òàêèõ ôîðìàëüíûõ ðåøåíèé. Â àñòíîñòè, â ýòîò êëàññ âõîäÿò ðÿäû ñ êîýôôèöèåíòàìè c k (x), êîòîðûå ñîäåðæàò êðàòíûå ëîãàðèôìû è ðàöèîíàëüíûå ôóíêöèè îò ñòåïåííûõ ôóíêöèé ñ êîìïëåêñíûìè ïîêàçàòåëÿìè ñòåïåíè.

ÊËÀÑÑÛ ÔÎÐÌÀËÜÍÛÕ ÐÅØÅÍÈÉ ÊÎÍÅ ÍÎÃÎ ÏÎÐßÄÊÀ... 67 Íàïîìíèì, òî ìåòîäû ïëîñêîé ñòåïåííîé ãåîìåòðèè îñíîâûâàþòñÿ íà ìåòîäå âûäåëåíèÿ âåäóùèõ ëåíîâ óðàâíåíèÿ ïî ìíîãîóãîëüíèêó ÍüþòîíàÁðþíî. Ìíîãîóãîëüíèê Íüþòîíà Áðþíî àíàëèòè åñêîãî îáûêíîâåííîãî äèôôåðåíöèàëüíîãî óðàâíåíèÿ ýòî ìíîãîóãîëüíîå ìíîæåñòâî íà ïëîñêîñòè, ãðàíèöà êîòîðîãî ñîñòîèò èç âåðøèí è ðåáåð. Êàæäàÿ âåðøèíà èëè ðåáðî ñîîòâåòñòâóþò îäíîìó, íåñêîëüêèì èëè ìíîãèì ñëàãàåìûì â èñõîäíîì îáûêíîâåííîì äèôôåðåíöèàëüíîì óðàâíåíèè. Ýòè ñëàãàåìûå ÿâëÿþòñÿ âåäóùèìè â îïðåäåëåííûõ îáëàñòÿõ íåçàâèñèìîé è çàâèñèìîé ïåðåìåííûõ è îáðàçóþò óêîðî åííûå óðàâíåíèÿ (ïðèáëèæåííûå óðàâíåíèÿ). óòü ïîäðîáíåé î ìåòîäå âûäåëåíèÿ óêîðî åííûõ óðàâíåíèé íàïèñàíî â ðàçäåëå 4. Áîëåå ïîäðîáíî ñì. [2] è [3, ãë. 1]. Äî ñèõ ïîð ñ èòàëîñü ñàìî ñîáîé ðàçóìåþùèìñÿ, òî ïåðâîå ïðèáëèæåíèå ôîðìàëüíîãî ðåøåíèÿ (2) (ïåðâûé ëåí c 0 (x)x s 0 ðÿäà (2)) ÿâëÿåòñÿ ðåøåíèåì óêîðî åííîãî óðàâíåíèÿ. Ýòî î åâèäíî äëÿ ñòåïåííûõ ðàçëîæåíèé ñ âåùåñòâåííûìè ïîêàçàòåëÿìè ñòåïåíè [2], íî òðåáóåò äîêàçàòåëüñòâà äëÿ îñòàëüíûõ òèïîâ, ïåðå èñëåííûõ ðàíåå. Äîêàçàòåëüñòâî òàêîãî óòâåðæäåíèÿ (òåîðåìà 1 èç ðàçäåëà 4) è ÿâëÿåòñÿ îñíîâíûì ðåçóëüòàòîì ñòàòüè.  ðàçäåëàõ 2 è 3 ïðåäúÿâëåíû ïîñòðîåíèÿ îáùèõ çàìêíóòûõ êëàññîâ ôîðìàëüíûõ ðåøåíèé êîíå íîãî ïîðÿäêà, ê êîòîðûì îòíîñÿòñÿ óêàçàííûå âûøå 5 òèïîâ (ëåììà 1 èç ðàçäåëà 3). Òåîðåìà 1 ðàçäåëà 4 äîêàçûâàåòñÿ äëÿ îáùèõ çàìêíóòûõ êëàññîâ ôîðìàëüíûõ ðåøåíèé êîíå íîãî ïîðÿäêà, ïîñòðîåííûõ â ðàçäåëå 2. Òàêæå ñòîèò ñêàçàòü, òî ïîêà íåèçâåñòíû äðóãèå ïðåäñòàâèòåëè ýòèõ îáùèõ êëàññîâ, êðîìå ôîðìàëüíûõ ðÿäîâ ýòèõ ïÿòè òèïîâ. Îòìåòèì, òî êëþ åâûì ïîíÿòèåì ïðè ïîñòðîåíèè ìíîãîóãîëüíèêà ÍüþòîíàÁðþíî ÿâëÿåòñÿ ïîíÿòèå ïîðÿäêà ôóíêöèè. Áóäåì ãîâîðèòü, òî ôóíêöèÿ ϕ(ξ) â òî êå ξ = 0 èìååò ïîðÿäîê ord 0 ϕ(ξ) R, åñëè ñóùåñòâóåò ïðåäåë ln ϕ(ξ) lim ξ 0 ln ξ ξ D = ord 0 ϕ(ξ). (3) Ïîäîáíûì îáðàçîì îïðåäåëèì ïîðÿäîê ôóíêöèè ϕ(ξ) â òî êå ξ =. Ôóíêöèÿ ϕ(ξ) â òî êå ξ = èìååò ïîðÿäîê ord ϕ(ξ) R, åñëè ñóùåñòâóåò ïðåäåë ln ϕ(ξ) lim = ord ϕ(ξ). (4) ln ξ ξ ξ D Ïîäîáíîå îïðåäåëåíèå ïîðÿäêà ôóíêöèè âñòðå àåòñÿ â òåîðèè öåëûõ ôóíêöèé, ñì., íàïðèìåð, ãë. V â [11]. Îòìåòèì, òî ñðåäè ôóíêöèé, ñîäåðæàùèõñÿ â ôîðìàëüíûõ ðåøåíèÿõ êîíå íîãî ïîðÿäêà óðàâíåíèé Ïåíëåâå, èìåþòñÿ ñòåïåííûå ôóíêöèè c êîìïëåêñíûìè ïîêàçàòåëÿìè ñòåïåíè è ëîãàðèôìû. Ëþáàÿ èç ýòèõ ôóíêöèé îáëàäàåò òåì ñâîéñòâîì, òî åå ïîðÿäêè â íóëå è áåñêîíå íîñòè ñîâïàäàþò. Áîëåå òîãî, åñëè ïðîäèôôåðåíöèðîâàòü ëþáóþ èç ýòèõ ôóíêöèé n ðàç, òî ïîëó àåòñÿ ôóíêöèÿ, ïîðÿäêè êîòîðîé â íóëå è â áåñêîíå íîñòè òàêæå ñîâïàäàþò (êîíå íî, åñëè òîëüêî ýòà ôóíêöèÿ íå ðàâíà òîæäåñòâåííî íóëþ), íî ìåíüøå íà n åäèíèö ïîðÿäêîâ â íóëå è â áåñêîíå íîñòè èñõîäíîé ôóíêöèè. Íàïðèìåð, ðàññìîòðèì ñòåïåííóþ ôóíêöèþ x α è åå n-óþ ïðîèçâîäíóþ α(α 1)... (α n + 1)x α n, α C \ Z. Äëÿ íèõ èìååì ñîîòíîøåíèÿ è α ord 0 x α = ord x = Re α α n ) ord 0 (α(α 1)... (α n + 1)x = ord (α(α 1)... (α n + 1)x = Re α n. Çàäà à ýòîé ðàáîòû: âûäåëèòü îáùèå êëàññû ôîðìàëüíûõ ðåøåíèé êîíå íîãî ïîðÿäêà àëãåáðàè åñêèõ îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé, êîòîðûå ìîãóò áûòü âû èñëåíû ñ ïîìîùüþ ìåòîäîâ ïëîñêîé ñòåïåííîé ãåîìåòðèè. α n )

68 È. Â. ÃÎÐÞ ÊÈÍÀ Ýòî çíà èò, òî íåîáõîäèìî âûäåëèòü êëàññû ôîðìàëüíûõ ôóíêöèîíàëüíûõ ðÿäîâ, êîòîðûå ñîäåðæàò òîëüêî òå ðÿäû, óêîðî åíèÿ ϕˆ (ïåðâûå ñëàãàåìûå) êîòîðûõ ÿâëÿþòñÿ òàêèìè ôîðìàëüíûìè ðÿäàìè êîìïëåêñíîé ïåðåìåííîé ξ, òî ïîðÿäêè â òî êàõ ξ = 0 è ξ = êàæäîãî èõ ëåíà ñîâïàäàþò ñ ïîðÿäêîì èñõîäíîãî ðÿäà (ïîíÿòèå ïîðÿäêà ôîðìàëüíîãî ðÿäà áóäåò äàíî â ðàçäåëå 2). Êðîìå òîãî, ïîñëå ïîäñòàíîâêè óêîðî åíèÿ ϕˆ â îäíîðîäíîå îáûêíîâåííîå äèôôåðåíöèàëüíîå óðàâíåíèå P 0 (u 0,..., u n ) = 0 (ãäå u j = x j u (j), P 0 (u 0,..., u n ) C[u 0,..., u n ]) ïîëó àåòñÿ ôîðìàëüíûé ðÿä P 0 ( ˆϕ 0,..., ˆn) ϕ (ϕˆj = x j ϕˆ(j) ) òîãî æå êëàññà, òî è èñõîäíûé ðÿä, èìåþùèé ë è á î íåíóëåâûå êîýôôèöèåíòû (òîãäà ïîðÿäêè ôóíêöèé ïðè ýòèõ êîýôôèöèåíòàõ â òî êàõ ξ = 0 è ξ = ðàâíû ïîðÿäêó ïîëó- èâøåãîñÿ ôîðìàëüíîãî ðÿäà è ïîðÿäêó èñõîäíîãî ðÿäà), ë è á î âñå åãî êîýôôèöèåíòû ýòî íóëè (òîãäà ïîðÿäîê ïîëó èâøåãîñÿ ðÿäà ðàâåí, ïîñêîëüêó ýòîò ðÿä ñîîòâåòñòâóåò òîæäåñòâåííîìó íóëþ è â ýòîì ñëó àå ãîâîðÿò, òî ôîðìàëüíûé ðÿä óäîâëåòâîðÿåò ýòîìó óðàâíåíèþ). Òàêæå íåîáõîäèìî äîêàçàòü, òî âûäåëåííûå êëàññû çàìêíóòû îòíîñèòåëüíî àëãåáðàè åñêèõ îïåðàöèé è äèôôåðåíöèðîâàíèÿ. Äëÿ ýòîãî îò ôóíêöèé, ñîäåðæàùèõñÿ â ôóíêöèîíàëüíûõ ðÿäàõ, ìû áóäåì òðåáîâàòü ïîíèæåíèÿ ïîðÿäêà íà åäèíèöó ïîñëå äèôôåðåíöèðîâàíèÿ è íàëîæèì îãðàíè åíèÿ íà èõ ðîñò (ñì. óñëîâèÿ 13 â ðàçäåëå 2). Òàê, íàïðèìåð, ôóíêöèè x è ln 1 x óäîâëåòâîðÿþò óñëîâèþ 1. Íî ïåðâàÿ èç íèõ óäîâëåòâîðÿåò òîëüêî óñëîâèþ 2, à âòîðàÿ òîëüêî óñëîâèþ 3. Êðîìå òîãî, êàê óæå áûëî ñêàçàíî ðàíåå, â ðàçäåëå 4 äîêàçûâàåòñÿ òåîðåìà 1 äëÿ ôîðìàëüíûõ ðåøåíèé (âûäåëåííîãî êëàññà) àëãåáðàè åñêèõ îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé. Â ðàáîòå [12, ãë. 1] äîêàçûâàåòñÿ àíàëîã òåîðåìû 1 èç ðàçäåëà 2 äëÿ ôîðìàëüíûõ ðåøåíèé, êîòîðûå ÿâëÿþòñÿ ðÿäàìè, ñîäåðæàùèìè âåùåñòâåííûå ñòåïåíè x è ln x. Ïðè ýòîì òîëüêî ïðåäïîëàãàåòñÿ, à íå äîêàçûâàåòñÿ çàìêíóòîñòü êëàññà ôîðìàëüíûõ ðåøåíèé äèôôåðåíöèàëüíûõ óðàâíåíèé îòíîñèòåëüíî àëãåáðàè åñêèõ îïåðàöèé è äèôôåðåíöèðîâàíèÿ. 2. Î êëàññàõ ôîðìàëüíûõ ðåøåíèé ÎÄÓ, êîòîðûå ìîãóò áûòü íàéäåíû ìåòîäàìè ïëîñêîé ñòåïåííîé ãåîìåòðèè 2.1. Ïîñòðîåíèå êëàññîâ Ðàññìîòðèì ïîñëåäîâàòåëüíîñòü ôóíêöèé {ϕ k0 (ξ 0 )} (5) k 0 =0, ãäå ξ 0 ýòî íåçàâèñèìàÿ êîìïëåêñíàÿ ïåðåìåííàÿ, ξ 0 D, ϕ k0 (ξ 0 ) O(D), ò. å. ϕ k0 +1 = o(ϕ k0 ). Ðàññìîòðèì ôîðìàëüíûé ðÿä k 0 =0 ϕ k0 +1(ξ 0 ) lim = 0, ξ 0 0 ϕ k0 (ξ 0 ) ξ 0 D α k0 ϕ k0 (ξ 0 ), α k0 = const C, (6) ñîîòâåòñòâóþùèé ñèñòåìå ôóíêöèé (5). Ïóñòü êëàññ C 0 îáðàçîâàí ôîðìàëüíûìè ðÿäàìè âèäà (6). Îïðåäåëèì ïî èíäóêöèè êëàññû C j ôîðìàëüíûõ ðÿäîâ j + 1 ïåðåìåííûõ j = 1,..., N, N N. Íà ïåðâîì øàãå èíäóêöèè îïðåäåëèì êëàññ C 1 = ϕ k0 (ξ 1 ) ϕ k0 (ξ 0 ) ϕ k0 C 0, k 0 =0 ξ 0,..., ξ j, ãäå

