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1: 26.12 Plane Partitions
A plane partition, π , of a positive integer n , is a partition of n in which the parts have been arranged in a 2-dimensional array that is weakly decreasing (nonincreasing) across rows and down columns. … It is useful to be able to visualize a plane partition as a pile of blocks, one block at each lattice point ( h , j , k ) π . … The plane partition in Figure 26.12.1 is an example of a cyclically symmetric plane partition. … A plane partition is totally symmetric if it is both symmetric and cyclically symmetric. … The example of a strict shifted plane partition also satisfies the conditions of a descending plane partition. …
2: 36.5 Stokes Sets
Stokes sets are surfaces (codimension one) in 𝐱 space, across which Ψ K ( 𝐱 ; k ) or Ψ ( U ) ( 𝐱 ; k ) acquires an exponentially-small asymptotic contribution (in k ), associated with a complex critical point of Φ K or Φ ( U ) . …where j denotes a real critical point (36.4.1) or (36.4.2), and μ denotes a critical point with complex t or s , t , connected with j by a steepest-descent path (that is, a path where Φ = constant ) in complex t or ( s , t ) space. In the following subsections, only Stokes sets involving at least one real saddle are included unless stated otherwise. … Red and blue numbers in each region correspond, respectively, to the numbers of real and complex critical points that contribute to the asymptotics of the canonical integral away from the bifurcation sets. …The distribution of real and complex critical points in Figures 36.5.5 and 36.5.6 follows from consistency with Figure 36.5.1 and the fact that there are four real saddles in the inner regions. …
3: 12.11 Zeros
If a 1 2 , then U ( a , x ) has no real zeros. … If a > 1 2 , then V ( a , x ) has no positive real zeros, and if a = 3 2 2 n , n , then V ( a , x ) has a zero at x = 0 . … When a > 1 2 the zeros are asymptotically given by z a , s and z a , s ¯ , where s is a large positive integer and … For large negative values of a the real zeros of U ( a , x ) , U ( a , x ) , V ( a , x ) , and V ( a , x ) can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). For example, let the s th real zeros of U ( a , x ) and U ( a , x ) , counted in descending order away from the point z = 2 a , be denoted by u a , s and u a , s , respectively. …
4: Bibliography P
  • P. Painlevé (1906) Sur les équations différentielles du second ordre à points critiques fixès. C.R. Acad. Sc. Paris 143, pp. 1111–1117.
  • PARI-GP (free interactive system and C library)
  • R. