Acta Mathematica

Analytic theory of linear differential equations

W. J. Trjitzinsky

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Acta Math., Volume 62 (1933), 167-226.

First available in Project Euclid: 31 January 2017

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1933 © Almqvist & Wiksells Boktryckeri-A.-B.


Trjitzinsky, W. J. Analytic theory of linear differential equations. Acta Math. 62 (1933), 167--226. doi:10.1007/BF02393604.

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  • H. Poincaré, American Journal of Mathematics, vol. 7 (1885), pp. 203–258.
  • 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.
  • G. D. Birkhoff and W. J. Trjitzinsky, Analytic Theory of Singular Difference Equations, Acta mathematica, vol. 60 (1933), pp. 1–89.
  • W. J. Trjitzinsky, Analytic Theory of Linear q-difference Equations, Acta mathematica, vol. 61 (1933), pp. 1–38.
  • Except with x and log x, superscripts here do not denote powers.
  • E. Fabry, Sur les intégrales des équations différentielles linéaires à coefficients rationnels, These, 1885, Paris.
  • Except, possibly, the values of r, associated with the same group, may differ by rational fractions.
  • If $c = a + \sqrt { - 1b,} {\text{ }}\Re c = a$ .
  • The members corresponding to terms in Q(z) not actually present are to be omitted.
  • 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 are r=ϱ1>o(ϱ1 being sufficiently great).
  • A curve will be said to be regular if it is representable by an equation of the form $0 = c_0 + c_1 r^{ - \frac{1}{{k_1 }}} + c_2 r^{ - \frac{2}{{k_2 }}} + \cdots {\text{ (}}k_1 {\text{ some integer)}}$ .
  • These directions can be always taken coincident with those of the corresponding Q curves.
  • The subscripts, here and in the sequel of this proof, should not be confused with the subscripts in (1).
  • For z on Q ${}_{1x}^{u}$ ℜ. Q1(z)=ℜQ1(x).
  • If a function ℜQ(z) does not vanish along any curve Q, possessing the same limiting direction as B (orB′), it increases indefinitely along every Cx (x and Cr in R), under consideration.
  • We take the unique curves QxQx* possessing the limiting directions of a boundary of Rk
  • One may select a suitable Qx or Q′x curve of the set of curves associated with (8).
  • Br can always be chosen, in R, with the same limiting direction at infinity as that of Br.
  • Provided that to(x) has sufficiently many terms, depending on the nature of Bt.
  • Cf., for instance, L. Schlesinger, Vorlesungen über lineare Differentialgleichungen, 1908.
  • Here Qσ, s=Qσ−Qs.
  • They will be independent of the path of Product-integration, inasmuch as the path extends to infinity and convergence conditions are satisfied.
  • In certain cases, as for instance when the n formal solutions of Ln(y)=0 are given by n determinations of the same series, p will certainly be different from k.
  • However, for the present, it cannot be asserted that the2ai(x) are independent of α2
  • Such a choice will give a suitably great value of $\bar \eta $ .
  • Here, R ${}_{σ}^{"}$ , if used, has Bσ, σ+1 for its left boundary.
  • Thus, Yσ+1(x) is to denote now a matrix possible distint from the matrix for which (2 a) had been asserted.
  • Compare with (4) and the sequel.
  • This is a matter of notation.
  • In this section we continue to assume that the regions (16a; § 2) are ordered in the counter clockwise direction.
  • The values Qi(x) will be the same in the corresponding elements of S(x) and S'(x).
  • In the case when the roots of the characteristic equation are all unequal L is of the form $\left( {\delta _{i,j} \cdot \exp \left( {2\pi r_i \sqrt { - 1} } \right)} \right)$ .
  • 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.
  • B., p. 548).
  • With the determinant of constant terms distinct from zero.
  • That is, the elements of S−1(x)S(1)(x) are formal series of the type (1; § 1o). In the case treated in (B) we would have T(x)(γi,j(x))=(γi,j,(x)xri, exp Qi(x)).
  • This theorem constitutes an extension to the unrestricted case of a theorem in—, pp. 548–550).
  • That is, we have a set of constants of the type of a set of characteristic constants of a system (B)
  • 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.
  • If bσ is the limiting direction of B ${}_{σ,σ+1}^{′}$ we shall take bσ+Φ for the limiting direction of $\bar B'_{\sigma ,{\text{ }}\sigma + 1} $ .
  • For the purposes at hand analyticity of the elements of Aσ(x) along the part of Kσ interior the circle |x|=ϖ (points of the circle included) is not necessary. At the points of the circle (and interior, as well) indefinite differentiability is sufficient.
  • The superscript in (14a) denotes the m-th derivative.
  • —, pp. 551–553.