ÊËÀÑÑÛ ÔÎÐÌÀËÜÍÛÕ ÐÅØÅÍÈÉ ÊÎÍÅ ÍÎÃÎ ÏÎÐßÄÊÀ... 69 íà âòîðîì êëàññ = C 2 k 0 =0 è. ò. ä., è íà N-îì øàãå êëàññ ãäå C N = k 0 =0 ϕ k0 ϕ k0 (ξ 1, ξ 2 ) ϕ ϕ k0 (ξ 0 ) k0 C 1, (ξ 1,..., ξ N ) ϕ k0 (ξ 0 ) ϕ k0 ϕ k0 (ξ 0 ) O(D), ϕ k0 +1 = o(ϕ k0 ), C N 1, ðÿä Îòìåòèì, òî ðÿä k 0 =0 ϕ k0 ξ 0 D, ξ i+1 = ξ i+1 (ξ i ), ξ i+1 O(D), i = 0,..., N 1. k 0 =0 ϕ k0 (ξ 1 )ϕ k0 (ξ 0 ) êëàññà C 1 ìîæíî çàïèñàòü â âèäå α k0 k 1 ϕ k0 k 1 (ξ 1 ) ϕ k0 (ξ 0 ), k =0 k =0 0 1 (ξ 1, ξ 2 )ϕ k0 (ξ 0 ) êëàññà C 2 â âèäå α k0 k 1 k 2 ϕ k0 k 1 k 2 (ξ 2 ) ϕ k0 k 1 (ξ 1 ) ϕ k0 (ξ 0 ), k 0 =0 k 1 =0 k 2 =0 è. ò. ä. Äàëåå àðãóìåíòû ôóíêöèé, óêàçàííûå â ñêîáêàõ, áóäåì îïóñêàòü â òåõ ñëó àÿõ, êîãäà ýòî íå ïðèâîäèò ê ïóòàíèöå, ñ öåëüþ îáëåã åíèÿ âîñïðèÿòèÿ òåêñòà. Côîðìóëèðóåì óñëîâèÿ, íåîáõîäèìûå äëÿ îïðåäåëåíèÿ ñïåöèàëüíûõ êëàññîâ ôîðìàëüíûõ ðÿäîâ C(ξ 0 ),..., C(ξ 0,..., ξ N ), N N, êîòîðûå ÿâëÿþòñÿ ðåøåíèÿìè àëãåáðàè åñêèõ îáûêíîâåííûõ äèôôåðåíöèàëüíûõ óðàâíåíèé è ìîãóò áûòü âû èñëåíû ñ ïîìîùüþ ìåòîäîâ ïëîñêîé ñòåïåííîé ãåîìåòðèè [2] (îñíîâàííûõ íà âûäåëåíèè ïðèáëèæåííûõ óðàâíåíèé ïîñðåäñòâîì ìíîãîóãîëüíèêà ÍüþòîíàÁðþíî). Óñëîâèå 1. Ïóñòü ôóíêöèÿ ϕ(ξ) íåïðåðûâíî äèôôåðåíöèðóåìà l ðàç â îáëàñòÿõ D è D, l N. Êðîìå òîãî, äëÿ íåå è åå ïðîèçâîäíûõ ñïðàâåäëèâû ñîîòíîøåíèÿ è åñëè ϕ (ξ) 0 è ϕ (ξ) 0, òî ord 0 ϕ(ξ) = ord ϕ(ξ) R, ord 0 ϕ (ξ) = ord ϕ (ξ) = ord 0 ϕ (ξ) 1 = ord ϕ (ξ) 1 R. Óñëîâèå 2. Ïóñòü ôóíêöèÿ ϕ(ξ) óäîâëåòâîðÿåò óñëîâèþ 1 è â îáëàñòÿõ D è D ïðè ϕ (ξ) 0 èìååò ìåñòî ðàâåíñòâî ξ l ϕ (ξ) = O (1). ϕ(ξ)

70 È. Â. ÃÎÐÞ ÊÈÍÀ Óñëîâèå 3. Ïóñòü ôóíêöèÿ φ(ξ) óäîâëåòâîðÿåò óñëîâèþ 1 è â îáëàñòÿõ D è D äëÿ íåå ñïðàâåäëèâû ñîîòíîøåíèÿ φ (ξ) 0, Ôîðìàëüíûé ðÿä êëàññà C 0 áóäåì îòíîñèòü ê êëàññó C(ξ 0 ), åñëè ôóíêöèè ϕ k0 óäîâëåòâîðÿþò óñëîâèþ 2. Ôîðìàëüíûé ðÿä êëàññà C j, j = 1,..., N, áóäåì îòíîñèòü ê êëàññó C(ξ 0,..., ξ j ), åñëè ξ l φ (ξ) = O(1). φ(ξ) Êðîìå òîãî, îáðàòíàÿ ôóíêöèÿ ξ(φ) íåïðåðûâíî äèôôåðåíöèðóåìà l ðàç â îáëàñòè E, ãäå E ýòî îáðàç îòîáðàæåíèÿ φ : D E, D D èëè D D, ìíîæåñòâî D òàêîå, òî ïðè ξ 0 èëè ïðè ξ, ξ D, âûïîëíÿåòñÿ φ(ξ) 0, φ(ξ) E. È èìååò ìåñòî ðàâåíñòâî lim φ l ξ (φ) = lim ξ(φ). φ 0 φ E Ïîëüçóÿñü óñëîâèÿìè 13, îïðåäåëèì èíäóêòèâíî ñïåöèàëüíûå êëàññû ôîðìàëüíûõ ðÿäîâ C(ξ 0 ),..., C(ξ 0,..., ξ N ). φ 0 φ E - ôóíêöèè ϕ k0 (ξ 0 ),..., ϕ k0...k j (ξ kj ) óäîâëåòâîðÿþò óñëîâèþ 2, - ôóíêöèè ξ i+1 (ξ i ) óäîâëåòâîðÿþò óñëîâèþ 3, i = 0,..., j 1, - ξ i = o(ξ i+1 ), ln ξ i+1 = o(ln ξ i ), - ϕ k0 = o(ϕ k0 k 1 ),..., ϕ k0...k j 1 = o(ϕ k0...k j ) äëÿ âñåõ ϕ k0,..., ϕ k0...k j, êðîìå òåõ, äëÿ êîòîðûõ ïðîèçâåäåíèå ϕ k0... ϕ k0...k j ïîñòîÿííî. Òàêæå áóäåì ïîëàãàòü, òî íóëü (íóëåâîé ðÿä) âêëþ åí â êëàññû C(ξ 0 ),..., C(ξ 0,..., ξ N ). 2.2. Ïðèìåðû ðÿäîâ âûäåëåííûõ êëàññîâ Ïðèìåð 1. Ðàññìîòðèì ðÿä c k0 k 1 (ln 1 x) k 1 x k 0 (7) k 0 =0 k 1 =0 êëàññà C ( x, ln 1 x ) (òàêæå ýòîò ðÿä îòíîñèòñÿ ê ñëîæíûì ðàçëîæåíèÿì â êëàññèôèêàöèè ðàçëîæåíèé, óêàçàííûõ â ðàáîòàõ [2] è [3]), ãäå c k0 k 1 (ln 1 x) k 1 C[[ln 1 x]], k 1 =0 x D, D îòêðûòûé ñåêòîð ñ âåðøèíîé â íóëå êîíå íîãî ðàñòâîðà è êîíå íîãî ðàäèóñà, êðîìå òîãî, èíòåðâàë (0, 1] D, à D ýòî ìíîæåñòâî, ïîëó åííîå èíâåðñèåé ìíîæåñòâà ) D. Ïðîâåðèì óñëîâèÿ ïðèíàäëåæíîñòè ôîðìàëüíîãî ðÿäà (7) êëàññó C (x, ln 1 x. ˆ Ôóíêöèè x k 0 óäîâëåòâîðÿþò óñëîâèÿì 1 è 2. Äåéñòâèòåëüíî, k 0 k 0 ord 0 x = ord x = k 0.

