Higher Transcendental Functions Research Articles

Non-linear dynamics of spinodal decomposition

A new approach is proposed to describe the spinodal decomposition, in particular, in polymer binary blends. In the framework of this approach, the spinodal decomposition is described as a relaxation of one-time structure factor S(q,t) treated as an independent dynamic object (a peculiar two-point order parameter). The dynamic equation for S(q,t), including the explicit expression for the corresponding effective kinetic coefficient, is derived. In the first approximation this equation is identical to the Langer equation. We first solved it both in terms of higher transcendental functions and numerically. The asymptotic behaviour of S(q,t) at large (from the onset of spinodal decomposition) times is analytically described. The values obtained for the power-law growth exponent for the large-time peak value and position of S(q,t) are in good agreement with experimental data and results of numerical integration of the Cahn-Hilliard equation.

Special Functions of the Isomonodromy Type

We introduce a new notion, a special function of the isomonodromy type, and show that most of the functions known in applied mathematics and mathematical physics as special functions belong to this type. This definition provides a unified approach to the theories of ‘linear’ special functions, i.e., classical higher transcendental functions, and ‘nonlinear’ special functions, i.e., functions of the Painleve type. We also show that our definition has not only a conceptual (methodological) value; many well-known properties of the single-variable special functions can be re-derived not only from the isomonodromy point of view, but also the practical one too: (1) the isomonodromy approach is already known as an extremely useful one in the theory of Painleve functions, (2) many properties (some of them can be new ones) of the multi-variable special functions can be obtained on a regular basis, and (3) the definition gives rise to several interesting mathematical questions which are discussed in the paper. We also make some remarks concerning the analogous description (via q-deformations) of q-special functions.

Remarks on a generalized beta function

We show that a certain generalized beta function B( x, y; b) which reduces to Euler's beta functions B( x, y) when its variable b vanishes and preserves symmetry in its parameters may be represented in terms of a finite number of well known higher transcendental functions except (possibly) in the case when one of its parameters is an integer and the other is not. In the latter case B( x, y; b) may be represented as an infinite series of either Wittaker functions or Laguerre polynomials. As a byproduct of this investigation we deduce representations for several infinite series containing Wittaker functions, Laguerre polynomials, and products of both.

On generalized gauss-hecke sums and theta constants

In the quadratic case, the analytic proof of the reciprocity law for a numbr field involves a connection between Gauss-Hecke sums and theta constants or Thetanullwerte. We propose a generalization of Gauss-Hecke sums to higher degrees and discuss some aspects of the problem of determining higher transcendental functions by suitable boundary conditions which involve these objects in a natural way.

Some formulas of L. Carlitz on Hermite polynomials

We have used the idea of ‘quasi inner product’ introduced by L. R. Bragg in 1986 to consider generating series urn:x-wiley:01611712:media:ijmm845912:ijmm845912-math-0001 studied by L. Carlitz in 1963. The pecularity of the series is that there is (n!) 2 in the denominator, which has a striking deviation from the usuaI generating series containing n! in the denominator. Our generating function for the said generating series is quite different from that of Carlitz, but somewhat analogous to generating integrals derived by G. N. Watson (Higher Transcendental function Vol.III, P 271‐272 for the case of Legendre, Gegenbauer and Jacobi polynomials.

Invariants of particle motion in one-dimensional time-dependent potentials

The author considers one-dimensional time-dependent Hamiltonians of the form H=p2/2+V(q,t). It is shown that the potentials determined in an implicit way by the relation Phi (V)=(a1V2+a2V+a3)t+q, where Phi is an arbitrary function and a1, a2 and a3 are arbitrary parameters, admit a single-valued constant of motion. This constant of motion is a higher transcendental function in the momentum.

