About the Project

central differences in imaginary direction

AdvancedHelp

(0.003 seconds)

11—20 of 949 matching pages

11: 18.2 General Orthogonal Polynomials
In the former case we also require …
§18.2(ii) x -Difference Operators
If the orthogonality discrete set X is { 0 , 1 , , N } or { 0 , 1 , 2 , } , then the role of the differentiation operator d / d x in the case of classical OP’s (§18.3) is played by Δ x , the forward-difference operator, or by x , the backward-difference operator; compare §18.1(i). … If the orthogonality interval is ( , ) or ( 0 , ) , then the role of d / d x can be played by δ x , the central-difference operator in the imaginary direction18.1(i)). … See Chihara (1978, pp. 86–89), and, in slightly different notation, Ismail (2009, §§2.3, 2.6, 2.10), where it is assumed that μ 0 = 1 . …
12: Annie A. M. Cuyt
She received her Doctorate in Science in 1982 from the same university, summa cum laude and with the felicitations of the jury. …Her main research interest is in the area of numerical approximation theory and its applications to a diversity of problems in scientific computing. As a consequence her expertise spans a wide range of activities from pure abstract mathematics to computer arithmetic and different engineering applications. A lot of her research has been devoted to rational approximations, in one as well as in many variables, and sparse interpolation. …She directs the HPC core facility of the University of Antwerp and is a member of the Flemish HPC steering committee. …
13: 18.39 Applications in the Physical Sciences
§18.39 Applications in the Physical Sciences
In what follows the radial and spherical radial eigenfunctions corresponding to (18.39.27) are found in four different notations, with identical eigenvalues, all of which appear in the current and past mathematical and theoretical physics and chemistry literatures, regarding this central problem. … noting that the ψ p , l ( r ) are real, follows from the fact that the Schrödinger operator of (18.39.28) is self-adjoint, or from the direct derivation of Dunkl (2003). … see Bethe and Salpeter (1957, p. 13), Pauling and Wilson (1985, pp. 130, 131); and noting that this differs from the Rodrigues formula of (18.5.5) for the Laguerre OP’s, in the omission of an n ! in the denominator. …
§18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences
14: Bibliography W
  • Z. Wang and R. Wong (2003) Asymptotic expansions for second-order linear difference equations with a turning point. Numer. Math. 94 (1), pp. 147–194.
  • Z. Wang and R. Wong (2005) Linear difference equations with transition points. Math. Comp. 74 (250), pp. 629–653.
  • G. Wolf (1998) On the central connection problem for the double confluent Heun equation. Math. Nachr. 195, pp. 267–276.
  • R. Wong and H. Li (1992a) Asymptotic expansions for second-order linear difference equations. II. Stud. Appl. Math. 87 (4), pp. 289–324.
  • R. Wong and H. Li (1992b) Asymptotic expansions for second-order linear difference equations. J. Comput. Appl. Math. 41 (1-2), pp. 65–94.
  • 15: 10.73 Physical Applications
    Laplace’s equation governs problems in heat conduction, in the distribution of potential in an electrostatic field, and in hydrodynamics in the irrotational motion of an incompressible fluid. … This equation governs problems in acoustic and electromagnetic wave propagation. …Consequently, Bessel functions J n ( x ) , and modified Bessel functions I n ( x ) , are central to the analysis of microwave and optical transmission in waveguides, including coaxial and fiber. … More recently, Bessel functions appear in the inverse problem in wave propagation, with applications in medicine, astronomy, and acoustic imaging. … In quantum mechanics the spherical Bessel functions arise in the solution of the Schrödinger wave equation for a particle in a central potential. …
    16: 18.25 Wilson Class: Definitions
    For the Wilson class OP’s p n ( x ) with x = λ ( y ) : if the y -orthogonality set is { 0 , 1 , , N } , then the role of the differentiation operator d / d x in the Jacobi, Laguerre, and Hermite cases is played by the operator Δ y followed by division by Δ y ( λ ( y ) ) , or by the operator y followed by division by y ( λ ( y ) ) . Alternatively if the y -orthogonality interval is ( 0 , ) , then the role of d / d x is played by the operator δ y followed by division by δ y ( λ ( y ) ) . … Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials W n ( x ; a , b , c , d ) , continuous dual Hahn polynomials S n ( x ; a , b , c ) , Racah polynomials R n ( x ; α , β , γ , δ ) , and dual Hahn polynomials R n ( x ; γ , δ , N ) . …
    18.25.8 h n = n !  2 π j < Γ ( n + a j + a ) .
    18.25.15 h n = n ! ( N n ) ! ( γ + δ + 2 ) N N ! ( γ + 1 ) n ( δ + 1 ) N n .
    17: 28.27 Addition Theorems
    Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. …
    18: Bibliography C
  • CEPHES (free C library)
  • R. Chattamvelli and R. Shanmugam (1997) Algorithm AS 310. Computing the non-central beta distribution function. Appl. Statist. 46 (1), pp. 146–156.
  • I. Cherednik (1995) Macdonald’s evaluation conjectures and difference Fourier transform. Invent. Math. 122 (1), pp. 119–145.
  • R. C. Y. Chin and G. W. Hedstrom (1978) A dispersion analysis for difference schemes: Tables of generalized Airy functions. Math. Comp. 32 (144), pp. 1163–1170.
  • A. G. Constantine (1963) Some non-central distribution problems in multivariate analysis. Ann. Math. Statist. 34 (4), pp. 1270–1285.
  • 19: Staff
    Frank W. J. Olver [December 15, 1924-April 23, 2013] served as Editor-in-Chief and Mathematics Editor for the DLMF project from its inception until his death on April 23, 2013. …
  • Ian J. Thompson, Lawrence Livermore National Laboratory, Chap. 33

  • Mourad E. H. Ismail, University of Central Florida

  • Ian J. Thompson, Lawrence Livermore National Laboratory, for Chap. 33

  • 20: Bibliography L
  • W. Lay and S. Yu. Slavyanov (1998) The central two-point connection problem for the Heun class of ODEs. J. Phys. A 31 (18), pp. 4249–4261.
  • N. L. Lepe (1985) Functions on a parabolic cylinder with a negative integer index. Differ. Uravn. 21 (11), pp. 2001–2003, 2024 (Russian).
  • N. A. Lukaševič and A. I. Yablonskiĭ (1967) On a set of solutions of the sixth Painlevé equation. Differ. Uravn. 3 (3), pp. 520–523 (Russian).
  • N. A. Lukaševič (1965) Elementary solutions of certain Painlevé equations. Differ. Uravn. 1 (3), pp. 731–735 (Russian).
  • N. A. Lukaševič (1971) The second Painlevé equation. Differ. Uravn. 7 (6), pp. 1124–1125 (Russian).