ÊËÀÑÑÛ ÔÎÐÌÀËÜÍÛÕ ÐÅØÅÍÈÉ ÊÎÍÅ ÍÎÃÎ ÏÎÐßÄÊÀ... 71 Åñëè k 0 l, òîãäà k 0 k 0 ord 0 x = ord x = k 0 l, ˆ x k 0 +1 = o(x k 0 ), (ln 1 x) k 1+1 = o (ln 1 x) k 1, x = o(ln 1 x), ln ln 1 x = o(ln x ), k 0 k 0 ord 0 x = ord x = k 0 l + 1 è l k 0 l k 0 x x x x lim = lim = x 0 xk 0 x xk 0 x D x D = k 0 (k 0 1)... (k 0 l + 1) N. k ˆ Ôóíêöèè ξ 1 1, ξ 1 = ln 1 x, óäîâëåòâîðÿþò óñëîâèÿì 1 è 2 îòíîñèòåëüíî ïåðåìåííîé ξ 1. Ðàññóæäåíèÿ òàêèå æå, êàê äëÿ ôóíêöèé x k 0. Ïîýòîìó èõ íå ïîâòîðÿåì. ˆ Ôóíêöèÿ ln 1 x óäîâëåòâîðÿåò óñëîâèÿì 1 è 3. Äåéñòâèòåëüíî, ˆ Êðîìå òîãî, ord 0 ln 1 x = ord ln 1 x = 0. ord 0 ln 1 x = ord ln 1 x = l, ord 0 ln 1 x = ord ln 1 x = l + 1, l l x ln 1 x x ln 1 x lim = lim = 0, x 0 ln 1 x x x D ln 1 x x D lim t l (exp (1/t)) = lim exp (1/t), (8) t 0 t E t 0 t E ãäå t = ln 1 x, E = {Re t < 0} ïðè x 0 èëè E = {Re t > 0} ïðè x. ˆ x k 0 = o (ln 1 x) k 1 äëÿ ëþáûõ k 0 è k 1. ( Âñå íåîáõîäèìûå ) óñëîâèÿ âûïîëíÿþòñÿ, ñëåäîâàòåëüíî, ðÿä (7) ïðèíàäëåæèò êëàññó C x, ln 1 x. Ïðèìåð 2. Ðàññìîòðèì ðÿä k 0 =0 R k0 x i x k 0 (9) (j) êëàññà C (x), ãäå R k x i x j ýòî ôóíêöèè, ðàöèîíàëüíûå îòíîñèòåëüíî ôóíêöèè x i 0 (j = 0,..., l Z + ) è ãîëîìîðôíûå â îáëàñòÿõ D è D, D ýòî îòêðûòûé ñåêòîð êîíå íîãî ðàñòâîðà è ìàëîãî ðàäèóñà â C ñ âåðøèíîé â íóëå, à D ýòî ìíîæåñòâî, ïîëó åííîå èíâåðñèåé ìíîæåñòâà D, óãîë ðàñòâîðà è ðàäèóñ ñåêòîðà D íàñòîëüêî ìàëû, òî îáà ìíîæåñòâà D è D íå ñîäåðæàò â ñåáå îñîáûõ òî åê x i = a = 0 (èëè, òî òî æå ñàìîå, ln x = arg a i ln a ), ðàñïîëîæåííûõ íà ëó àõ arg x = ln a, a ýòî ïîëþñ èëè íóëü (j) ðàöèîíàëüíûõ ôóíêöèé R i k x x j. Îòìåòèì, òî ýòîò ðÿä ìîæåò îòíîñèòüñÿ ê ñòåïåííûì, 0 ïîëóýêçîòè åñêèì èëè ýêçîòè åñêèì ðàçëîæåíèÿì â êëàññèôèêàöèè ðàçëîæåíèé, óêàçàííûõ â ðàáîòàõ [2] è [3]. Îòìåòèì, òî ôóíêöèÿ x i = exp( arg x) exp(i ln x ) îãðàíè åíà â D è â D (çíà åíèå arg x îãðàíè åíî, à çíà åíèÿ ôóíêöèè exp(i ln x ) ëåæàò íà åäèíè íîé îêðóæíîñòè). Ïðîâåðèì óñëîâèÿ ïðèíàäëåæíîñòè ôîðìàëüíîãî ðÿäà (9) êëàññó C (x).

72 È. Â. ÃÎÐÞ ÊÈÍÀ ˆ Ôóíêöèè R k0 x i x k 0 óäîâëåòâîðÿþò óñëîâèþ 1. Äåéñòâèòåëüíî, ( ) ( ) i k 0 i k ord = x 0 0 R k0 x x ord R k0 x = k 0. Ïîðÿäîê ïðîèçâåäåíèÿ ôóíêöèé ðàâåí ñóììå ïîðÿäêîâ ýòèõ ôóíêöèé è â íóëå, è â áåñêîíå íîñòè, ïîýòîìó à Óáåäèìñÿ â ýòîì. Ðàññìîòðèì ôóíêöèîíàë ( ) i k 0 i ord 0 R k0 x x = ord 0 R k0 x + k 0, ( ) i k 0 i ord R k0 x x = ord R k0 x + k 0, i i ord 0 R k0 x = ord R k0 x = 0. ( ) ln R k0 x i i ord 0 R k0 x = lim. x 0 ln x Ïîñêîëüêó ôóíêöèÿ R k0 x i íå îáðàùàåòñÿ íè â íóëü, íè â áåñêîíå íîñòü â îáëàñòè D, ìîäóëü x i = exp( arg x ( ) îãðàíè åí ) ñ äâóõ ñòîðîí ïîëîæèòåëüíûìè êîíñòàíòàìè, ñëåäîâàòåëüíî, ìîäóëü R i k0 x òàêæå îãðàíè åí ñ äâóõ ñòîðîí ( ïîëîæèòåëüíûìè ) êîíòàêæå ñòàíòàìè â îáëàñòè D. Îòñþäà ñëåäóåò, òî ôóíêöèÿ ln R k0 x i îãðàíè åíà â îáëàñòè D ïðè íåîãðàíè åííîì ñòðåìëåíèè x ê íóëþ, ò. å. ord 0 R k0 x i = 0. Àíàëîãè íî äîêàçûâàåòñÿ, òî ord R k0 x i = 0. ( ) i k R 0 k0 x x  ñèëó òîãî, òî 0, ãäå ξ D ( ) l i k ln 0 k 0 l (j) i R k0 x x = ln x α j R x = ln x j=0 j α j = C l j (k 0 j + 1), t=1 C l áèíîìèíàëüíûå êîýôôèöèåíòû, ìîäóëü j l (j) i α j Rk 0 x ( ) ( ) i k 0 i k ord 0 0 R k0 x x = ord R k0 x x = k 0 l, l (j) + ln α j R x i, k 0 l k 0 k 0 j=0 j=0 îãðàíè åí ñ äâóõ ñòîðîí ïîëîæèòåëüíûìè êîíñòàíòàìè â îáëàñòÿõ D è D (ýòî ñëåäóåò èç òîãî, òî ìîäóëü x i = exp( arg x) îãðàíè åí ñ äâóõ ñòîðîí ïîëîæèòåëüíûìè êîíñòàíòàìè, à ôóíêöèè (j) R i k x j 0 x íå îáðàùàþòñÿ íè â íóëü, íè â áåñêîíå íîñòü â îáëàñòÿõ D è D ) è â ýòèõ îáëàñòÿõ ñïðàâåäëèâû ðàâåíñòâà ( ) ( ) i k 0 i k ord 0 0 R k0 x x = ord R k0 x x = k 0 l + 1, ( xl R i k 0 k0 x x ) = O (1). R k0 (x i ) x k 0

ÊËÀÑÑÛ ÔÎÐÌÀËÜÍÛÕ ÐÅØÅÍÈÉ ÊÎÍÅ ÍÎÃÎ ÏÎÐßÄÊÀ... 73 ( ) i k 0 +1 i k 0 ˆ Rk0 +1 x x = o R k0 x x. Âñå íåîáõîäèìûå óñëîâèÿ âûïîëíÿþòñÿ, ñëåäîâàòåëüíî, ðÿä (9) ïðèíàäëåæèò êëàññó C (x). Ëåììà 1. Óêàçàííûå âî ââåäåíèè ïÿòü òèïîâ ôîðìàëüíûõ ðåøåíèé, à èìåííî, ñòåïåííûå, ñòåïåííî-ëîãàðèôìè åñêèå, ñëîæíûå, ïîëóýêçîòè åñêèå è ýêçîòè åñêèå ðàçëîæåíèÿ, îòíîñÿòñÿ ê âûäåëåííûì êëàññàì C(x, ln 1 x) è C(x). Äîêàçàòåëüñòâî ëåììû 1 ïðîâîäèòñÿ àíàëîãè íî äîêàçàòåëüñòâó ïðèíàäëåæíîñòè ôîðìàëüíûõ ðÿäîâ èç ïðèìåðîâ 1 è 2 ê êëàññàì C(x, ln 1 x) è C(x). Îòìåòèì, òî â áîëüøèíñòâå ñëó àåâ âû èñëåíèå ôîðìàëüíûõ ðÿäîâ âûäåëåííûõ êëàññîâ (äàæå â óêàçàííîì âî ââåäåíèè ñìûñëå) ÿâëÿåòñÿ äîâîëüíî ñëîæíîé çàäà åé. Èçó åíèå ñâîéñòâ (àñèìïòîòè íîñòü, ñõîäèìîñòü, ñóììèðóåìîñòü) òàêèõ ðÿäîâ åùå áîëåå ñëîæíàÿ çàäà à. Èíîãäà âû èñëèòü äàæå âòîðîå ñëàãàåìîå ôîðìàëüíîãî ðåøåíèÿ óðàâíåíèÿ â âèäå ïðîèçâåäåíèÿ ñòåïåííîé ôóíêöèè ñ âåùåñòâåííûì ïîêàçàòåëåì ñòåïåíè è ôóíêöèè íóëåâîãî ïîðÿäêà íåâîçìîæíî. Íî ìîæíî èññëåäîâàòü (îáû íî ëèíåéíîå) îáûêíîâåííîå äèôôåðåíöèàëüíîå óðàâíåíèå íà âòîðîå ñëàãàåìîå âî âñåõ åãî îñîáûõ òî êàõ ( òîáû ïîëó èòü èíôîðìàöèþ î íàñòîÿùåì ðåøåíèè ýòîãî óðàâíåíèÿ) è ïðåäúÿâèòü åãî ôîðìàëüíîå ðåøåíèå â âèäå ïðîèçâåäåíèÿ ñòåïåííîé ôóíêöèè ñ òåì æå ñàìûì âåùåñòâåííûì ïîêàçàòåëåì ñòåïåíè è ñòåïåííîãî ðÿäà ïî ôóíêöèè íóëåâîãî ïîðÿäêà â îêðåñòíîñòè íóëÿ (èëè â îêðåñòíîñòè äðóãîé îñîáîé èëè íåîñîáîé òî êè, â çàâèñèìîñòè îò ïîñòàâëåííîé çàäà è). Òî æå ñàìîå èìååò ìåñòî è äëÿ ñëåäóþùèõ ñëàãàåìûõ ðÿäà. Ïî ýòîé ïðè èíå âû èñëèòü îáëàñòè D è D â êîòîðûõ âñå ñëàãàåìûå áåñêîíå íîãî ðÿäà áóäóò âåñòè ñåáÿ ðåãóëÿðíî âîçìîæíî â èñêëþ èòåëüíûõ ñëó àÿõ. Ìíîãèå èçâåñòíûå ôîðìàëüíûå ðÿäû âûäåëåííûõ êëàññîâ îáúåêòû íîâûå è êðàéíå ñëîæíûå. Èõ òàêæå ìîæíî îòíåñòè ê ðÿäàì, íàçûâàåìûì â èíîñòðàííîé ëèòåðàòóðå, grid-based series è transseries (ñì., íàïðèìåð, [13][16]). Ñòðîãèå îïðåäåëåíèÿ ïîíÿòèé grid-based series è transseries òðåáóþò íåñêîëüêèõ îòäåëüíûõ ïàðàãðàôîâ. Ïîñêîëüêó â ýòîé ðàáîòå grid-based series è transseries íå ÿâëÿþòñÿ ïðåäìåòîì èññëåäîâàíèÿ, à â ðàáîòàõ [13][16] ìîæíî íàéòè ñòðîãèå îïðåäåëåíèÿ è îñíîâû ìåòîäîâ èõ ïîñòðîåíèÿ è èçó åíèÿ, òî ìû îïðåäåëèì èõ íåôîðìàëüíî. Grid-based series ýòî ðÿäû Ëîðàíà ìíîãèõ ïåðåìåííûõ â îêðåñòíîñòè íóëÿ ñ êîíå íîé ãëàâíîé àñòüþ, êàæäàÿ ïåðåìåííàÿ ðÿäà ýòî ôóíêöèÿ íåçàâèñèìîé ïåðåìåííîé, êîòîðàÿ â íåêîòîðîé îêðåñòíîñòè íóëÿ ÿâëÿåòñÿ áåñêîíå íî ìàëîé âåëè èíîé. Transseries ýòî õîðîøî óïîðÿäî åííûå ôîðìàëüíûå ñóììû ëåíîâ, îáðàçîâàííûõ êîìáèíàöèÿìè è êîìïîçèöèÿìè ñòåïåííûõ ôóíêöèé, ëîãàðèôìîâ è ýêñïîíåíò ñ êîíå íûì èñëîì îáðàçóþùèõ ôóíêöèé. Ïîä õîðîøî óïîðÿäî åííûìè ñóììàìè ìîæíî ïîíèìàòü àñèìïòîòè åñêóþ óïîðÿäî åííîñòü, íî ïîñêîëüêó ñðàâíèâàþòñÿ ëåíû transseries, êîòîðûå ìîãóò ñîäåðæàòü ðàñõîäÿùèåñÿ ðÿäû (ò. å. ëåíû transseries ìîãóò íå ÿâëÿòüñÿ íàñòîÿùèìè ôóíêöèÿìè, äëÿ êîòîðûõ ìîæíî ðàññìàòðèâàòü ïðåäåë àñòíîãî), òî ïðàâèëüíåå áóäåò íàçûâàòü òàêóþ óïîðÿäî åííîñòü ôîðìàëüíîé àñèìïòîòè åñêîé óïîðÿäî åííîñòüþ. Ïðèìåðîì transseries ÿâëÿåòñÿ ñóììà k,m=0 k mx x e c îáðàçóþùèìè x 1 è e x, x R, x +. À ðÿäû (ñì. [13]) k ekx x è e, x R, x +, k=0 k=0 íå ÿâëÿþòñÿ transseries ïîñêîëüêó ïåðâûé ðÿä ïðîòèâîðå èò ôîðìàëüíûì ïðàâèëàì àñèìïòîòè åñêîé óïîðÿäî åííîñòè, à âòîðîé èìååò áåñêîíå íîå èñëî îáðàçóþùèõ. Èçó åíèå ôîðìàëüíûõ àñèìïòîòèê è ðÿäîâ ïðåäñòàâëÿåò áîëüøîé èíòåðåñ, ïîñêîëüêó îíè ïðåòåíäóþò íà

