Reccurent sequenses in solving the Schrödinger equation

Faà di Bruno's formula and inversion of power series

A numerical routine for the crossed vertex diagram with a massive-particle loop

The Cauchy problem for a linear second-order uniformly parabolic equation with variable coefficients depending only on the spatial variable is considered. For two- and three-dimensional spaces, asymptotic expansions of the solution as t → +0 are written. It is shown how these expansions can be used in the study of inverse problems of determining the coefficients of a parabolic equation.

In this paper we consider the confluent Heun equation, which is a linear differential equation of second order with three singular points --- two of them are regular and the third one is irregular of rank 1. The purpose of the work is to propose a procedure for numerical evaluation of the equation's solutions (confluent Heun functions). A scheme based on power series, asymptotic expansions and analytic continuation is described. Results of numerical tests are given.

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

Explicit analytical expressions to represent the radiative properties of spherical particles in the small size-parameter range were obtained. These expressions were deduced following Penndorf’s approach of expanding the Mie coefficients in power series on the size parameter. However, in opposition to Penndorf’s original work—in which some errors were found and corrected—the Mie coefficients were expanded to the eighth power of the size parameter, which results in a five-term approximation to the extinction and scattering efficiencies. Also, expressions for the evaluation of the asymmetry factor and both polarized and unpolarized phase functions were deduced and presented. The results so obtained have proved to be very accurate, even for size parameters beyond the limit of validity of the approach utilized.

In the process of mechanism design and trajectory planning of in-parallel actuated manipulators, it is necessary to know the locations of all singular points in the working space. We propose a new method for numerically locating the singular points of in-parallel actuated manipulators based on the pressure angle. A singular-point curve is defined as a set of singular points. Kinematic and static characteristics at singular points are analyzed by pressure angles and instantaneous screw axes. Singular points of the Stewart Platform are investigated using the proposed method. The neighborhood of singular points is determined from output error, isotropy of output force and driving force, and thus the practical working space is defined.

CHAPTER 1. AN INTRODUCTION TO DATA AND FUNCTIONS. 1.1 Describing Single-Variable Data. 1.2 Describing Relationships between Two Variables. 1.3 An Introduction to Functions. 1.4 The Language of Functions. 1.5 Visualizing Functions. CHAPTER 2. RATES OF CHANGE AND LINEAR FUNCTIONS. 2.1 Average Rates of Change. 2.2 Change in the Average Rate of Change. 2.3 The Average Rate of Change Is a Slope. 2.4 Putting a Slant on Data. 2.5 Linear Functions: When Rates of Change Are Constant. 2.6 Visualizing Linear Functions. 2.7 Constructing Graphs and Equations of Linear Functions. 2.8 Special Cases. 2.9 Breaking the Line: Piecewise Linear Functions. 2.10 Constructing Linear Models of Data. 2.11 Looking for Links between Education and Earnings: A Case Study on Using Regression Lines. CHAPTER 3. WHEN LINES MEET: LINEAR SYSTEMS. 3.1 Interpreting Intersection Points: Linear and Nonlinear Systems. 3.2 Visualizing and Solving Linear Systems. 3.3 Reading between the Lines: Linear Inequalities. 3.4 Systems with Piecewise Linear Functions: Tax Plans. CHAPTER 4. THE LAWS OF EXPONENTS AND LOGARITHMS: MEASURING THE UNIVERSE. 4.1 The Numbers of Science: Measuring Time and Space. 4.2 Positive Integer Exponents. 4.3 Zero, Negative, and Fractional Exponents. 4.4 Converting Units. 4.5 Orders of Magnitude. 4.6 Logarithms as Numbers. CHAPTER 5. GROWTH AND DECAY: AN INTRODUCTION TO EXPONENTIAL FUNCTIONS. 5.1 Exponential Growth. 5.2 Exponential Decay. 5.3 Comparing Linear and Exponential Functions. 5.4 Visualizing Exponential Functions. 5.5 Exponential Functions: A Constant Percent Change. 5.6 More Examples of Exponential Growth and Decay. 5.7 Compound Interest and the Number e. 5.8 Semi-Log Plots of Exponential Functions. CHAPTER 6. LOGARITHMIC LINKS: LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 6.1 Using Logarithms to Solve Exponential Equations. 6.2 Using Natural Logarithms to Solve Exponential Equations Base e. 6.3 Visualizing and Applying Logarithmic Functions. 6.4 Using Semi-Log Plots to Construct Exponential Models for Data. C H A P T E R 7. POWER FUNCTIONS. 7.1 The Tension between Surface Area and Volume. 7.2 Direct Proportionality: Power Functions with Positive Powers. 7.3 Visualizing Positive Integer Power Functions. 7.4 Comparing Power and Exponential Functions. 7.5 Inverse Proportionality: Power Functions with Negative Powers. 7.6 Visualizing Negative Integer Power Functions. 7.7 Using Logarithmic Scales to Find the Best Functional Model. CHAPTER 8. QUADRATICS AND THE MATHEMATICS OF MOTION. 8.1 An Introduction to Quadratic Functions: The Standard Form. 8.2 Visualizing Quadratics: The Vertex Form. 8.3 The Standard Form vs. the Vertex Form. 8.4 Finding the Horizontal Intercepts: The Factored Form. 8.5 The Average Rate of Change of a Quadratic Function. 8.6 The Mathematics of Motion. CHAPTER 9. NEW FUNCTIONS FROM OLD. 9.1 Transformations. 9.2 The Algebra of Functions. 9.3 Polynomials: The Sum of Power Functions. 9.4 Rational Functions: The Quotient of Polynomials. 9.5 Composition and Inverse Functions. 9.6 Exploring, Extending & Expanding. APPENDIX Student Data Tables for Exploration 2.1. Data Dictionary for FAM1000 Data. SOLUTIONS For all Algebra Aerobics and Check Your Understanding problems for odd-numbered problems in the Exercises and Chapter Reviews. All solutions are grouped by chapter ANS-1. INDEX.

