A power series inverse in time of the universal Kepler equation
Series inverse in powers of time and radius of convergence for universal form of Kepler equation, discussing recursion formulas for coefficients
Estes, R. H.↗
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Solution of the N-body problem with recurrent power series.
Adaptation of Steffensen's method of solution of both the restricted and general three-body problems in terms of power series in time to the solution of the equations of motion of N planets around the sun. The key feature of this adaptation lies in the introduction of N new variables and in the use of new relations which contain the time derivatives and can thus be treated as differential equations.
Broucke, R.↗
Power series solutions of ordinary differential equations in MACSYMA
A program is described which extends the differential equation solving capability of MACSYMA to power series solutions and is available via the SHARE library. The program is directed toward those classes of equations with variable coefficients (in particular, those with singularities) and uses the method of Frobenius. Probably the most important distinction between this package and others currently available or being developed is that, wherever possible, this program will attempt to provide a complete solution to the equation rather than an approximation, i.e., a finite number of terms. This solution will take the form of a sum of infinite series.
Lafferty, E. L.↗
An application of the power series method to solution of the first boundary value problem for an equation of a parabolic type
Solution of first boundary value problem for parabolic equation using power series method
Filchakov, P. F.↗
Comparison of circular orbit and Fourier power series ephemeris representations for backup use by the upper atmosphere research satellite onboard computer
The Upper Atmosphere Research Satellite (UARS) is a three-axis stabilized Earth-pointing spacecraft in a low-Earth orbit. The UARS onboard computer (OBC) uses a Fourier Power Series (FPS) ephemeris representation that includes 42 position and 42 velocity coefficients per axis, with position residuals at 10-minute intervals. New coefficients and 32 hours of residuals are uploaded daily. This study evaluated two backup methods that permit the OBC to compute an approximate spacecraft ephemeris in the event that new ephemeris data cannot be uplinked for several days: (1) extending the use of the FPS coefficients previously uplinked, and (2) switching to a simple circular orbit approximation designed and tested (but not implemented) for LANDSAT-D. The FPS method provides greater accuracy during the backup period and does not require additional ground operational procedures for generating and uplinking an additional ephemeris table. The tradeoff is that the high accuracy of the FPS will be degraded slightly by adopting the longer fit period necessary to obtain backup accuracy for an extended period of time. The results for UARS show that extended use of the FPS is superior to the circular orbit approximation for short-term ephemeris backup.
Kast, J. R.↗
Rational approximations from power series of vector-valued meromorphic functions
Let F(z) be a vector-valued function, F: C yields C(sup N), which is analytic at z = 0 and meromorphic in a neighborhood of z = 0, and let its Maclaurin series be given. In this work we developed vector-valued rational approximation procedures for F(z) by applying vector extrapolation methods to the sequence of partial sums of its Maclaurin series. We analyzed some of the algebraic and analytic properties of the rational approximations thus obtained, and showed that they were akin to Pade approximations. In particular, we proved a Koenig type theorem concerning their poles and a de Montessus type theorem concerning their uniform convergence. We showed how optical approximations to multiple poles and to Laurent expansions about these poles can be constructed. Extensions of the procedures above and the accompanying theoretical results to functions defined in arbitrary linear spaces was also considered. One of the most interesting and immediate applications of the results of this work is to the matrix eigenvalue problem. In a forthcoming paper we exploited the developments of the present work to devise bona fide generalizations of the classical power method that are especially suitable for very large and sparse matrices. These generalizations can be used to approximate simultaneously several of the largest distinct eigenvalues and corresponding eigenvectors and invariant subspaces of arbitrary matrices which may or may not be diagonalizable, and are very closely related with known Krylov subspace methods.
Sidi, Avram↗
Operations on power series and higher derivatives of functions
Equations are derived for the general terms of the series obtained when certain operations are carried out on one or several given series. Explicit and recurrence relations are given for the literal groups which are common to the given series. A general method is shown to obtain the multipliers which depend on the operation being performed, and the method is illustrated for several examples.
Chang, F.-C.↗
The rate of convergence of Chebyshev polynomials for functions which have asymptotic power series about one endpoint
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Boyd, J. P.↗
Power series expansions for the frequency and period of the limit cycle of the van der Pol equation
An equation reported by van der Pol (1926) in connection with relaxation-oscillations studies is considered. The equation contains the factor epsilon which can assume values in the range from zero to infinity. The period T(epsilon), or equivalently the frequency nu(epsilon) of the limit cycle has been studied. However, to date there has been little success in discovering the analytical structure of T(epsilon) as a function of epsilon. The present investigation has the objectives to present the Taylor series expansion of nu(epsilon), to locate the singularities which determine the radius of convergence of that expansion, to introduce a new damping variable in terms of which the expansion converges for all epsilon, to form a new expansion for the period T(epsilon) which improves the rate of convergence of the series, to attempt to 'complete' the series, and to compare the obtained results with the numerically determined values of T(epsilon) and with the asymptotic approximation valid for large epsilon.