74 È. Â. ÃÎÐÞ ÊÈÍÀ ðîëü àñèìïòîòèê è àñèìïòîòè åñêèõ ðàçëîæåíèé íàñòîÿùèõ ðåøåíèé óðàâíåíèé. Íàïðèìåð, øåñòîå óðàâíåíèå Ïåíëåâå ïðè íåêîòîðûõ çíà åíèÿõ ïàðàìåòðîâ óðàâíåíèÿ èíòåãðèðóåòñÿ â ýëëèïòè åñêèõ ôóíêöèÿõ [17], è åãî ðåøåíèå èìååò ïîâåäåíèå âáëèçè îñîáûõ è íåîñîáûõ òî- åê, ïîäîáíîå ïîâåäåíèþ ôîðìàëüíûõ àñèìïòîòèê, âû èñëåííûõ ñ ïîìîùüþ ìåòîäîâ ïëîñêîé ñòåïåííîé ãåîìåòðèè. È, êàê óæå áûëî ñêàçàíî, ýòè ôîðìàëüíûå àñèìïòîòèêè ïðîäîëæàþòñÿ â ðàçëîæåíèÿ. 2.3. Î ïîðÿäêå ôîðìàëüíîãî ðÿäà âûäåëåííûõ êëàññîâ Ôîðìàëüíûé ðÿä ϕˇ, ïðèíàäëåæàùèé êëàññó C(ξ 0,..., ξ N ), çàïèñûâàåòñÿ â âèäå ϕˇ = α k0 k 1...k N ϕ k0 k 1...k N (ξ N )... ϕ k0 k 1 (ξ 1 ) ϕ k0 (ξ 0 ). (10) k 0,k 1,...,k N =0 Ïîðÿäêîì p( ˇϕ) R íåíóëåâîãî ôîðìàëüíîãî ðÿäà (10) êëàññà C(ξ 0,..., ξ N ) áóäåì ñ èòàòü ïîðÿäîê åãî ïåðâîãî íåíóëåâîãî ëåíà â òî êå ξ 0 = 0 (èëè â òî êå ξ =, ïîñêîëüêó ïîðÿäêè ïåðâîãî ñëàãàåìîãî ðÿäà, óêàçàííîãî êëàññà, ñîâïàäàþò è â íóëå, è â áåñêîíå íîñòè), ò. å. p( ˇϕ) = ord 0 (α 0...0 ϕ 0...0... ϕ 0 ) = ord (α 0...0 ϕ 0...0... ϕ 0 ), α 0...0 0. Áîëåå òîãî, îòìåòèì, òî p( ˇϕ) = ord 0 (ϕ 0 (ξ 0 )) = ord (ϕ 0 (ξ 0 )). Äåéñòâèòåëüíî, ln α 0...0 ϕ 0...0... ϕ 0 p( ˇϕ) = lim = ξ 0 0 ln ξ 0 ξ 0 D [ ] ln α 0...0 ln ϕ 0 ln ϕ 00 ln ξ 1 ln ϕ 0...0 ln ξ N = lim + + + + = ξ 0 0 ln ξ 0 ln ξ 0 ln ξ 1 ln ξ 0 ln ξ N ln ξ 0 ξ 0 D ln ϕ 0 = lim = ord 0(ϕ 0 (ξ 0 )) = ord (ϕ 0 (ξ 0 )). ξ 0 0 ln ξ 0 ξ 0 D Ïîðÿäêîì íóëåâîãî ðÿäà êëàññà C(ξ 0,..., ξ N ) áóäåì ñ èòàòü. Íàïîìíèì, òî íóëåâîé ðÿä êëàññà C(ξ 0,..., ξ N ) ýòî ðÿä âèäà (10), ãäå âñå α k0...k N = 0. Çàìå àíèå 1. Ïðè äèôôåðåíöèðîâàíèè ïîñòîÿííîãî ëåíà ðÿäà, ïîëó àåòñÿ ðÿä ñ íóëåâûì ëåíîì. Åñòåñòâåííî áûëî áû åãî èñêëþ èòü èç áåñêîíå íîé ñóììû, ïåðåíóìåðîâàâ ëåíû ðÿäà. Íî ìû ýòîãî äåëàòü íå áóäåì, ïîñêîëüêó ýòî ïðèâåäåò ê óñëîæíåíèþ èçëîæåíèÿ. Íàîáîðîò, ìû ñîõðàíèì íóìåðàöèþ â ïîëó åííîì ïîñëå äèôôåðåíöèðîâàíèÿ ðÿäå. Ïðè ýòîì íóëü áóäåì çàïèñûâàòü, êàê íåêîòîðóþ íàì óäîáíóþ ôóíêöèþ ñ íóëåâûì êîýôôèöèåíòîì. 3. Äîêàçàòåëüñòâî çàìêíóòîñòè âûäåëåííîãî êëàññà îòíîñèòåëüíî àëãåáðàè åñêèõ îïåðàöèé è äèôôåðåíöèðîâàíèÿ 3.1. Äîêàçàòåëüñòâî çàìêíóòîñòè îòíîñèòåëüíî äèôôåðåíöèðîâàíèÿ dϕˇ Ïðåäëîæåíèå 1. Ôîðìàëüíûé ðÿä, ïîëó åííûé â ðåçóëüòàòå äèôôåðåíöèðîâàíèÿ dξ 0 ïî ξ 0 ðÿäà ϕˇ ïîðÿäêà p( ˇϕ) êëàññà C(ξ 0,..., ξ N ): 1) òàêæå ïðèíàäëåæèò êëàññó C(ξ 0,..., ξ N ), è 2) åãî ïîðÿäîê áîëüøå èëè ðàâåí p( ˇϕ) 1.

ÊËÀÑÑÛ ÔÎÐÌÀËÜÍÛÕ ÐÅØÅÍÈÉ ÊÎÍÅ ÍÎÃÎ ÏÎÐßÄÊÀ... 75 dϕˇ Çàìå àíèå 2.  óñëîâèÿõ ïðåäëîæåíèÿ 1 ïîðÿäîê ôîðìàëüíîãî ðÿäà áóäåò áîëüdξ 0 øå, åì p( ˇϕ) 1, òîëüêî â òîì ñëó àå, åñëè ðÿä ϕˇ íà èíàåòñÿ ñ ïîñòîÿííîãî ëåíà è íå ñîäåðæèò äðóãèõ ëåíîâ íóëåâîãî ïîðÿäêà, âî âñåõ îñòàëüíûõ ñëó àÿõ èìååòñÿ ðàâåíñòâî dϕˇ p = p( ˇϕ) 1. dξ 0 Äîêàçàòåëüñòâî ïðåäëîæåíèÿ 1. Äîêàæåì ñíà àëà ïåðâóþ àñòü ïðåäëîæåíèÿ 1. Äëÿ ïðîñòîòû èçëîæåíèÿ áóäåì ðàññìàòðèâàòü ñëó àé N = 1, ò. å. ϕˇ C(ξ 0, ξ 1 ). Çàïèøåì ðÿä ϕˇ â âèäå ϕˇ = α k0 k 1 ϕ k0 k 1 (ξ 1 ) ϕ k0 (ξ 0 ). k,k =0 0 1 Ïðîäèôôåðåíöèðóåì åãî ïî ξ 0 è ïîëó èì ðÿä ãäå dϕˇ ϕ k0 (ξ 0 ) = dξ 0 ξ 0 k 0,k 1 =0 α k0 k 1 Φ k0 k 1 (ξ 1 ), (11) k 0 (ξ 0 ) ξ 0 Φ k0 k 1 (ξ 1 ) = ϕ (ξ 1 ) ξ 1 (ξ 0 ) ξ 0 + ϕ k0 k 1 (ξ 1 ) ϕ k0 (ξ 0 ) ϕ ïðè ϕ (ξ 0 ) 0 èëè ϕ (ξ 1 ) 0, è k 0 ϕk0k1 (ξ1) Φ k0 k 1 (ξ 1 ) = ξ 1 (ξ 0 ) ξ 0, à α k0 k 1 = 0, ξ 1 ïðè ϕ k0 (ξ 0 )ϕ k0 k 1 (ξ 1 ) = const. Ïîêàæåì, òî ðÿä (11) ïðèíàäëåæèò êëàññó C(ξ 0, ξ 1 ), ò. å. ôóíêöèè Φ k0 k 1 (ξ 1 ) è ϕ k0 (ξ 0 ) îáëàäàþò ñâîéñòâàìè 1. Φ k0 k 1 +1 = o(φ k0 k 1 ), Φ (ξ 1 ) ξ l 1 2. = O (1), åñëè Φ (ξ 1 ) 0, Φ k0 k 1 (ξ 1 ) 3. ord 0 Φ (ξ 1 ) = ord 0 Φ (ξ 1 ) 1 è åñëè Φ (ξ 1 ), Φ (ξ 1 ) 0, ord Φ (ξ 1 ) = ord Φ (ξ 1 ) 1, 4. ϕ k0 = o (Φ k0 k 1 ) äëÿ âñåõ ϕ k0 è Φ k0 k 1, êðîìå òåõ, äëÿ êîòîðûõ ïðîèçâåäåíèÿ ϕ k0 Φ k0 k 1 ïîñòîÿííû. Ñâîéñòâî 1. Ðàññìîòðèì îòíîøåíèå Φ k0 k 1 +1/Φ k0 k 1. Ïîñêîëüêó ôóíêöèè ϕ k0 k 1 (ξ 1 ) è ϕ k0 (ξ 0 ) óäîâëåòâîðÿþò óñëîâèþ 2, à ôóíêöèÿ ξ 1 (ξ 0 ) óäîâëåòâîðÿåò óñëîâèþ 3, òî Φ k0 s(ξ 1 ) = O (ϕ k0 s(ξ 1 )), åñëè ϕ k 0 (ξ 0 ) 0, (12) ξ 1 (ξ 0) ξ 0 ξ 1 (ξ 0 ) O ϕ k0 s(ξ 1 ), åñëè ϕ k 0 (ξ 0 ) 0,

