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Analytic theory of linear differential equations. (English) Zbl 0008.25501

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[1] H. Poincaré, American Journal of Mathematics, vol. 7 (1885), pp. 203–258. · JFM 17.0290.01
[2] G. D. Brikhoff, Trans. Am. Math. Soc., vol. 10 (1909), pp. 436–470. Cf. also J. Horn, Math. Zeit., vol. 21 (1924), pp. 85–95; here many references are given. · JFM 40.0352.02
[3] G. D. Birkhoff and W. J. Trjitzinsky,Analytic Theory of Singular Difference Equations, Acta mathematica, vol. 60 (1933), pp. 1–89. · Zbl 0006.16802
[4] W. J. Trjitzinsky,Analytic Theory of Linear q-difference Equations, Acta mathematica, vol. 61 (1933), pp. 1–38. · Zbl 0007.21103
[5] Except withx and logx, superscripts here do not denote powers.
[6] E. Fabry, Sur les intégrales des équations différentielles linéaires à coefficients rationnels, These, 1885, Paris.
[7] Except, possibly, the values ofr, associated with the same group, may differ by rational fractions.
[8] If \(c = a + \sqrt { - 1b,} {\text{ }}\operatorname{Re} c = a\) .
[9] The members corresponding to terms inQ(z) not actually present are to be omitted.
[10] Speaking of various regions extending to infinity, the shape of the boundary near the origin is immaterial. We may always consider this part of the boundary as consisting of a circular arer= 1>o( 1 being sufficiently great).
[11] A curve will be said to be regular if it is representable by an equation of the form \(0 = c_0 + c_1 r^{ - \frac{1}{{k_1 }}} + c_2 r^{ - \frac{2}{{k_2 }}} + \cdots {\text{ (}}k_1 {\text{ some integer)}}\) .
[12] These directions can be always taken coincident with those of the correspondingQ curves.
[13] The subscripts, here and in the sequel of this proof, should not be confused with the subscripts in (1).
[14] Forz onQ 1x u .Q 1(z)=Q 1(x).
[15] If a function Q(z) does not vanish along any curveQ, possessing the same limiting direction asB (orB’), it increases indefinitely along everyC x (x andC r inR), under consideration.
[16] We take the unique curvesQ x Q x * possessing the limiting directions of a boundary ofR k
[17] One may select a suitableQ x orQ’ x curve of the set of curves associated with (8).
[18] Br can always be chosen, inR, with the same limiting direction at infinity as that ofBr.
[19] Provided thatt o(x) has sufficiently many terms, depending on the nature ofBt.
[20] Cf., for instance, L. Schlesinger, Vorlesungen über lineare Differentialgleichungen, 1908.
[21] HereQ {\(\sigma\)}, s =Q {\(\sigma\)} s .
[22] They will be independent of the path of Product-integration, inasmuch as the path extends to infinity and convergence conditions are satisfied.
[23] In certain cases, as for instance when then formal solutions ofL n (y)=0 are given byn determinations of the same series,p will certainly be different fromk.
[24] However, for the present, it cannot be asserted that the2ai(x) are independent of {\(\alpha\)}2
[25] Such a choice will give a suitably great value of \(\bar \eta \) .
[26] Here,R {\(\sigma\)} ” , if used, hasB {\(\sigma\)}, {\(\sigma\)}+1 for its left boundary.
[27] Thus,Y {\(\sigma\)}+1 (x) is to denote now a matrix possible distint from the matrix for which (2 a) had been asserted.
[28] Compare with (4) and the sequel.
[29] This is a matter of notation.
[30] In this section we continue to assume that the regions (16a; § 2) are ordered in the counter clockwise direction.
[31] The valuesQ i(x) will be the same in the corresponding elements ofS(x) andS’(x).
[32] In the case when the roots of the characteristic equation are all unequalL is of the form \(\left( {\delta _{i,j} \cdot \exp \left( {2\pi r_i \sqrt { - 1} } \right)} \right)\) .
[33] G. D. Birkhoff,The Generalized Riemann Problem for Linear Differential Equations and the Allied Problems for Linear Difference and q-Difference Equations, Proc. Am. Acad. Arts and Sciences, vol. 49 (1913), pp. 521–568. This paper will be referred to as (B). In using or quoting the results of (B) the notation will be used conforming with that of the present paper.
[34] B., p. 548).
[35] With the determinant of constant terms distinct from zero.
[36] That is, the elements ofS (x)S (1) (x) are formal series of the type (1; § 1o). In the case treated in (B) we would haveT(x)({\(\gamma\)} i,j (x))=({\(\gamma\)} i,j, (x)x ri , expQ i (x)).
[37] This theorem constitutes an extension to the unrestricted case of a theorem in–, pp. 548–550).
[38] That is, we have a set of constants of the type of a set of characteristic constants of a system (B)
[39] G. D. Birkhoff,The Generalized Riemann Problem for Linear Differential Equations and the Allied Problems for Linear Difference and q-Difference Equations, Proc. Am. Acad. Arts and Sciences, vol. 49 (1913), pp. 533–534. · JFM 44.0391.03
[40] Ifb {\(\sigma\)} is the limiting direction ofB {\(\sigma\)},{\(\sigma\)}+1 ’ we shall takeb {\(\sigma\)}+{\(\Phi\)} for the limiting direction of \(\bar B'_{\sigma ,{\text{ }}\sigma + 1} \) .
[41] For the purposes at hand analyticity of the elements ofA {\(\sigma\)} (x) along the part ofK {\(\sigma\)} interior the circle |x|= (points of the circle included) is not necessary. At the points of the circle (and interior, as well) indefinite differentiability is sufficient.
[42] The superscript in (14a) denotes them-th derivative.
[43] –, pp. 551–553.
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