The idealized model of the geomagnetic field-solar wind interaction yields a singular (neutral) point on the magnetopause at which the magnetic field vanishes. If we expand the fields in a power series around this point, including quadratic terms, we can derive an approximate equation of the magnetopause in a small neighborhood of the neutral point which is consistent with the idealized boundary conditions to fourth order. We then consider the additional pressure due to multiple reflections of particles in this neighborhood and show that less than a 4 per cent correction to the single-reflection pressure condition is necessary.

The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found

Computer computation of the method of multiple scales—dirichlet problem for a class of system of nonlinear differential equations

The tail decay of the hitting probabilities on level zero in M/G/1-type Markov renewal processes (MRPs) is studied. The Markov renewal process is transformed into a Markov chain so that the problem of tail decay is reformulated in terms of the decay of the coefficients of a suitable power series. The latter problem is reduced to analyze the analyticity domain of the power series. Two numerical methods are introduced to compute the convergence radius of the latter power series. Some numerical experiments are presented that show the effectiveness of the proposed algorithms.

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.

and that the corresponding limit at x = e(O), where 0 is not congruent (mod 1) to an integral multiple of a, is zero. P. Sziisz has pointed out that for the proof of (1) and (2) Davenport's conditions can be reduced to the following: (a) limN_. EInIN ane(nt) exist for each real t, and (b) I E jnI,N a.e(nt) I <K < co uniformly in N and t. That is, the convergence of Ean and condition (6) in Davenport's paper can be replaced by (a). The purpose of this note is to generalize the above results in a couple of directions. It is to be noticed that the power series coefficients in (2), A,=g(va+y), constitute in some general sense an almost periodic sequence with the Fourier expansion

We present a new method for visualizing implicit real algebraic curves inside a bounding box in the $2$-D or $3$-D ambient space based on numerical continuation and critical point methods. The underlying techniques work also for tracing space curve in higher-dimensional space. Since the topology of a curve near a singular point of it is not numerically stable, we trace only the curve outside neighborhoods of singular points and replace each neighborhood simply by a point, which produces a polygonal approximation that is $\epsilon$-close to the curve. Such an approximation is more stable for defining the numerical connectedness of the complement of the projection of the curve in $\mathbb{R}^2$, which is important for applications such as solving bi-parametric polynomial systems. The algorithm starts by computing three types of key points of the curve, namely the intersection of the curve with small spheres centered at singular points, regular critical points of every connected components of the curve, as well as intersection points of the curve with the given bounding box. It then traces the curve starting with and in the order of the above three types of points. This basic scheme is further enhanced by several optimizations, such as grouping singular points in natural clusters, tracing the curve by a try-and-resume strategy and handling "pseudo singular points". The effectiveness of the algorithm is illustrated by numerous examples. This manuscript extends our preliminary results that appeared in CASC 2018.