76 È. Â. ÃÎÐÞ ÊÈÍÀ s = k 1, k 1 + 1. Òîãäà Φ k0 k 1 +1 ϕ k0 k 1 +1 = O. Φ k0 k 1 ϕ k0 k 1 À ïîñêîëüêó ϕ k0 k 1 +1 = o(ϕ k0 k 1 ), òî ñâîéñòâî 1 âûïîëíåíî. Ñâîéñòâî 2. Òðåáóåòñÿ äîêàçàòü, òî åñëè Φ k k 1 (ξ 1 ) 0, òî Φ k (ξ 1 ) ξ l 0 k 1 1 Φ k0 k 1 (ξ 1 ) (p) p! (m1 + +m ξ p) 1 (ξ 0 ) ξ 0 = ξ 1 (ξ 0 ) ξ ξ 1 0 m ξ 1!...m p! 0 t! 0 = O (1). Ïðîäèôôåðåíöèðóåì l ðàç ïî ξ 1 ôóíêöèþ Φ k0 k 1. Áóäåì ïîëàãàòü, òî ïðîèçâîäíàÿ Φ (ξ 1 ) 0. Ïðè ýòîì, åñëè ϕ (ξ 0 ) 0 èëè ϕ (ξ 1 ) 0, òî ïîëó àåì âûðàæåíèå k 0 [ (p) ] l (l p+1) ϕ (p) (l p) k0 (ξ 0 ) ξ 0 C p ξ Φ (ξ 1 ) = l ϕ (ξ 1 ) 1(ξ 0 ) ξ 0 + ϕ (ξ 1 ), ξ 1 ϕ k0 (ξ 0 ) p=0 ξ 1 à åñëè ϕ (ξ 0 ) 0 è ϕ k 0 (ξ 1 ) 0, òî âûðàæåíèå ϕ k0 k 1 (ξ 1 ) Φ (ξ 1 ) = ( 1) l l! ξ 1 (ξ 0 ) ξ 0. ξ l+1 ξ 1 1 Äàëåå âîñïîëüçóåìñÿ ôîðìóëîé Ôàà-äè-Áðóíî ïðîèçâîäíîé ñëîæíîé ôóíêöèè (ñì. [18]) p (t) mt d p f(g(x)) p! (x) f (m g = 1+ +m p) (g), dx p m p 1!...m p! t! t=1 tm t =p t=1 p N, m 1,..., m p Z +. Ïðèìåíÿÿ ýòó ôîðìóëó ê ëåíàì âûðàæåíèé âûøå, ïîëó àåì ñîîòíîøåíèÿ è p t=1 tm t =p t=1 p (ξ (t) (ξ1 )) mt 0 (p) (m1 + +m p) p (t) ϕ (ξ ϕ 0 ) ξ 0 p! (ξ 0 ) ξ 0 k (ξ (ξ1 )) mt 0 k 0 0 =. ϕ k0 (ξ 0 ) m 1!...m p! ϕ k0 (ξ 0 ) t! ξ 1 p tm t =p ξ 0 t=1 t=1 Ðàñïèñàâ áîëåå ïîäðîáíî âûðàæåíèÿ âûøå è ó èòûâàÿ òîò ôàêò, òî ôóíêöèè ϕ k0 (ξ 0 ), ϕ k0 k 1 (ξ 1 ) óäîâëåòâîðÿþò óñëîâèþ 2, à ôóíêöèÿ ξ 1 (ξ 0 ) óäîâëåòâîðÿåò óñëîâèþ 3, ïîëó àåì ðàâåíñòâà l p+1 (l p+1) l p (l p) ξ1 ϕ (ξ 1 ) = O (ϕ k0 k 1 (ξ 1 )), ξ ϕ (ξ 1 ) = O 1 (ϕ k0 k 1 (ξ 1 )), m 1 + +m p (m1 + +m ξ p) ξ1 (ξ 0 ) ξ 0 = O (ξ 1 (ξ 0 )), 0 ξ 0 (m1 + +m ϕ (ξ p) m 1 + +m p k 0 0 )ξ 0 ξ = O 0 (1), ϕ k0 (ξ 0 ) ξ 0 p p mt (t) (t) mt ξ p 1 ξ 0 (ξ 1 ) = ξ 0 (ξ 1 ) ξ1 t, (13) t=1 t=1

ÊËÀÑÑÛ ÔÎÐÌÀËÜÍÛÕ ÐÅØÅÍÈÉ ÊÎÍÅ ÍÎÃÎ ÏÎÐßÄÊÀ... 77 ãäå p (t) m t lim ξ0 (ξ 1 ) ξ1 t = lim ξ m 0+ +m p 0 (ξ 1 ). ξ 1 0 ξ 1 0 t=1 Òàêèì îáðàçîì, ïîëó àåì, òî ôóíêöèÿ O (ϕ k0 k 1 (ξ 1 )), åñëè ϕ k 0 (ξ 0 ) 0, Φ (ξ 1 ) ξ l 1 = (14) ξ 1 (ξ 0) ξ 0 O ϕ k0 k 1 (ξ 1 ), åñëè ϕ k 0 (ξ 0 ) 0. ξ 1 (ξ 0 ) Íàêîíåö, èç ôîðìóë (12) è (14) ñëåäóåò, òî ñâîéñòâî 2 âûïîëíÿåòñÿ. Ñâîéñòâî 3. Ðàâåíñòâî ord 0 (Φ (ξ 1 )) = ord 0 (Φ (ξ 1 )) 1 ïðè Φ (ξ 1 ) 0 è Φ (ξ 1 ) 0, ðàâíîñèëüíî ðàâåíñòâó Φ (ξ 1 ) lim ln (ξ 1 )/Φ = lim ln ξ 1. (15) ξ1 0 Φ (ξ 1 ) = ξ 1 0 Ïðè ýòîì ðàâåíñòâî (15) âûïîëíÿåòñÿ, åñëè âûïîëíÿåòñÿ ðàâåíñòâî lim (ξ 1 )/Φ lim ξ 1 (= 0). (16) ξ1 0 Àíàëîãè íûå ôîðìóëû ïîëó àåì ïðè ξ 1. Ïðèìåíÿÿ ôîðìóëó (14) ê ëåâîé àñòè ðàâåíñòâà (16), âèäèì, òî ôóíêöèè Φ k0 k 1 îáëàäàþò ñâîéñòâîì 3. Ñâîéñòâî 4. Ïîëüçóÿñü ôîðìóëîé (12), íåòðóäíî óáåäèòüñÿ, òî ϕ k0 = o(φ k0 k 1 ). Ïåðâàÿ àñòü ïðåäëîæåíèÿ 1 äîêàçàíà, ðÿä (11) ïðèíàäëåæèò êëàññó C(ξ 0, ξ 1 ). Äîêàæåì âòîðóþ àñòü ïðåäëîæåíèÿ 1. Ïîñêîëüêó ðÿä (11) ïðèíàäëåæèò êëàññó C(ξ 0, ξ 1 ), òî åãî ïîðÿäîê îïðåäåëÿåòñÿ ïî ïåðâîìó íåíóëåâîìó ëåíó ðÿäà, ò. å. ξ 1 0 dϕˇ p = ord 0 (α κ0 κ 1 Φ κ0 κ 1 (ξ 1 )ϕ κ0 (ξ 0 )), α κ0 κ 1 0. dξ 0 Ó èòûâàÿ ôîðìóëó (12), è òî, òî ôóíêöèÿ ξ 1 (ξ 0 ) óäîâëåòâîðÿåò óñëîâèþ 1, ïîëó àåì ðàâåíñòâî lim ξ 0 0 ln α κ 0 κ 1 Φ κ0 κ 1 ϕ κ0 ξ 0 ln α κ0 κ 1 ln Φ κ0 κ 1 ln ϕ κ0 + + ln ξ = lim ln ξ0 1 = 0 ξ 0 0 ln ξ 0 ln ξ 0 = ord 0 ϕ κ0 (ξ 0 ) 1 ord 0 ϕ 0 (ξ 0 ) 1 = p( ˇϕ) 1. 3.2. Äîêàçàòåëüñòâî çàìêíóòîñòè îòíîñèòåëüíî ñóììèðîâàíèÿ Ïðåäëîæåíèå 2. Ñóììà ϕˇ + ψˇ 0 ôîðìàëüíûõ ðÿäîâ ϕˇ è ψˇ êëàññà C(ξ 0,..., ξ N ) ïîðÿäêîâ p( ˇϕ) è p(ψˇ) R ñîîòâåòñòâåííî ÿâëÿåòñÿ ôîðìàëüíûì ðÿäîì θˇ êëàññà C(ξ 0,..., ξ N ) ïîðÿäêà p(θˇ) R, ãäå p(θˇ) min (p( ˇϕ), p(ψˇ)). Çàìå àíèå 3.  óñëîâèÿõ ïðåäëîæåíèÿ 2 âìåñòî íåðàâåíñòâà p(θˇ) min (p( ˇϕ), p(ψˇ)) ìîæíî íàïèñàòü ðàâåíñòâî p(θˇ) = min (p( ϕ), ˇ p( ψ)), ˇ åñëè p( ϕ) ˇ = p(ψˇ) èëè åñëè p( ˇϕ) = p(ψˇ), íî íå âñå ñëàãàåìûå ñ k 0 = 0 â ýòèõ ðÿäàõ âçàèìíîîáðàòíûå.

78 È. Â. ÃÎÐÞ ÊÈÍÀ Äîêàçàòåëüñòâî ïðåäëîæåíèÿ 2. Ñíîâà äëÿ ïðîñòîòû èçëîæåíèÿ áóäåì ðàññìàòðèâàòü ñëó àé N = 1, ò. å. ϕ, ˇ ψ ˇ C(ξ 0, ξ 1 ). Çàïèøåì ðÿäû ϕˇ è ψ ˇ â âèäå à èõ ñóììó ˇθ â âèäå ˇϕ = ϕ k0 (ξ 1 ) ϕ k0 (ξ 0 ), ϕ k0 (ξ 1 ) = k 0 =0 k 1 =0 ˇψ = ψ k0 (ξ 1 ) ψ k0 (ξ 0 ), ψk0 (ξ 1 ) = k 0 =0 k 1 =0 ˇθ = θ k0 (ξ 1 ) θ k0 (ξ 0 ), θk0 (ξ 1 ) = k 0 =0 k 1 =0 Óïîðÿäî èì ëåíû ðÿäà (19) ñëåäóþùèì îáðàçîì: θ 0 = åñëè k 0 1, òî θ k0 = ϕ m0 ïðè ϕ m0 = o(θ k0 1), ϕ 0 ïðè ψ 0 = o(ϕ 0 ), ψ 0 ïðè ϕ 0 = o(ψ 0 ), ϕ 0 ïðè ψ 0 = O (ϕ 0 ), ψ p0 = o(ϕ m0 ) äëÿ p 0 òàêèõ, òî ψ p0 = o(θ k0 1), ϕ m0 1 o(θ k0 1) ïðè m 0 1, α k0 k 1 ϕ k0 k 1 (ξ 1 ), (17) β k0 k 1 ψ k0 k 1 (ξ 1 ), (18) γ k0 k 1 θ k0 k 1 (ξ 1 ). (19) θ k0 = ψ p0 èëè ïðè ϕ m0 = O (ψ p0 ), ϕ m0 = o(θ k0 1), íî ϕ m0 1 = o(θ k0 1) ïðè m 0 1; ψ p0 = o(θ k0 1), ϕ m0 = o(ψ p0 ) äëÿ m 0 òàêèõ, òî ϕ m0 = o(θ k0 1), ψ p0 1 o(θ k0 1) ïðè p 0 1; θ k 0 = ϕ m0 θ k 0 = ψ p 0 ïðè ïðè θ k0 = ϕ m0, θ k0 = ψ p0, ψ p0 O (ϕ m0 ) äëÿ p 0 ; ψ p0 = O (ϕ m0 ) äëÿ m 0 ;

ÊËÀÑÑÛ ÔÎÐÌÀËÜÍÛÕ ÐÅØÅÍÈÉ ÊÎÍÅ ÍÎÃÎ ÏÎÐßÄÊÀ... 79 ãäå θ k0 = ϕ m0 + ψ p 0 ψ p0 ϕ m0 ïðè θ k0 = ϕ m0, ϕm0 = O (ψ p0 ), íî ϕ k0 ϕ k0 = ψ k 0 ψ k0, θ k0 0 = ϕ m0 0 ïðè ψ p0 0 = o(ϕ m0 0) èëè ϕ m0 0 = O (ψ p0 0), β p0 0 = 0; ψ p0 θk0 0 = ψ p0 0 ϕm0 ïðè ϕ m0 0 = o(ψ p0 0) èëè ϕ m0 0 = O (ψ p0 0), α m0 0 = 0; β p0 0 ψ p0 ïðè ϕ m0 0 = O (ψ p0 0) è α m0 0, β p0 0 C ; åñëè k 0 + k 1 1, òî θ k0 k 1 = ϕ m0 m 1 ïðè èëè èëè θk0 k 1 = ψ p0 p 1 ϕ m0 m 1 = o(θ k0 k 1 1), θk0 0 = ϕ m0 0 + ψ p0 0 αm0 0 ϕ m0 ψ p0 p 1 = o(ϕ m0 m 1 ) äëÿ p 1 òàêèõ, òî ψ p0 p 1 = o(θ k0 k 1 1), ϕ m0 m 1 1 o(θ k0 k 1 1) ïðè m 1 1, ϕ m0 m 1 = O (ψ p0 p 1 ), β p0 p 1 = 0, ϕ m0 m 1 = o(θ k0 k 1 1), íî ϕ m0 m 1 1 = o(θ k0 k 1 1) ïðè m 1 1; ψ p0 ϕ m0 ïðè ψ p0 p 1 = o(θ k0 k 1 1), ϕ m0 m 1 = o(ψ p0 p 1 ) äëÿ m 1 òàêèõ, òî ϕ m0 m 1 = o(θ k0 k 1 1), ψ p0 p 1 1 o(θ k0 k 1 1) ïðè m 1 1, ϕ m0 m 1 = O (ψ p0 p 1 ), α m0 m 1 = 0, ϕ m0 m 1 = o(θ k0 k 1 1), íî ϕ m0 m 1 1 = o(θ k0 k 1 1) ïðè m 1 1; β p0 p 1 ψ p0 ψp0 p 1 ϕ m0 ïðè θ k0 k 1 = ϕ m0 m 1 + αm0 m 1 ϕ m0 m 1 = O (ψ p0 p 1 ), ϕ m0 m 1 = o(θ k0 k 1 1), íî ϕ m0 m 1 1 o(θ k0 k 1 1) ïðè m 1 1, α m0 m 1, β p0 p 1 0. Òåïåðü ïîêàæåì, òî ðÿä (19) ïðèíàäëåæèò êëàññó C(ξ 0, ξ 1 ), ò. å. ôóíêöèè θ k0 k 1 (ξ 1 ) è θ k0 (ξ 0 ) îáëàäàþò ñâîéñòâàìè

80 È. Â. ÃÎÐÞ ÊÈÍÀ 1. θ k0 k 1 +1 = o(θ k0 k 1 ), θ k (ξ 1 ) ξ l 0 k 1 1 2. = O (1), åñëè θ k0 k 1 (ξ 1 ) 0, θ k0 k 1 (ξ 1 ) 3. ord 0 θ (ξ 1 ) = ord 0 θ (ξ 1 ) 1 è åñëè θ (ξ 1 ) 0 è θ (ξ 1 ) 0, 4. θ k0 = o (θ k0 k 1 ) äëÿ âñåõ θ k0 θ k0 k 1 = const. ord θ (ξ 1 ) = ord θ (ξ 1 ) 1, Íåòðóäíî âèäåòü, òî ôóíêöèè θ k0 (ξ 0 ) è θ k0 k 1 (ξ 1 ) îáëàäàþò ñâîéñòâàìè 1 è 4 â ñèëó ââåäåííîé óïîðÿäî åííîñòè. Äîêàæåì, òî ôóíêöèè θ k0 k 1 (ξ 1 ) îáëàäàþò ñâîéñòâîì 2. Äîñòàòî íî ðàññìîòðåòü ñëó àé ψ p0 θk0 = ϕ m0, θ k0 k 1 = ψ p0 p 1, ψ p0 = O (ϕ m0 ), ϕ m0 òàê êàê âî âñåõ îñòàëüíûõ ñëó àÿõ ñïðàâåäëèâîñòü ñâîéñòâà 2 ëèáî î åâèäíà, ëèáî î åâèäíî ñëåäóåò èç ýòîãî ñëó àÿ. ψ Äèôôåðåíöèðóÿ p0 ψ p0 p 1 ïî ξ 1 l ðàç, ïîëó àåì âûðàæåíèÿ ϕ m0 ïðè j 1 l ψ (j) p0 (ξ 0 ) j ψ p0 (ξ 0 ) ψ p0 p 1 (ξ 1 ) = C (ψ p0 p 1 (ξ 1 )) (l j), ϕ m0 (ξ 0 ) l ϕ m0 (ξ 0 ) ξ 1 j=0 ξ 1 (j) (m1 + +m j ) j (t) ψ p0 (ξ 0 ) j! ψ p0 (ξ 0 ) (ξ 0 (ξ 1 )) mt =, ϕ m0 (ξ 0 ) m 1!... m j! ϕ m0 (ξ 0 ) t! ξ 1 j tm t =j ξ 0 t=1 t=1 m 1,..., m j Z +. Ïðèìåíÿÿ ñíîâà ôîðìóëû ïðîèçâîäíîé ïðîèçâåäåíèÿ, ïðîèçâîäíîé ñëîæíîé ôóíêöèè è ó èòûâàÿ òî, òî ôóíêöèè ϕ m0 (ξ 0 ), ψ p0 (ξ 0 ) è ψ p0 p 1 (ξ 1 ) óäîâëåòâîðÿþò óñëîâèþ 2, ïîëó àåì âûðàæåíèÿ l j ψ (l j) ξ (ξ 1 ) = O (ψ p0 p 1 (ξ 1 )), 1 p 0 p 1 (m1 + +m j ) m 1 + +m j ψ p0 (ξ 0 ) ψ p0 (ξ 0 ) ξ = O. 0 ϕ m0 (ξ 0 ) ξ 0 ϕ m0 (ξ 0 ) j (ξ (t) (ξ1 )) mt t! Êðîìå òîãî, äëÿ âûðàæåíèÿ 0 ñïðàâåäëèâà ôîðìóëà (13). t=1 Òîãäà èìååì ñîîòíîøåíèå ψ p0 (ξ 0 ) ψ p0 (ξ 0 ) ψ p0 p 1 (ξ 1 ) ξ ϕ 1 m0 (ξ 0 ) ξ 1 ϕ m0 (ξ 0 ) l = O ψ p0 p 1 (ξ 1 ). (20)

ÊËÀÑÑÛ ÔÎÐÌÀËÜÍÛÕ ÐÅØÅÍÈÉ ÊÎÍÅ ÍÎÃÎ ÏÎÐßÄÊÀ... 81 Ñëåäîâàòåëüíî, ðàâåíñòâî θ k0 k 1 (ξ 1 ) ξ l 1 = O (1) θ k0 k 1 (ξ 1 ) âûïîëíÿåòñÿ. Íåòðóäíî ïðîâåðèòü, òî ïðè θ (ξ 1 ) 0 è θ (ξ 1 ) 0 ñâîéñòâî 3 âûïîëíÿåòñÿ, ò. å. èìåþò ìåñòî ñîîòíîøåíèÿ è ord 0 θ (ξ 1 ) = ord 0 θ (ξ 1 ) 1 ord θ (ξ 1 ) = ord θ (ξ 1 ) 1. Äîêàçàòåëüñòâî ýòèõ ñîîòíîøåíèé íå ïðèâîäèì, ïîñêîëüêó îíî àíàëîãè íî äîêàçàòåëüñòâó âûïîëíåíèÿ ñâîéñòâà 3 ïðåäëîæåíèÿ 1 ñ ó åòîì ôîðìóëû (13). Èòàê, ìû äîêàçàëè, òî ðÿä θˇ = ϕˇ + ψˇ ïðèíàäëåæèò êëàññó C(ξ 0, ξ 1 ). Èç ââåäåííîé óïîðÿäî åííîñòè ëåíîâ ðÿäà θˇ î åâèäíî ñëåäóåò, òî åãî ïîðÿäîê p(θˇ) R è âûïîëíÿåòñÿ íåðàâåíñòâî p(θˇ) min (p( ˇϕ), p(ψˇ)). 3.3. Äîêàçàòåëüñòâî çàìêíóòîñòè îòíîñèòåëüíî ïðîèçâåäåíèÿ Ïðåäëîæåíèå 3. Ïðîèçâåäåíèå ϕˇ ψˇ 0 ôîðìàëüíûõ ðÿäîâ ϕˇ è ψˇ êëàññà C(ξ 0,..., ξ N ) ïîðÿäêîâ p( ˇϕ) è p(ψˇ) R ñîîòâåòñòâåííî ÿâëÿåòñÿ ôîðìàëüíûì ðÿäîì ϑˇ êëàññà C(ξ 0,..., ξ N ) ïîðÿäêà p(ϑˇ) = p( ˇϕ) + p(ψˇ). Äîêàçàòåëüñòâî. Ñíîâà äëÿ ïðîñòîòû èçëîæåíèÿ áóäåì ðàññìàòðèâàòü ñëó àé N = 1. Ðàññìîòðèì ðÿäû (17) è (18). Èõ ïðîèçâåäåíèå ϑˇ çàïèøåì â âèäå ãäå ϑˇ = ϑ k 0 (ξ 1 ) ϑ k0 (ξ 0 ), ϑ k 0 (ξ 1 ) = δ k0 k 1 ϑ k0 k 1 (ξ 1 ). (21) k 0 =0 k 1 =0 ϑ 0 = ϕ 0 ψ 0, δ 00 ϑ 00 = α 00 β 00 ϕ 00 ψ 00, α 00, β 00 C, ϑ k0 k 1 = ϕ m0 m 1 ψ p0 p 1 ϕ m0 ψ p0 = o(ϑ k0 1), íî ϑ k0 = ϕ m0 ψ p0 ïðè k 0 1 è ϕ m0 1ψ p0 o(ϑ k0 1) ïðè m 0 1, ïðè k 0 + k 1 1 è ϑ k0 = ϕ m0 ψ p0, ϕ m0 ψ p0 1 = o(ϑ k0 1) ïðè p 0 1, ϕ m0 m 1 ψ p0 p 1 = o(ϑ k0 k 1 1), íî ϕ m0 m 1 1ψ p0 p 1 o(ϑ k0 k 1 1) ïðè m 1 1, ϕ m0 m 1 ψ p0 p 1 1 o(ϑ k0 k 1 1) ïðè p 1 1. Ïîêàæåì, òî ðÿä (21) ïðèíàäëåæèò êëàññó C(ξ 0, ξ 1 ), ò. å. ôóíêöèè ϑ k0 k 1 (ξ 1 ) è ϑ k0 (ξ 0 ) îáëàäàþò ñâîéñòâàìè