Andersen, C. M.↗
A power-series solution for the unsteady laminar boundary-layer flow in an expansion wave of finite width moving through a gas initially at rest
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Cohen, Nathaniel B↗
Some comments on series solutions
The use of power series and truncated power series in the MACSYMA system for algebraic manipulation is illustrated. Algebraic and differential equations are solved using Taylor series or asymptotic series. Deficiencies of the current scheme are noted, and remedies suggested.
Fateman, R. J.↗
Application of Rosseland approximation and solution based on series expansion of the emission power to radiation problems.
Emission-power series expansion solution and Rosseland approximation applied to radiation problems, discussing errors introduced by approximate methods
Hartnett, J. P.↗
Modeling the full-bridge series-resonant power converter
A steady state model is derived for the full-bridge series-resonant power converter. Normalized parametric curves for various currents and voltages are then plotted versus the triggering angle of the switching devices. The calculations are compared with experimental measurements made on a 50 kHz converter and a discussion of certain operating problems is presented.
King, R. J.↗
On the Character of the Zeros of Polynomials Relevant to the Small Curvature Approximation in Asymptotic Diffraction Theory
In dealing with the problem of estimating the high frequency coupling between two antennas mounted on a non-metallic aircraft skin, one is faced with approximation of the spectral integrals representing the propagation of rays along the geodesics of the surface between the antennas. When the antennas are sufficiently separated, the integrals can be conveniently represented as a rapidly convergent residue series. On the other hand, when the antennas are in close proximity, the residue series fails to converge rapidly and a power series representation proves to be efficacious. [Paknys and Wang, IEEE Trans. AP-35(3), 1987, 293-298]. When the effective surface impedance is not small, an intermediate region of separation appears in which neither the residue series nor the power series is effective. Recently, an asymptotic formalism was presented [Pogorzelski, National Radio Science Meeting, Boulder, CO, January 1995] which extends the earlier work of Bremmer [IRE Trans. AP-6, 1958, 267-272] and Wait curvature approximation' to the case of general (non-azimuthal) ray directions on the surface of a cylinder (excluding only axial propagation). Based on the formulation of Pearson [Radio Sci. 21(4), 1986] this asymptotic formalism provided a means of approximating the spectral integrals in the intermediate region of separation.
Zeros Polynomials↗
Comparison of various methods for mathematical analysis of the Foucault knife edge test pattern to determine optical imperfections
The linearized integral equation for the Foucault test of a solid mirror was solved by various methods: power series, Fourier series, collocation, iteration, and inversion integral. The case of the Cassegrain mirror was solved by a particular power series method, collocation, and inversion integral. The inversion integral method appears to be the best overall method for both the solid and Cassegrain mirrors. Certain particular types of power series and Fourier series are satisfactory for the Cassegrain mirror. Numerical integration of the nonlinear equation for selected surface imperfections showed that results start to deviate from those given by the linearized equation at a surface deviation of about 3 percent of the wavelength of light. Several possible procedures for calibrating and scaling the input data for the integral equation are described.