Integrable Hamiltonians with higher transcendental invariants

A new family of integrable two-dimensional Hamiltonians is found. The corresponding potentials V(x,y) are determined by the relation Phi (V)-y-x/(a1V+a2)=0, where Phi is an arbitrary function and a1, a2 are arbitrary parameters. The second constants of motion are higher transcendental functions in momenta. These results are found by deriving non-separable solutions of the Hamilton-Jacobi equation.

Technical memorandum. Kummer and Tricomi functions for analysis of circular gyrotropic waveguides with azimuthal magnetisation

The wave functions for propagation of rotationally symmetric modes in circular gyrotropic waveguides with azimuthal magnetisation (the complex Kummer and Tricomi confluent hypergeometric functions) are studied analytically and numerically. Based on specially compiled tables, families of curves are presented, illustrating the behaviour of both higher transcendental functions. The analysis allows a comprehensive treatment of closed and opened circular guiding structures with radially stratified gyrotropic layers, magnetised azimuthally.

Recurrence Relations for the Coefficients in Jacobi Series Solutions of Linear Differential Equations

Next article Recurrence Relations for the Coefficients in Jacobi Series Solutions of Linear Differential EquationsStanisław LewanowiczStanisław Lewanowiczhttps://doi.org/10.1137/0517074PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractA method is presented for obtaining recurrence relations for the coefficients in Jacobi series solutions of linear ordinary differential equations with polynomial coefficients.[1] C. W. Clenshaw, The numerical solution of linear differential equations in Chebyshev series, Proc. Cambridge Philos. Soc., 53 (1957), 134–149 18,516a 0077.32503 CrossrefGoogle Scholar[2] David Elliott, The expansion of functions in ultraspherical polynomials, J. Austral. Math. Soc., 1 (1959/1960), 428–438 23:A1997 0099.28603 CrossrefGoogle Scholar[3] A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, New York, 1953 Google Scholar[4] L. Fox, Chebyshev methods for ordinary differential equations, Comput. J., 4 (1961/1962), 318–331 24:B2554 0103.34203 CrossrefISIGoogle Scholar[5] L. Fox and , I. B. Parker, Chebyshev polynomials in numerical analysis, Oxford University Press, London, 1968ix+205, England 37:3733 Google Scholar[6] K. O. Geddes, Symbolic computation of recurrence equations for the Chebyshev series solution of linear ODE'S, Proc. 1977 MACSYMA Users' Conference, Univ. of California, Berkeley, CA, 1977, 405–423, NASA CP-2012 Google Scholar[7] T. S. Horner, Recurrence relations for the coefficients in Chebyshev series solutions of ordinary differential equations, Math. Comp., 35 (1980), 893–905 81d:65038 0446.65040 CrossrefISIGoogle Scholar[8] S. Lewanowicz, Construction of a recurrence relation of the lowest order for coefficients of the Gegenbauer series, Zastos. Mat., 15 (1976), 345–396 54:6527 0357.33006 Google Scholar[9] S. Lewanowicz, Construction of the lowest-order recurrence relation for the Jacobi coefficients, Zastos. Mat., 17 (1983), 655–675 85d:33030 0591.65089 Google Scholar[10] Stanisław Lewanowicz, Recurrence relations for hypergeometric functions of unit argument, Math. Comp., 45 (1985), 521–535 86m:33004 0583.33005 CrossrefISIGoogle Scholar[11] Y. L. Luke, The Special Functions and their Approximations, Academic Press, New York, 1969 Google Scholar[12] A. Magnus, Application des récurrences au calcul d'une classe d'intégrales, Rep., 71, Inst. Math. Pure Appl., Univ. de Louvain, 1974 Google Scholar[13] A. G. Morris and , T. S. Horner, Chebyshev polynomials in the numerical solution of differential equations, Math. Comp., 31 (1977), 881–891 56:1729 0386.65040 CrossrefISIGoogle Scholar[14] O. Oluremi Olaofe, On the Tchebyschev method of solution of ordinary differential equations, J. Math. Anal. Appl., 60 (1977), 1–7 10.1016/0022-247X(77)90043-9 56:1724 0363.65065 CrossrefISIGoogle Scholar[15] Stefan Paszkowski, Zastosowania numeryczne wielomianów i szeregów Czebyszewa, Państwowe Wydawnictwo Naukowe, Warsaw, 1975, 481– 56:13534 0423.65012 Google Scholar[16] N. Robertson, An ALTRAN program for finding a recursion formula for the Gegenbauer coefficients of a function, Spec. Rep., SWISK 11, Nat. Res. Inst. for Math. Sci., Pretoria, 1979 Google Scholar[17] Jet Wimp, Computation with recurrence relations, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984xii+310 85f:65001 Google ScholarKeywordsJacobi seriesJacobi coefficientsrecurrence relationsdifference operatorslinear differential equation Next article FiguresRelatedReferencesCited byDetails Descriptions of fractional coefficients of Jacobi polynomial expansions18 April 2022 | The Journal of Analysis, Vol. 30, No. 4 Cross Ref On Jacobi polynomials and fractional spectral functions on compact symmetric spaces4 January 2021 | The Journal of Analysis, Vol. 29, No. 3 Cross Ref Spectral Solutions for Differential and Integral Equations with Varying Coefficients Using Classical Orthogonal Polynomials17 July 2018 | Bulletin of the Iranian Mathematical Society, Vol. 45, No. 2 Cross Ref On the coefficients of differentiated expansions and derivatives of chebyshev polynomials of the third and fourth kindsActa Mathematica Scientia, Vol. 35, No. 2 Cross Ref On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials6 January 2004 | Journal of Physics A: Mathematical and General, Vol. 37, No. 3 Cross Ref On the coefficients of differentiated expansions and derivatives of Jacobi polynomials8 April 2002 | Journal of Physics A: Mathematical and General, Vol. 35, No. 15 Cross Ref The ultraspherical coefficients of the moments of a general-order derivative of an infinitely differentiable functionJournal of Computational and Applied Mathematics, Vol. 89, No. 1 Cross Ref On the legendre coefficients of the moments of the general order derivative of an infinitely differentiable functionInternational Journal of Computer Mathematics, Vol. 56, No. 1-2 Cross Ref Evaluation of Bessel function integrals with algebraic singularitiesJournal of Computational and Applied Mathematics, Vol. 37, No. 1-3 Cross Ref Properties of the polynomials associated with the Jacobi polynomials1 January 1986 | Mathematics of Computation, Vol. 47, No. 176 Cross Ref Volume 17, Issue 5| 1986SIAM Journal on Mathematical Analysis History Submitted:15 April 1985Published online:17 July 2006 InformationCopyright © 1986 Society for Industrial and Applied MathematicsKeywordsJacobi seriesJacobi coefficientsrecurrence relationsdifference operatorslinear differential equationMSC codes42C1039A7065L0565L10PDF Download Article & Publication DataArticle DOI:10.1137/0517074Article page range:pp. 1037-1052ISSN (print):0036-1410ISSN (online):1095-7154Publisher:Society for Industrial and Applied Mathematics

Correction [to “Lateral waves: New formulas for E1ϕ and E1z” by T. T. Wu and R. W. P. King

Radio ScienceVolume 19, Issue 5 p. 1422-1422 CorrectionsFree Access Correction [to “Lateral waves: New formulas for E1ϕ and E1z” by T. T. Wu and R. W. P. King] First published: September-October 1984 https://doi.org/10.1029/RS019i005p01422Citations: 3AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat No abstract is available for this article. References Bateman, H., Higher Transcendental Functions, II, 396, McGraw-Hill, New York, 1953. Gradshteyn, I. S., I. M. Ryzhik, Tables of Integrals, Series and Products, 1160, Academic, New York, 1980. Citing Literature Volume19, Issue5September-October 1984Pages 1422-1422 ReferencesRelatedInformation

Asymptotische Darstellungen der hypergeometrischen Funktionen für grosse Werte eines Parameters