82 È. Â. ÃÎÐÞ ÊÈÍÀ 1. ϑ k0 +1 = o(ϑ k0 ) è ϑ k0 k 1 +1 = o(ϑ k0 k 1 ), k 0 0 1 ϑ (ξ 0 ) ξ l ϑ (ξ 1 ) ξ l 2. = O (1) è = O (1), ϑ k0 (ξ 0 ) ϑ k0 k 1 (ξ 1 ) åñëè ϑ k 0 (ξ 0 ) 0 è ϑ (ξ 1 ) 0, 3. ord 0 ϑ (ξ 0 ) = ord 0 ϑ (ξ 0 ) 1 è è k 0 k 0 ord 0 ϑ (ξ 1 ) = ord 0 ϑ (ξ 1 ) 1, ord ϑ (ξ 0 ) = ord ϑ (ξ 0 ) 1 k 0 k 0 ord ϑ (ξ 1 ) = ord ϑ (ξ 1 ) 1, åñëè ϑ (ξ 1 ) 0 è ϑ (ξ 1 ) 0, 4. ϑ k0 = o (ϑ k0 k 1 ) äëÿ âñåõ ϑ k0 ϑ k0 k 1 = const. Äëÿ ôóíêöèé ϑ k0 (ξ 0 ) è ϑ k0 k 1 (ξ 1 ) ñâîéñòâà 1 è 4 âûïîëíåíû ñîãëàñíî ââåäåííîé óïîðÿäî- åííîñòè. Íåòðóäíî ïðîâåðèòü, òî äëÿ ôóíêöèé ϕ 1 (ξ) è ϕ 2 (ξ), óäîâëåòâîðÿþùèõ óñëîâèþ 2, ñïðàâåäëèâî ðàâåíñòâî (ϕ 1 (ξ)ϕ 2 (ξ)) ξ l = O (1), (22) ϕ 1 (ξ)ϕ 2 (ξ) åñëè ϕ 1 (ξ)ϕ 2 (ξ) = const. Èç ðàâåíñòâà (22) ñëåäóåò, òî ôóíêöèè ϑ k0 (ξ 0 ) è ϑ k0 k 1 (ξ 1 ) îáëàäàþò ñâîéñòâîì 2. Ïîâòîðÿÿ ðàññóæäåíèÿ äîêàçàòåëüñòâà ñâîéñòâà 3 ïðåäëîæåíèÿ 1 ñ ó åòîì ôîðìóëû (22), ïîëó àåì, òî ñâîéñòâî 3 ïðåäëîæåíèÿ 3 òàêæå ñïðàâåäëèâî. Èòàê, ðÿä ϑˇ = ϕˇ ψˇ ïðèíàäëåæèò êëàññó C(ξ 0, ξ 1 ). Èç ââåäåííîé óïîðÿäî åííîñòè ëåíîâ ðÿäà ϑˇ î åâèäíî ñëåäóåò, òî åãî ïîðÿäîê p(ϑˇ) ðàâåí p( ˇϕ) + p(ψˇ). 4. Î ôîðìàëüíûõ ðÿäàõ êëàññà C(ξ 0,..., ξ N ), ÿâëÿþùèõñÿ ôîðìàëüíûìè ðåøåíèÿìè àëãåáðàè åñêîãî ÎÄÓ Ðàññìîòðèì ôîðìàëüíûé ðÿä âèäà (10) êëàññà C(ξ 0,..., ξ N ). Îáîçíà èì åãî ïîðÿäîê p( ˇϕ) = ρ 0 R. Ïóñòü ðÿä (10) ÿâëÿåòñÿ ôîðìàëüíûì ðåøåíèåì (ñì. çàìå àíèå 1) àëãåáðàè åñêîãî îáûêíîâåííîãî äèôôåðåíöèàëüíîãî óðàâíåíèÿ j ãäå y j = ξ 0 d j y f(ξ 0, y 0,..., y n ) = 0, (23) j, f(ξ 0, y 0,..., y n ) ìíîãî ëåí ñâîèõ ïåðåìåííûõ, ò. å. dξ 0 j d j ϕ f(ξ 0, ϕˇ0,..., ϕˇn) 0, ϕˇj = ξ 0 ˇ, j dξ 0

ÊËÀÑÑÛ ÔÎÐÌÀËÜÍÛÕ ÐÅØÅÍÈÉ ÊÎÍÅ ÍÎÃÎ ÏÎÐßÄÊÀ... 83 è â ñèëó ïðåäëîæåíèé 13 Ñîãëàñíî ïðåäëîæåíèþ 1 Êàæäîìó ñëàãàåìîìó óðàâíåíèÿ (23) âèäà f(ξ 0, ϕˇ0,..., ϕˇn) C(ξ 0,..., ξ N ). p( ˇϕ 0 ) = ρ 0,..., p( ˇϕ n ) = ρ 0. λ ξ 0 q 1 q 20 q y... y 2n 0 n, ãäå λ C, q 1, q 20,..., q 2n Z +, ìîæíî ïîñòàâèòü â ñîîòâåòñòâèå òî êó (âåêòîðíûé ïîêàçàòåëü ñòåïåíè ýòîãî ñëàãàåìîãî) (q 1, q 2 ), q 2 = q 20 + = q 2n. Ñîãëàñíî ïðåäëîæåíèÿì 2 è 3 p(ξ q 1 ϕ q 20 ϕ q 2n ˇ... ˇ ) = q 1 + q 2 ρ 0. 0 0 n Ïóñòü ìíîæåñòâî {Q i = (q 1 i, q 2 i ), i = 0,..., m} ýòî íîñèòåëü óðàâíåíèÿ (23), ò. å. ìíîæåñòâî, ñîäåðæàùåå âñå âåêòîðíûå ïîêàçàòåëè ñòåïåíè âñåõ ñëàãàåìûõ ýòîãî óðàâíåíèÿ, à R ýòî âåêòîð (1, ρ 0 ). Ðàññìîòðèì âñå âîçìîæíûå ñêàëÿðíûå ïðîèçâåäåíèÿ Q i, R = c i R. Ïóñòü c = min c i. Ñóììà ñëàãàåìûõ óðàâíåíèÿ (23) ñ òàêèìè âåêòîðíûìè ïîêàçàòåëÿi=0,...,m ìè ñòåïåíè, òî Q i, R = c R, íàçûâàåòñÿ óêîðî åííîé ñóììîé (ñì. [2]), îáîçíà èì åå fˆ(ξ 0, y 0,..., y n ), à óðàâíåíèå fˆ(ξ 0, y 0,..., y n ) = 0 (24) óêîðî åííûì óðàâíåíèåì. Óðàâíåíèå (24) ÿâëÿåòñÿ êâàçèîäíîðîäíûì îáûêíîâåííûì äèôôåðåíöèàëüíûì óðàâíåíèåì, ýòî îçíà àåò, òî fˆ(ξ 0, y 0,..., y n ) = ρ y=ξ0 0 u c ξ 0 P 0 (u 0,..., u n ), j (j) ãäå P 0 (u 0,..., u n ) ýòî ìíîãî ëåí ñ ïîñòîÿííûìè êîýôôèöèåíòàìè, u j = ξ 0 u. Òåîðåìà 1. Åñëè óðàâíåíèå (23) èìååò ôîðìàëüíîå ðåøåíèå ϕˇ C(ξ 0,..., ξ N ) âèäà (10), òîãäà óêîðî åííîå óðàâíåíèå (24) èìååò óêîðî åííîå (ôîðìàëüíîå) ðåøåíèå y = ϕ, ˆ ˆϕ = ϕ 0(ξ 1,..., ξ N ) ϕ 0 (ξ 0 ). (25) Äîêàçàòåëüñòâî. Åñëè ρ 0 = 0, òî â óðàâíåíèè (23) è åãî ôîðìàëüíîì ðåøåíèè ϕ ˇ âèäà (10) ñäåëàåì ñòåïåííîå ïðåîáðàçîâàíèå Ïîëó àåì óðàâíåíèå ρ y = ξ 0 0 u. (26) F (ξ 0, u 0,..., u n ) = 0, (27) p(φ) ˇ ρ ñ ðåøåíèåì u = Φˇ, ϕˇ = ξ 0 0 Φˇ, = 0. Óðàâíåíèå (27) óæå íå ïîëèíîì, ïîñêîëüêó ñîäåðæèò ξ 0 â âåùåñòâåííûõ ñòåïåíÿõ. Äåéñòâèòåëüíî, ïîñëå ñòåïåííîãî ïðåîáðàçîâàíèÿ (26) êàæäîå ñëàãàåìîå óðàâíåíèÿ (23) âèäà ãäå β C, q 20 q 2n 0 y 0 n β ξ q 1... y, q 1, q 20,..., q 2n Z +, ïåðåõîäèò â ñóììó ñëàãàåìûõ âèäà q 20 q 2n β ξ q 1+q 2 ρ 0 u... u, (28) 0 0 n ãäå β C, q 20,..., q 2n Z +, q 20 + + q 2n = q 2. Ñëåäîâàòåëüíî, ïîä äåéñòâèåì ñòåïåííîãî ïðåîáðàçîâàíèÿ (26) âñÿ óêîðî åííàÿ ñóììà fˆ(ξ 0, y 0,..., y n ) ïåðåõîäèò â âûðàæåíèå

84 È. Â. ÃÎÐÞ ÊÈÍÀ ξc 0 P 0 (u 0,..., u n ), ãäå c ýòî ìèíèìóì âñåõ çíà åíèé q 1 + q 2 ρ 0, P 0 (u 0,..., u n ) ýòî ìíîãî- ëåí ñ ïîñòîÿííûìè êîýôôèöèåíòàìè, u j = ξ j 0 u (j), à îñòàëüíûå ñëàãàåìûå ïåðåéäóò â ñóììó ñëàãàåìûõ âèäà (28) ñ q 1 + q 2 ρ 0 > c. Ïîýòîìó óðàâíåíèå (27) èìååò âèä c v ξ 0 [P 0 (u 0,..., u n ) + ξ 1 v 0 P 1 (u 0,..., u n ) + + ξ t 0 P t (u 0,..., u n )] = 0, ãäå P 0 (u 0,..., u n ),..., P t (u 0,..., u n ) ýòî ìíîãî ëåíû ñ ïîñòîÿííûìè êîýôôèöèåíòàìè, èñëà v 1,..., v n R +, v 1,..., v n 0. Ðàçäåëèì ýòî óðàâíåíèå íà ξ 0 c, ïîëó èì óðàâíåíèå P 0 (u 0,..., u n ) + ξ v 1 P 1 (u 0,..., u n ) + + ξ vt P t (u 0,..., u n ) = 0, (29) 0 0 êîòîðîå ( îáîçíà èì g(ξ 0 ), u 0,..., u n ) = 0. Èç çàïèñè (29) è ïðåäëîæåíèé 13 ñëåäóåò, òî ïîðÿäîê p g(ξ 0, Φˇ 0,..., Φˇ n) 0. Ðåøåíèå óðàâíåíèÿ (29) çàïèøåì â âèäå u = Φ, ˇ Φ ˇ = Φ k0 (ξ 1,..., ξ N ) Φ k0 (ξ 0 ) k 0 =0 ˆ ˆ u = Φ + w, Φ = Φ 0 (ξ 1,..., ξ N ) Φ 0 (ξ 0 ). (30) Ïîäñòàâèì ðåøåíèå (30) â óðàâíåíèå (29), à çàòåì (ó èòûâàÿ òî, òî óðàâíåíèå (29) ïîëèíîìèàëüíî ïî u 0,..., u n ) â âûðàæåíèè ïîñëå ïîäñòàíîâêè ðàñêðîåì ñêîáêè ïî ôîðìóëå áèíîìà Íüþòîíà, ïîëó èì âûðàæåíèå [ ] P 0 (Φˆ 0,..., Φˆ n) P 0 (Φˆ 0,..., Φˆ n) P 0 (Φˆ 0,..., Φˆ n) + w 0 + + w n + u 0 u n j j (j) + + ξ v 1 0 P 1 (Φˆ 0,..., Φˆ n) + = 0, Φˆ j = ξ0 Φˆ (j), w = ξ0 w. (31) Îòäåëüíî îöåíèâàÿ ïîðÿäîê êàæäîãî ñëàãàåìîãî óðàâíåíèÿ (31), óáåæäàåìñÿ, òî íàèìåíüøèé ïîðÿäîê ôóíêöèé â íóëå è â áåñêîíå íîñòè èìåþò ëåíû ðÿäà P 0 (Φˆ 0,..., Φˆ n), ïðè ýòîì ïîðÿäîê ðÿäà p(p 0 (Φˆ 0,..., Φˆ n)) ëèáî ðàâåí íóëþ, ëèáî ðàâåí (åñëè P 0 (Φˆ 0,..., Φˆ n) = 0). À ïîðÿäêè îñòàëüíûõ íåíóëåâûõ ñëàãàåìûõ ðàâåíñòâà (31) áîëüøå íóëÿ. Äåéñòâèòåëüíî, ïîðÿäîê p(w j ) > p(φˆ j) = 0, ìíîãî ëåíû P 0 (u 0,..., u n ),..., P t (u 0,..., u n ) è èõ ïðîèçâîäíûå ïî u 0,..., u n (êîòîðûå òîæå ìíîãî ëåíû) ïðè u j = Φˆ j ëèáî ðàâíû íóëþ, ëèáî èìåþò ïîðÿäîê, ðàâíûé íóëþ â ñèëó ïðåäëîæåíèé 2 è 3, à èñëà v n > > v 1 > 0. Íî ïîñêîëüêó ðÿä Φˇ ôîðìàëüíî óäîâëåòâîðÿåò óðàâíåíèþ (29), òî ñóììà âñåõ ñëàãàåìûõ ìèíèìàëüíîãî ïîðÿäêà â âûðàæåíèè (31) ðàâíà íóëþ, ò. å. P 0 (Φˆ 0,..., Φˆ n) = 0. Äåëàÿ â óðàâíåíèè P 0 (u 0,..., u n ) = 0 îáðàòíîå ïðåîáðàçîâàíèå ê ïðåîáðàçîâàíèþ (26) è äîìíîæàÿ ðåçóëüòàò íà ξ c 0, ïîëó àåì, òî fˆ(ξ 0, ϕˆ0,..., ϕˆn) = 0, ϕˆj = ξ j ϕˆ(j), ϕˆ = ξ ρ 0 Φˆ. 0 0 Îòìåòèì, òî îáðàòíîå óòâåðæäåíèå ê óòâåðæäåíèþ òåîðìû 1 íåâåðíî. Óðàâíåíèå (23) ìîæåò íå èìåòü ôîðìàëüíîãî ðåøåíèÿ, êîòîðîå íà èíàåòñÿ ñ ðåøåíèÿ óêîðî åííîãî óðàâíåíèÿ (24). Íî ìåòîäû ïëîñêîé ñòåïåííîé ãåîìåòðèè ïîçâîëÿþò àëãîðèòìè íî îïðåäåëèòü ðåøåíèÿ óêîðî åííûõ óðàâíåíèé, êîòîðûå íå ÿâëÿþòñÿ ôîðìàëüíûìè àñèìïòîòèêàìè. À èìåííî, ñðåäè âñåõ ðåøåíèé óðàâíåíèÿ (24) âûáèðàþòñÿ òîëüêî òå, êîòîðûå èìåþò ïîðÿäîê ρ 0. èñëî ρ 0 òàêîå, òî ìèíèìóì (èëè ìàêñèìóì, åñëè x ) ñêàëÿðíûõ ïðîèçâåäåíèé q 1 + q 2 ρ 0 ïî âñåì q 1, q 2 óðàâíåíèÿ (23) äîñòèãàåòñÿ èìåííî íà ëåíàõ óðàâíåíèÿ (24). Òîëüêî òàêèå ðåøåíèÿ óêîðî åííîãî óðàâíåíèÿ ìîãóò ïðîäîëæàòüñÿ â ðÿäû, êîòîðûå ÿâëÿþòñÿ ôîðìàëüíûìè ðåøåíèÿìè ïîëíîãî óðàâíåíèÿ.