Gatewood, B. E.↗
UV Spectral Variability and the Lyman-Alpha Forest in the Lensed Quasar Q0957+561
Far-ultraviolet spectra of the gravitational lens components Q0957+561 A and B were obtained with the Hubble Space Telescope Faint Object Spectrograph (FOS) at five equally spaced epochs, one every two weeks. We confirm the flux variability of the quasar's Lyman-alpha and O VI lambda 1037 emission lines reported in IUE (International Ultraviolet Explorer) spectra. The fluxes in these lines vary on a time scale of weeks in the local rest frame (LRF), independently of each other and of the surrounding continuum. The individual spectra of each image were co-added to investigate the properties of the Lyman-alpha forest along the two lines of sight to the quasar. Absorption lines having equivalent width W (sub lambda) greater than or equal to 0.3 Angstroms in the LRF not previously identified as interstellar lines, metal lines, or higher order Lyman lines were taken to be LY-alpha forest lines. The existence of each line in this consistently selected set was then verified by its presence in two archival FOS spectra with approx. 1.5 times higher signal to noise than our co-added spectra. Ly-alpha forest lines with W (sub lambda) greater than or equal to 0.3 Angstroms appear at 42 distinct wavelengths in the spectra of the two images. Two absorption lines in the spectrum of image A have no counterpart at that wavelength in the spectrum of image B, and two lines in image B have no counterpart in image A. Based on the separation of the lines of sight at the redshift of the absorption lines appearing in only one spectrum, the density of the absorbing clouds in the direction of Q0957+561 must change significantly over a distance of 79 (+34, -26) h (sub 50) (sup -1) kpc in the simplified model where the absorbers are treated as spherical clouds and the characteristic dimension is the radius. (We adopt H (sub 0)= 50 h (sub 50) km s (sup -1) kpc (sup -1), q (sub 0) = 1/2, and LAMBDA = 0 throughout the paper.) The two limits define the 68% confidence interval on the characteristic dimension, equivalent to the 1 sigma confidence interval for a Gaussian distribution. The 95% confidence interval extends from (32 - 250) h (sub 50) (sup -1) kpc. We show in the Appendix that the fraction of Ly-alpha forest lines that appear in only one spectrum can be expressed as a rapidly converging power series in 1/r, where r the ratio of the radius of the cloud to the separation of the two LOS at the redshift of the cloud. This power series can be rewritten to give r in terms of the fraction of Ly-alpha forest wavelengths that appear in the spectrum of only one image. A simple linear approximation to the solution which everywhere agrees with the power series solution to better than 0.8% for r greater than or equal to 2 is derived in the Appendix.
Dolan, J. F.↗
The Lyman-Alpha Forest in the Lensed Quasar Q0957+561 and the Characteristic Dimension of the Absorbing Clouds
Far-ultraviolet spectra of the gravitational lens components Q0957+561 A and 9 were obtained with the Hubble Space Telescope Faint Object Spectrograph (FOS) at five equally spaced epochs, one every two weeks. We confirm the flux variability of the quasar's Lyman-alpha and 0 VI emission lines reported by Dolan et al. (1995) in IUE spectra. The fluxes in these lines vary on a time scale of weeks in the observer's rest frame, independently of each other and of the surrounding continuum. The individual spectra of each image were co-added to investigate the properties of the Lyman-alpha forest along the two lines of sight to the quasar. Absorption lines having equivalent width W > 0.3 A in the observer's frame not previously identified by Michalitsianos et al. (1997) as interstellar lines, metal lines, or higher order Lyman lines were taken to be Ly-alpha forest lines. The existence of each line in this consistently selected set was then verified by its presence in two archival FOS spectra with -1.5 times higher signal to noise than our co-added spectra. Ly-alpha forest lines with W > 0.3 A appear at 41 distinct wavelengths in the spectra of the two images. one absorption line in the spectrum of image A has no counterpart in the spectrum of image B and one line in image B has no counterpart in image A. Based on the separation of the lines of sight over the redshift range searched for Ly-forest lines, the density of the absorbing clouds in the direction of Q0957+561 must change significantly over a distance R = 160 (+120, -70)/ h(sub 50) kpc in the simplified model where the absorbers are treated as spherical clouds and the characteristic dimension, R, is the radius. (We adopt H(sub 0) = 50 h(sub 50) km/s/ kpc, q(sub 0) = 1/2, and lambda = 0 throughout the paper.) The 95% confidence interval on R extends from (50 - 950)/h(sub 50) kpc We show in the Appendix that the fraction of Ly-alpha forest lines that appear in only one spectrum can be expressed as a rapidly converging power series in 1/r, where r the ratio of the radius of the cloud to the separation of the two lines of sight at the redshift of the cloud. This power series can be rewritten to give r in terms of the fraction of Ly-forest wavelengths that appear in the spectrum of only one image. A simple linear approximation to the solution that everywhere agrees with the power series solution to better than 0.8% for r > 2 is derived in the Appendix.
Dolan, Joseph↗
Analytic approximations to the modon dispersion relation
Three explicit analytic approximations are given to the modon dispersion relation developed by Flierl et al. (1980) to describe Gulf Stream rings and related phenomena in the oceans and atmosphere. The solutions are in the form of k(q), and are developed in the form of a power series in q for small q, an inverse power series in 1/q for large q, and a two-point Pade approximant. The low order Pade approximant is shown to yield a solution for the dispersion relation with a maximum relative error for the lowest branch of the function equal to one in 700 in the q interval zero to infinity.
Boyd, J. P.↗