Asymptotic expansions of the hypergeometric function F(a, b, c; z) for |b| \to \infty where a, c, z are fixed complex numbers given in well-known tables (e.g. Bateman/Erdélyi: Higher transcendental functions I. New York 1953) are incorrect. In the present paper asymptotic representations of the hypergeomctric function F for a (complex) \to \infty are derived where b, c (c \neq 0, -1, -2,\dots) and z (z \neq 0 , |Arg (1- z) | < \pi are fixed complex numbers. By change of a and b appropriate asymptotic representations of F for |b| \to \infty are obtained.

Mathieu functions of general order: connection formulae, base functions and asymptotic formulae. III. The Liouville-Green method and its extensions

An account is given of the Liouville-Green method for the approximate solution, with error estimates, of linear second-order differential equations, together with certain extensions of the method. The purpose is to make readily available a range of techniques for use in the two final parts of the present series. The topics treated include: ( a ) the construction of approximations in terms of both elementary and higher transcendental functions, ( b ) the relations between approximations of the same solution in terms of different functions, ( c ) the identification of solutions and the estimation of connection coefficients, ( d ) uniform estimation of the error-control function in problems with more than one widely ranging parameter, ( e ) the construction of majorants for approximating functions, the last two being required for the derivation of satisfactory error estimates. There is little in this part that is new, though a method of constructing approximations in terms of Bessel functions is developed specifically for application to the Mathieu equation. Apart from this, some aspects of the presentation are thought to be novel.

Mathieu functions of general order: connection formulae, base functions and asymptotic formulae. V. Approximations in terms of higher transcendental functions

The methods of part III are further applied to the construction of approximations for the fundamental solution and base functions of part II in terms of higher transcendental functions. The domain of validity is now the complete half-strip {z; 0 ≤ Re z ≤ ½π, Im z ≥. 0} without exceptional point. Relative remainder estimates are again uniformly valid provided they are bounded. Specifically, approximations are obtained in terms of: ( a ) Airy functions, applicable if A ≠ ± 2h 2 ; ( b ) parabolic cylinder functions, applicable if |A ≥ 4h 2 , including A = ± 2h 2 ; ( c ) Bessel functions, applicable if |A| ≥ 4h 2 ; these formulae have maximum relative error A - 3/2 h 2 O (l) on the half-strip, even if h is arbitrarily small, provided only that A -1 is bounded. This is significantly better when A/h 2 is large than the corresponding estimate, A -½ 0(1), for the Airy function approximations. Certain more refined estimates for the auxiliary parameters introduced in part II are also obtained.

Mathieu functions of general order: connection formulae, base functions and asymptotic formulae. I. Introduction

This is the first, introductory, paper of a series devoted to the derivation of a comprehensive set of approximate formulae for solutions of Mathieu’s equation with real parameters, in terms both of elementary and of higher transcendental functions. Order-of-magnitude error-estimates are obtained; these in every case reflect faithfully the behaviour of the actual error over the appropriate range of parameters and of independent variable. The general scope of the work is outlined in this Introduction, and is compared with that of previous work, in particular that of Langer (1934 b ). There then follows a description of the plan of the work and of the content of the several parts.

On Asymptotic Average Properties of Zeros of Orthogonal Polynomials

Distribution of zeros of orthogonal polynomials is investigated by means of the coefficients of the three-term recurrence relation which generates the orthogonal polynomials.