ÊËÀÑÑÛ ÔÎÐÌÀËÜÍÛÕ ÐÅØÅÍÈÉ ÊÎÍÅ ÍÎÃÎ ÏÎÐßÄÊÀ... 85 5. Çàêëþ åíèå Îïåðàöèè ñëîæåíèÿ, óìíîæåíèÿ è äèôôåðåíöèðîâàíèÿ ôîðìàëüíûõ ðÿäîâ êëàññà C(ξ 0,..., ξ N ) íå âûâîäÿò èõ èç ýòîãî êëàññà. Êðîìå òîãî, åñëè òàêîé ôîðìàëüíûé ðÿä óäîâëåòâîðÿåò àëãåáðàè åñêîìó îáûêíîâåííîìó äèôôåðåíöèàëüíîìó óðàâíåíèþ, òî åãî óêîðî åíèå (ïåðâîå ïðèáëèæåíèå ðÿäà) ÿâëÿåòñÿ ôîðìàëüíûì ðåøåíèåì óêîðî åííîãî óðàâíåíèÿ (ïåðâîãî ïðèáëèæåíèÿ óðàâíåíèÿ). Ðåçóëüòàòû ýòîé ðàáîòû ÿâëÿþòñÿ òåîðåòè åñêîé áàçîé äëÿ ìåòîäîâ ïëîñêîé ñòåïåííîé ãåîìåòðèè [2], ïîçâîëÿþùèõ âû èñëÿòü ôîðìàëüíûå ðÿäû âûäåëåííîãî êëàññà. Ãëóáîêî áëàãîäàðþ ðåöåíçåíòîâ çà çàìå àíèÿ ê òåêñòó ñòàòüè. ÑÏÈÑÎÊ ÖÈÒÈÐÎÂÀÍÍÎÉ ËÈÒÅÐÀÒÓÐÛ 1. Ýðäåéè À. Àñèìïòîòè åñêèå ðàçëîæåíèÿ, Ì.: Ãîñóäàðñòâåííîå èçäàòåëüñòâî ôèçèêî-ìàòåìàòè åñêîé ëèòåðàòóðû, 1962, 128 ñ. 2. Áðþíî À. Ä. Àñèìïòîòèêè è ðàçëîæåíèÿ ðåøåíèé îáûêíîâåííîãî äèôôåðåíöèàëüíîãî óðàâíåíèÿ // Óñïåõè ìàòåìàòè åñêèõ íàóê, 2004, ò. 59, 3, ñ. 429480. 3. Áðþíî À. Ä., Ãîðþ êèíà È. Â. Àñèìïòîòè åñêèå ðàçëîæåíèÿ ðåøåíèé øåñòîãî óðàâíåíèÿ Ïåíëåâå // Òðóäû ÌÌÎ, 2010, ò. 71, ñ. 6118. 4. Áðþíî À. Ä., Ïåòðîâè Â. Þ. Îñîáåííîñòè ðåøåíèé ïåðâîãî óðàâíåíèÿ Ïåíëåâå // Ïðåïðèíòû ÈÏÌ èì. Ì. Â. Êåëäûøà, 2004, 75, 17 ñ. 5. Áðþíî À. Ä., Ãðèäíåâ À. Â. Ñòåïåííûå è ýêñïîíåíöèàëüíûå ðàçëîæåíèÿ ðåøåíèé òðåòüåãî óðàâíåíèÿ Ïåíëåâå // Ïðåïðèíòû ÈÏÌ èì. Ì. Â. Êåëäûøà, 2003, 51, 19 ñ. 6. Áðþíî À. Ä., Ãðèäíåâ À. Â. Íåñòåïåííûå ðàçëîæåíèÿ ðåøåíèé òðåòüåãî óðàâíåíèÿ Ïåíëåâå // Ïðåïðèíòû ÈÏÌ èì. Ì. Â. Êåëäûøà, 2010, 10, 18 ñ. 7. Bruno A. D. Power Geometry as a new calculus, Analysis and Applications - ISAAC 2001, Eds. H. G. W. Begehr, R. P. Gilbert and M. W. Wong, Kluwer Academic Publishers: Dordrecht/ Boston/ London, 2003, pp. 51-71. 8. Áðþíî À. Ä., Ïàðóñíèêîâà À. Â. Ëîêàëüíûå ðàçëîæåíèÿ ðåøåíèé ïÿòîãî óðàâíåíèÿ Ïåíëåâå // Äîêëàäû ÀÍ, 2011, ò. 438, 4, ñ. 439443. 9. Áðþíî À. Ä. Ýêñïîíåíöèàëüíûå ðàçëîæåíèÿ ðåøåíèé îáûêíîâåííîãî äèôôåðåíöèàëüíîãî óðàâíåíèÿ // Äîêëàäû ÀÍ, 2012, ò. 443, 5, ñ. 539544. 10. Áðþíî À. Ä. Ýëåìåíòû íåëèíåéíîãî àíàëèçà, Ìàòåìàòè åñêèé ôîðóì, ñåðèÿ Èòîãè íàóêè. Þã Ðîññèè, ÞÌÈÂÍÖ, 2015, ñ. 1333. 11. Øàáàò Á. Â. Ââåäåíèå â òåîðèþ àíàëèòè åñêèõ ôóíêöèé. àñòü 1., Ì.: Íàóêà, 1985, 336 ñ. 12. Áðþíî À. Ä., Øàäðèíà Ò. Â. Îñåñèììåòðè íûé ïîãðàíè íûé ñëîé íà èãëå // Òðóäû ÌÌÎ, 2007, ò. 68, ñ. 224287. 13. Costin O. Asymptotics and Borel Summability, CRC Press. London, 2009.

86 È. Â. ÃÎÐÞ ÊÈÍÀ 14. Van der Hoeven J. Transseries and Real Dierential Algebra // Lecture Notes in Mathematics. Springer. New York, 2006, vol. 1888. 15. Aschenbrenner M., Van den Dries L., Van der Hoeven J. Asymptotic Differential Algebra and Model Theory of Transseries, 2015. Available at: http: //arxiv.org/abs/1509.02588. 16. Edgar G. A. Transseries for beginners // Real Analysis Exchange, 2010, no. 35, pp. 253310. 17. Ãîðþ êèíà È. Â. Òî íîå ðåøåíèå øåñòîãî óðàâíåíèÿ Ïåíëåâå è ýêçîòè åñêàÿ àñèìïòîòèêà // Äîêëàäû ÀÍ, 2013, ò. 450, 4, ñ. 381-383. 18. Êóäðÿâöåâ Â. À. Îáùàÿ ôîðìóëà äëÿ ïðîèçâîäíîé n-ãî ïîðÿäêà ñòåïåíè íåêîòîðîé ôóíêöèè // Ìàòåìàòè åñêîå Ïðîñâåùåíèå. Ì.-Ë.: ÎÍÒÈ, Âûïóñê 1, 1934, c. 24-27. 19. Grigor'ev D.Yu., Singer M. F. Solving ordinary dierential equations in terms of series with real exponents // Trans. Amer. Math. Soc., 1991, vol. 327, no. 1, pp. 329-351. 20. Ðàìèñ Æ.-Ï. Ðàñõîäÿùèåñÿ ðÿäû è àñèìïòîòè åñêèå òåîðèè, Ìîñêâà-Èæåâñê: Èíñòèòóò êîìïüþòåðíûõ èñëåäîâàíèé, 2002. 21. Balser W. From divergent power series to analytic functions, Springer-Verlag, 1994. 22. Sibuya Y. Linear Dierential Equations in the Complex Domain: Problems of Analytic Continuation // Transl. Math. Monographs. A.M.S., 1990, vol. 82. REFERENCES 1. Erd elyi, A., 1955, Asymptotic expansions, Dover, New York, 108 p. 2. Bruno, A. D., 2004, Asymptotic behaviour and expansions of solutions of an ordinary dierential equation, Uspekhi Mat. Nauk, vol. 59, No. 3, pp. 3180. 3. Bruno, A. D., Goryuchkina, I. V., 2010, Asymptotic expansions of the solutions of the sixth Painlev e equation, Trans. Moscow Math. Soc., pp. 1104. 4. Bruno, A. D., Petrovich, V. Yu., 2004, Singularities of solutions to the rst Painleve equation Keldysh Institute preprints, no. 75, 17 pp. 5. Bruno, A. D., Gridnev, A. V., 2003, Power and exponential expansions of solutions to the third Painlev e equation, Keldysh Institute preprints, No. 51, 19 pp. 6. Bruno, A. D., Gridnev, A. V., 2010, Nonpower expansions of solutions to the third Painleve equation, Keldysh Institute preprints, No. 10, 18 pp. 7. Bruno, A. D., 2003, Power Geometry as a new calculus, Analysis and Applications - ISAAC 2001, Eds. H. G. W. Begehr, R. P. Gilbert and M. W. Wong, Kluwer Academic Publishers: Dordrecht/ Boston/ London, pp. 51-71. 8. Bruno, A. D., Parusnikova, A. V., 2011, Local expansions of solutions to the Fifth Painlev e equation, Doklady Math., vol. 438, no. 4, pp. 348-352. 9. Bruno, A. D., 2012, Exponential expansions of solutions to an ordinary dierential equation, Doklady Math., vol. 85, no. 2, pp. 259-264. 10. Bruno, A. D., 2015, Elements of Nonlinear Analisys, Mathematical Forum, Ser. Itogi nauki. Yug Rossii. Yuzh. Math. Inst. Vladikavkaz Scien. Center, pp. 13-33.

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