Inversion of Power Series

Faà di Bruno's formula and inversion of power series

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

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.

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} .

It is shown here the compositional inversion of formal power series has a natural formulation in terms of umbral operators. This fact is used to derive a new proof of the Lagrange inversion theorem by operator theoretic methods.

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.

Variations on computing reciprocals of power series

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

For a ring [Formula: see text] with a derivation [Formula: see text], we determine the Jacobson radical of the pseudo-differential operator ring [Formula: see text] for a large class of rings with a [Formula: see text]-condition. Various types of examples of these rings are provided.

This article draws attention to a comparative study of different members within the inverse Weibull Power Series (IWPS) to analyze the COVID-19 data from South Africa for the period from 27 March to 23 August 2020. A new sibling of the IWPS is introduced, namely the inverse Weibull negative binomial. An EM algorithm is developed for computing the maximum likelihood estimates of the model parameters. The IWPS growth curve model and its special cases are used for prediction of the COVID-19 spread in South Africa. It is found that the IWPS model fits the disease growth of the COVID-19 confirmed cases well with worthy long-term predictions. The IWPS growth curve modeling of South African predicts that the number of confirmed new cases will decrease at the end of November 2020.

Variational calculation of dynamic polarizabilities

MODIS is one of the key instruments for the NASA's EOS mission, currently operated on both Terra and Aqua spacecraft making continuous and complementary observations in 36 spectral bands from 0.4 to 14.2 micrometers. Among them are long-wave infrared (LWIR) bands (11-14.2 micrometers) with Photoconductive (PC) HgCdTe detectors. The Terra MODIS pre-launch thermal vacuum calibration and characterization have shown clear evidence of an optical leak presented by the 11 micron band (band 31) into other PC bands (bands 32-36). A correction algorithm has been designed and implemented in the MODIS Level 1B (L1B) code using coefficients determined from the lunar observations made in year 2000. In this work we evaluate the optical leak correction coefficients derived from Terra MODIS lunar observations from January 2001 to December 2004. Our method assumes that in the absence of an optical leak the detector's signal would decay away from its peak response to the moon as an inverse power series function of the frame offset number. The least Chi-squared minimization method is used to calculate the parameters in the fitting function. Analysis results from the lunar and Earth view observations show that the Terra MODIS PC optical leak has been effectively removed. The Level 1B algorithm has been shown to be stable.

As a continuation to the preceding paper in which a general method of analysis of the separated flows past bluff bodies attached to a plane wall was presented within the framework of the free-streamline theory, the present paper applies the method to the cases of semielliptical projection and bluff bodies of arbitrary shapes. The theoretical pressure distribution on the wetted surface of the semielliptical projection is found to show good agreement with an experimental measurement. An analytical solution for arbitrary bluff bodies is obtained by assuming an inverse power series for mapping functions. Numerical calculations showed that reasonable pressure distributions could be obtained for a few profiles which were arbitrarily chosen.

Asymptotic approximations ($n \to \infty$) to the truncation errors $r_n = - \sum_{\nu=0}^{\infty} a_{\nu}$ of infinite series $\sum_{\nu=0}^{\infty} a_{\nu}$ for special functions are constructed by solving a system of linear equations. The linear equations follow from an approximative solution of the inhomogeneous difference equation $\Delta r_n = a_{n+1}$. In the case of the remainder of the Dirichlet series for the Riemann zeta function, the linear equations can be solved in closed form, reproducing the corresponding Euler-Maclaurin formula. In the case of the other series considered -- the Gaussian hypergeometric series ${}_2 F_1 (a, b; c; z)$ and the divergent asymptotic inverse power series for the exponential integral $E_1 (z)$ -- the corresponding linear equations are solved symbolically with the help of Maple. The practical usefulness of the new formalism is demonstrated by some numerical examples.