A new expression for umbral operators and power series inversion

Faà di Bruno's formula and inversion of power series

An explicit numerical-analytical method is demonstrated for accurate solving the Schrödinger equation in those cases when this equation reducible to a system of n coupled ordinary differential equations with singular points. Fundamental system of solutions is constructed as algebraic combinations of power series, power functions and logarithmic function in the neighbourhood of the regular singular point and as asymptotic expansions of solutions in the neighbourhood of the irregular singular point. The method is based on the calculation of recurrent sequences of constant matrices of coefficients in power series and in inverse power series in asymptotic expansions using derived recurrent relations, that makes possible to calculate solutions at any given point using only algebraic computations and elementary functions. In turn it makes possible to solve accurately the eigenvalue problem and scattering problem and to derive analytical expressions for the wavefunctions. The method is applied to calculations of energies and wavefunctions of the discrete spectrum and wave functions of the continuous spectrum of the hydrogen-like atoms and of acceptors in semiconductors.

Next article Factorization of Lagrange’s Expansion by Means of Exponential Generating FunctionsR. A. SackR. A. Sackhttps://doi.org/10.1137/0114001PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] P. Appell and , J. Kampé De Fériet, Fonctions hypergéometriques et hypersphériquesPolynomes d' Hermite, Gauthier-Villars, Paris, 1926, Appendix 4 Google Scholar[2] H. Bateman, An extension of Lagrange's expansion, Trans. Amer. Math. Soc., 28 (1926), 346–356 MR1501349 Google Scholar[3] Richard Birkeland, Über die Auflösung algebraischer Gleichungen durch hypergeometrische Funktionen, Math. Z., 26 (1927), 566–578 10.1007/BF01475474 MR1544875 CrossrefGoogle Scholar[4] M. M. Carter and , R. A. Sack, Inverse perturbation expansions, to be published Google Scholar[5] A. Cayley, Note sur une formule pour la réversion des séries, J. Reine Angew. Math., 52 (1856), 276–284, Collected Papers, vol. 4, Cambridge, 1891, pp. 30–37 CrossrefGoogle Scholar[6] A. Cayley, Deuxième note sur une formule pour la réversion des séries, J. Reine Angew. Math., 54 (1857), 156–161, Collected Papers, vol. 4, Cambridge, 1891, pp. 54–59 CrossrefGoogle Scholar[7] A. Cayley, On Wronski's theorem, Quart. J. Pure Appl. Math., 12 (1873), 221–228, Collected Papers, vol. 9, Cambridge, 1896, pp. 96–102 Google Scholar[8] J. S. Frame, Power series expansions for inverse functions, Amer. Math. Monthly, 64 (1957), 236–240 MR0085330 0077.28402 CrossrefGoogle Scholar[9] I. J. Good, Generalizations to several variables of Lagrange's expansion, with applications to stochastic processes, Proc. Cambridge Philos. Soc., 56 (1960), 367–380 MR0123021 0135.18802 CrossrefGoogle Scholar[10] E. Goursat, Cours d' analyse mathématique, Vol. 1, Gauthier-Villars, Paris, 1917, Chap. 9, vol. 2, Chaps. 14, 17; English transl. A Course in Mathe-matical Analysis, E. R. Hedrick and O. Dunkel, translators, New York, 1959 Google Scholar[11] C. G. J. Jacobi, De resolutione aequationum per series infinitas, J. Reine Angew. Math., 6 (1830), 257–286, Ges. Werke, vol. 6, Berlin, 1891, pp. 26–61 CrossrefGoogle Scholar[12] Franz Kamber, Formules exprimant les valeurs des coefficients des séries de puissances inverses, Acta Math., 78 (1946), 193–204 MR0014493 0060.15402 CrossrefISIGoogle Scholar[13] Hr. Mellin, Ein allgemeiner Satz über algebraische Gleichungen, Ann. Acad. Sci. Fenn. Ser. A. I, 7 (1915), Google Scholar[14] R. A. Sack, Interpretation of Lagrange's expansion and its generalization to several variables as integration formulas, J. Soc. Indust. Appl. Math., 13 (1965), 47–59 10.1137/0113004 MR0181723 0143.27402 LinkISIGoogle Scholar[15] R. A. Sack, Generalization of Lagrange's expansion for functions of several implicitly defined variables, J. Soc. Indust. Appl. Math., 13 (1965), 913–926 10.1137/0113059 MR0196020 0143.27403 LinkISIGoogle Scholar[16] R. A. Sack, Rayleigh-Schrödinger perturbation expansions to arbitrary order, to be published Google Scholar[17] P. A. Sturrock, Generalization of the Lagrange expansion with application to physical problems, J. Mathematical Phys., 1 (1960), 405–408 10.1063/1.1703671 MR0116836 0091.43503 CrossrefISIGoogle Scholar[18] J. J. Sylvester, Note on Burman's law for the inversion of the independent variable, Philos. Mag., 8 (1854), 535–540, Collected Papers, vol. 2, Cambridge, 1908, pp. 44–49 CrossrefGoogle Scholar[19] J. J. Sylvester, On the change of systems of independent variables, Quart. J. Math., 1 (1857), 42–56, 126–134, Collected Papers, vol. 2, Cambridge, 1908, pp. 65–85 Google Scholar[20] E. T. Whittaker and , G. N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1927, Chaps. 7, 16 Google Scholar Next article FiguresRelatedReferencesCited ByDetails Degenerate RS perturbation theoryThe Journal of Chemical Physics, Vol. 60, No. 3 | 1 Feb 1974 Cross Ref Perturbation Methods and Lagrange’s ExpansionCooperative Phenomena | 1 Jan 1973 Cross Ref Generalization of Lagrange’s Expansion for Functions of Several Implicitly Defined VariablesR. A. SackJournal of the Society for Industrial and Applied Mathematics, Vol. 13, No. 4 | 13 July 2006AbstractPDF (1277 KB)Chapter 3: Approximate MethodsNonlinear Partial Differential Equations in Engineering | 1 Jan 1965 Cross Ref Volume 14, Issue 1| 1966SIAM Journal on Applied Mathematics History Submitted:10 September 1964Published online:13 July 2006 InformationCopyright © 1966 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0114001Article page range:pp. 1-15ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics

In this chapter we prepare the ground for the proof of the Jung-Abhyankar theorem in section 2 and the study of quasiordinary power series in section 4 of chapter V. We assume that the reader is acquainted with the notion of power series over a field; in section 1, for the convenience of the reader, we give some background, introduce convergent power series over the real and complex numbers in section 2, and prove Weierstraß division and preparation theorem in section 3. The category of formal (resp. analytic) algebras—these are homomorphic images of formal (resp. convergent) rings of power series—is treated in section 4. The last section considers, in particular, the completion of an analytic algebra; these results are needed to prove the convergent case of the Jung-Abhyankar theorem.

Variations on computing reciprocals of power series

Inversion of Power Series

This note shows, using elementary properties of ribbon trees, that the universal formula for the inversion of power series can be obtained by counting strata in the compactified moduli space \overline{\mathcal M}_{0,n} .

Presents algorithms for planning feasible and minimum energy paths. The algorithms rely on series expansions to characterize planning problems for polynomial control systems. The resulting inversion problem is solved through an iterative contraction or a power series inversion. While the design methodology is general, our focus is on a class of polynomial control systems for which we provide explicit convergence guarantees. We demonstrate performance and numerical characteristics using one dimensional and six dimensional systems.

The Nottingham group at 2 is the group of (formal) power series t+a2t2+a3t3+⋯ in the variable t with coefficients ai from the field with two elements, where the group operation is given by composition of power series. The depth of such a series is the largest d⩾1 for which a2=…=ad=0.Only a handful of power series of finite order (forcedly a power of 2) are explicitly known through a formula for their coefficients. We argue in this paper that it is advantageous to describe such series in closed computational form through automata, based on effective versions of proofs of Christol's theorem identifying algebraic and automatic series.Up to conjugation, there are only finitely many series σ of order 2n with fixed break sequence (i.e. the sequence of depths of σ∘2i). Starting from Witt vector or Carlitz module constructions, we give an explicit automaton-theoretic description of: (a) representatives up to conjugation for all series of order 4 with break sequence (1,m) for m<10; (b) representatives up to conjugation for all series of order 8 with minimal break sequence (1,3,11); and (c) an embedding of the Klein four-group into the Nottingham group at 2.We study the complexity of the new examples from the algebro-geometric properties of the equations they satisfy. For this, we generalise the theory of sparseness of power series to a four-step hierarchy of complexity, for which we give both Galois-theoretic and combinatorial descriptions. We identify where our different series fit into this hierarchy. We construct sparse representatives for the conjugacy class of elements of order two and depth 2μ±1(μ⩾1). Series with small state complexity can end up high in the hierarchy. This is true, for example, for a new automaton we found, representing a series of order 4 with 5 states (the minimal possible number for such a series).

Operator Theoretic Methods in the Control of Distributed and Nonlinear Systems

The techniques of nuclear operators and duality theory are applied to the problem of time-varying control of time-invariant systems. In this paper the problem of assessing the advatages of time-varying compensation will be studied using operator theoretic methods. In specific, the problems of disturbance rejection, H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , and H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> /H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> , will be analyzed.

Normal matrices over hermitian discrete valuation rings

Abstract. Let G be a complex, affine algebraic group and a meromorphic connection on the trivial G-bundle over , with a pole of order 2 at zero and a pole of order 1 at infinity. We show that the map taking the residue of at zero to the corresponding Stokes factors is given by an explicit, universal Lie series whose coefficients are multilogarithms. Using a non-commutative analogue of the compositional inversion of formal power series, we show that the same holds for the inverse of , and that the corresponding Lie series coincides with the generating function for counting invariants in abelian categories constructed by D. Joyce.

The combinatorial power of the companion matrix

Formal power series and regular operations on fuzzy languages