On the Distribution of the Time up to the First Occurrence of a Given Number of Different $\ell $-Tuple Series

Previous article Next article On the Distribution of the Time up to the First Occurrence of a Given Number of Different $\ell $-Tuple SeriesV. A. Ivanov and A. E. NovikovV. A. Ivanov and A. E. Novikovhttps://doi.org/10.1137/1122063PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1A] Arthur Erdélyi, , Wilhelm Magnus, , Fritz Oberhettinger and , Francesco G. Tricomi, Higher transcendental functions. Vol. I, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953xxvi+302 MR0058756 (15,419i) 0051.30303 Google Scholar[1B] Arthur Erdélyi, , Wilhelm Magnus, , Fritz Oberhettinger and , Francesco G. Tricomi, Higher transcendental functions. Vol. II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953xvii+396 MR0058756 (15,419i) 0052.29502 Google Scholar[1C] Arthur Erdélyi, , Wilhelm Magnus, , Fritz Oberhettinger and , Francesco G. Tricomi, Higher transcendental functions. Vol. III, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955xvii+292 MR0066496 (16,586c) 0064.06302 Google Scholar[2] V. F. Kolchin, , B. A. Sevast'ianov and , V. P. Chistiakov, Random mixing, Nauka, Moscow, 1976, (in Russian) Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails References7 February 2012 Cross Ref Bibliography14 October 2011 Cross Ref On the Waiting Time of Appearance of Given Patterns Cross Ref Poisson approximations for runs and patterns of rare events1 July 2016 | Advances in Applied Probability, Vol. 23, No. 4 Cross Ref The Occurrence of Sequence Patterns in Repeated Dependent ExperimentsO. Chrysaphinou and S. Papastavridis17 July 2006 | Theory of Probability & Its Applications, Vol. 35, No. 1AbstractPDF (655 KB) Volume 22, Issue 3| 1978Theory of Probability & Its Applications History Submitted:22 January 1976Published online:28 July 2006 InformationCopyright © 1978 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1122063Article page range:pp. 533-542ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Computer-aided steady-state and transient solutions of field problems in induction devices

The paper describes a method for solving steady-state and transient electromagnetic-field problems in certain multi-region induction devices, which uses only a very small amount of analysis. In general, the device is modelled by means of a nonuniform transmission line consisting of a large number of T-circuits in cascade. Circuit-analysis computer packages can be used to solve for the circuit quantities, and hence the field quantities, at all nodes in the line. Neither higher transcendental functions nor operational methods are required.

An Exponential Stress-Strain Law for Cyclic Plasticity

A previously proposed nonlinear differential constitutive equation for creep-plasticity interaction under a uniaxial state of stress is specialized for the time independent case. The characteristics of the second derivative of the stress-strain diagram are matched by an exponential function. The integration yields higher transcendental functions. For the matching of the stress-strain diagram, four easily obtainable constants are necessary at each cycle which are fed into a newly developed FORTRAN computer program. A plotting routine yields stress-strain diagrams and hysteresis loops. The procedure gives good matches for stress-strain diagrams of Type 304 stainless steel. Specifically, stress-strain diagrams for various product forms and the initial cyclic hardening of this material are reproduced quite accurately without the usual decomposition into elastic and plastic strains.

Wave impedance of induction devices using the scalar Riccati equation

A frequently recurring problem in induction device theory involves the determination of the input wave impedance to a device. This is commonly done by solving for the field components. In this paper, a completely different method based on the scalar Riccati equation is described. It is shown that the wave impedance can be obtained by the numerical integration of a Riccati-type differential equation, having wave impedance as the unknown variable. This is illustrated with three simple models each having an unknown field distribution in one direction only. Extensive computer testing was carried out and the results obtained were found to agree very closely with published analytical methods. The method is very simple, easily programmed and avoids the use of higher transcendental functions.

Gravitational perturbations of equatorial orbits

Asymptotic solutions are developed for the motion of a geocentric satellite in the equatorial plane due to gravitational perturbations such as nonsphericity (especially oblateness) of the primary body. Axisymmetric potentials are considered. A class of transformations is developed and the equations of motion are solved by the method of generalized multiple scales. Further it is shown that the equations of motion can be transformed into the required form to within any specified degree of accuracy. The transformations form an Abelian group of infinite order which leaves the differential equations of motion invariant. Solutions are developed in terms of elementary functions instead of elliptic or other higher transcendental functions and are shown to agree with known results.