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1: Foreword
Particular attention is called to the generous support of the National Science Foundation, which made possible the participation of experts from academia and research institutes worldwide. … November 20, 2009 …
2: 3.8 Nonlinear Equations
Solutions are called roots of the equation, or zeros of f . … and the solutions are called fixed points of ϕ . … Consider x = 20 and j = 19 . We have p ( 20 ) = 19 ! and a 19 = 1 + 2 + + 20 = 210 . … For an arbitrary starting point z 0 , convergence cannot be predicted, and the boundary of the set of points z 0 that generate a sequence converging to a particular zero has a very complicated structure. …
3: 11.2 Definitions
Particular solutions: … Particular solutions: …
§11.2(iii) Numerically Satisfactory Solutions
When z and ν 0 , numerically satisfactory general solutions of (11.2.7) are given by … When ν 0 , numerically satisfactory general solutions of (11.2.9) are given by …
4: Frank W. J. Olver
He is particularly known for his extensive work in the study of the asymptotic solution of differential equations, i. …, the behavior of solutions as the independent variable, or some parameter, tends to infinity, and in the study of the particular solutions of differential equations known as special functions (e. …
5: Bibliography M
  • A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.
  • Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.
  • R. Metzler, J. Klafter, and J. Jortner (1999) Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.
  • D. S. Moak (1981) The q -analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.
  • N. Mohankumar and A. Natarajan (1997) The accurate evaluation of a particular Fermi-Dirac integral. Comput. Phys. Comm. 101 (1-2), pp. 47–53.
  • 6: 18.38 Mathematical Applications
    Differential Equations: Spectral Methods
    Each of these typically require a particular non-classical weight functions and analysis of the corresponding OP’s. … However, by using Hirota’s technique of bilinear formalism of soliton theory, Nakamura (1996) shows that a wide class of exact solutions of the Toda equation can be expressed in terms of various special functions, and in particular classical OP’s. … has a solutionDunkl type operators and nonsymmetric polynomials have been associated with various other families in the Askey scheme and q -Askey scheme, in particular with Wilson polynomials, see Groenevelt (2007), and with Jacobi polynomials, see Koornwinder and Bouzeffour (2011, §7). …
    7: 27.15 Chinese Remainder Theorem
    The Chinese remainder theorem states that a system of congruences x a 1 ( mod m 1 ) , , x a k ( mod m k ) , always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod m ), where m is the product of the moduli. … Their product m has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result ( mod m ) , which is correct to 20 digits. …
    8: 15.11 Riemann’s Differential Equation
    The most general form is given by … The complete set of solutions of (15.11.1) is denoted by Riemann’s P -symbol: …In particular, …denotes the set of solutions of (15.10.1).
    §15.11(ii) Transformation Formulas
    9: 3.6 Linear Difference Equations
    §3.6 Linear Difference Equations
    §3.6(ii) Homogeneous Equations
    Thus the asymptotic behavior of the particular solution 𝐄 n ( 1 ) is intermediate to those of the complementary functions J n ( 1 ) and Y n ( 1 ) ; moreover, the conditions for Olver’s algorithm are satisfied. …
    10: 10.25 Definitions
    Its solutions are called modified Bessel functions or Bessel functions of imaginary argument. …
    §10.25(ii) Standard Solutions
    In particular, the principal branch of I ν ( z ) is defined in a similar way: it corresponds to the principal value of ( 1 2 z ) ν , is analytic in ( , 0 ] , and two-valued and discontinuous on the cut ph z = ± π . …
    §10.25(iii) Numerically Satisfactory Pairs of Solutions
    Table 10.25.1 lists numerically satisfactory pairs of solutions2.7(iv)) of (10.25.1). …