New Development in Poincaré’s Problem of Irregular Integrals

Dong Ming-de

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 1985 > 6(4): 303-315

Dong Ming-de. New Development in Poincaré’s Problem of Irregular Integrals[J]. Applied Mathematics and Mechanics, 1985, 6(4): 303-315.

Citation:

Dong Ming-de. New Development in Poincaré’s Problem of Irregular Integrals[J]. Applied Mathematics and Mechanics, 1985, 6(4): 303-315.

Citation:

Dong Ming-de. New Development in Poincaré’s Problem of Irregular Integrals[J]. Applied Mathematics and Mechanics, 1985, 6(4): 303-315.

PDF( 802 KB)

New Development in Poincaré’s Problem of Irregular Integrals

Dong Ming-de

Institute of Theoretical Physics, Academia Sinica, Beijing

Received Date: 1983-09-09
Publish Date: 1985-04-15

Abstract

Abstract

In connection with non-Fuchsian equations Poincaré has made an important conclusion; It is impossible to obtain explicit expressions of irregular integrals. To elucidate the essence of Poincaré's problem, we establish correspondence theorem, Irregular integrals are analytic functions of new kind, possessing tree structure, part of which can be represented by conventional recursive series, while its remaining part is expressed by the so-called tree series, not subjecting to any recursive relation at all. In contrast to the numerical solution calculated by infinite determinant of classical theory (Hill-Poincaré-von Koch),our method yields naturally exact analytic solution in explicit form, The method proposed map be used to;construct a unifying theory for general equations with variable coefficients, having varioas kinds of singularities as singular lines. The significance of Poincaré conjecture is discussed. The tree series obtained belong to higher automorphic functions.

FullText(HTML)

References(21)

References

[1]	Fuchs,L.,Zur Theorie der linearen Differential gleichungen mit verandlichen Coefficienten.J.fur math.,66(1866),121;Gesammelte Werke,T.1(1904).111-240.
[2]	Hill,G.W.,On the part of the motion of lunar perigee which is a function of the mean motions of the sun and moon.Am.J.Math,T.1(1878):Acta Math.,T.Ⅷ(1886),1-36.
[3]	Poincaré,H.,(a)Sur les équations linéaires aux différentielles ordinaires et aux différences finies Am.J.of Math.Ⅶ(1885),1-56,Oeuvres,T.1.226-289.(b)Sur les intégrals irréguliéres des équations linéaires,Acta Math.T.8(1886).295-344,Oeuvres,T.1.290-332.(c)Remmarques sur les intégrales irréguires des équations linéaires(Réponse σ M.Thomé)Acta Math.,T.10(1887).310-312.Oeuvres.T.1.333-335.336-373.
[4]	Thomé.L,W.,Crelle.T,XCV(1883).75.
[5]	Fabry,Theése,Gauthler-Villars,Paris(1885).
[6]	von Koch,H.,Acta Math.,T.15(1891),T.16(1892-3).217-295.T.24(1901).89-122.
[7]	Liapounoff,A.,C.R.,T.CXXⅢ(1896),1248-1252.
[8]	Brikhoff,G.D.,Proc.,Am.Acad,49(1913),54-568.Col Math.Paper.Vol Ⅰ.
[9]	Brillouin L.,Quart.Appl.Math.,6(1948).
[10]	Schlesinger,L.,Handbuch der Theorie der Differentialgleichungen,T.Ⅰ-Ⅲ(1898).
[11]	Forsyth,A.R.,Theory of Differential Equations Vol,Ⅳ.ch 7.8.
[12]	Голубев В.В.,(a)Лекчuu no Аналumuчесоu Теорuu Пuфференчuалъых Урабненuu,2-е изд.,Гостехиздат,москва(1950);(b)Лекчuu no Инmерuробанuю Урабненuu Дбuженuu Дбuженuя Тажелоio Тбербоiо Тела Ококло Неnoббuжноu Точкu,Гостехиздат,москва(1953).
[13]	Erdelyi,A.,Higher Trascendental Functions,Vol,Ⅰ-Ⅲ(1953).
[14]	Bieberbach,L.,Theorie der Gewohnliche Diff(1965).
[15]	Van der Pol,B.,H.Bremmer,Operational Calculus on the Two-Sided Laplace Integral(1947).
[16]	Wiener,N.and R.Payley,Fourier Transform in the Complex Domain(1934).
[17]	Dong Ming-de,Intrinsic Structure of Hill Function,Report on the Congress of Chinese Astronomical Society(1978 Aug.,Shanghai).A Summary was published in Acta Astronomica Sinica,21,1(1980).
[18]	Dong Ming-de,On Poincare’s problem of irregular integrals(to be published)..
[19]	Dong Ming-de,New Developments in the Mathematical Theory and Physical Application for Equations of Fuchsian and Non-Fuchsiann Type(1980,Lecture Notes.unpublished).
[20]	Hagihara,Y.,Celestial Mechanics,Vol.Ⅰ-Ⅲ,M,I.T,Press(1970).
[21]	Harry,F.,Graph Theory(1971).