About the Project

fixed points

AdvancedHelp

(0.002 seconds)

11—20 of 31 matching pages

11: 2.4 Contour Integrals
For a coalescing saddle point and endpoint see Olver (1997b, Chapter 9) and Wong (1989, Chapter 7); if the endpoint is an algebraic singularity then the uniform approximants are parabolic cylinder functions with fixed parameter, and if the endpoint is not a singularity then the uniform approximants are complementary error functions. …
12: 20.2 Definitions and Periodic Properties
For fixed τ , each θ j ( z | τ ) is an entire function of z with period 2 π ; θ 1 ( z | τ ) is odd in z and the others are even. … The four points ( 0 , π , π + τ π , τ π ) are the vertices of the fundamental parallelogram in the z -plane; see Figure 20.2.1. The points …are the lattice points. The theta functions are quasi-periodic on the lattice: …
13: 2.8 Differential Equations with a Parameter
For a coalescing turning point and simple pole see Nestor (1984) and Dunster (1994b); in this case the uniform approximants are Whittaker functions (§13.14(i)) with a fixed value of the second parameter. …
14: Bibliography U
  • F. Ursell (1972) Integrals with a large parameter. Several nearly coincident saddle-points. Proc. Cambridge Philos. Soc. 72, pp. 49–65.
  • F. Ursell (1980) Integrals with a large parameter: A double complex integral with four nearly coincident saddle-points. Math. Proc. Cambridge Philos. Soc. 87 (2), pp. 249–273.
  • F. Ursell (1984) Integrals with a large parameter: Legendre functions of large degree and fixed order. Math. Proc. Cambridge Philos. Soc. 95 (2), pp. 367–380.
  • 15: 31.15 Stieltjes Polynomials
    The system (31.15.2) determines the z k as the points of equilibrium of n movable (interacting) particles with unit charges in a field of N particles with the charges γ j / 2 fixed at a j . …
    16: 21.7 Riemann Surfaces
    Consider the set of points in 2 that satisfy the equation …Equation (21.7.1) determines a plane algebraic curve in 2 , which is made compact by adding its points at infinity. …This compact curve may have singular points, that is, points at which the gradient of P ~ vanishes. … The zeros λ j , j = 1 , 2 , , 2 g + 1 of Q ( λ ) specify the finite branch points P j , that is, points at which μ j = 0 , on the Riemann surface. Denote the set of all branch points by B = { P 1 , P 2 , , P 2 g + 1 , P } . …
    17: 10.2 Definitions
    This solution of (10.2.1) is an analytic function of z , except for a branch point at z = 0 when ν is not an integer. … For fixed z ( 0 ) each branch of J ν ( z ) is entire in ν . … Whether or not ν is an integer Y ν ( z ) has a branch point at z = 0 . … For fixed z ( 0 ) each branch of Y ν ( z ) is entire in ν . … Each solution has a branch point at z = 0 for all ν . …
    18: 10.25 Definitions
    It has a branch point at z = 0 for all ν . … For fixed z ( 0 ) each branch of I ν ( z ) and K ν ( z ) is entire in ν . …
    19: 14.15 Uniform Asymptotic Approximations
    §14.15(i) Large μ , Fixed ν
    For the interval 1 < x < 1 with fixed ν , real μ , and arbitrary fixed values of the nonnegative integer J , … For the interval 1 < x < the following asymptotic approximations hold when μ , with ν ( 1 2 ) fixed, uniformly with respect to x : …
    §14.15(iii) Large ν , Fixed μ
    For ν and fixed μ ( 0 ), …
    20: 11.6 Asymptotic Expansions
    §11.6(i) Large | z | , Fixed ν
    §11.6(ii) Large | ν | , Fixed z
    For fixed λ ( > 1 ) …and for fixed λ ( > 0 ) For fixed λ ( > 0 )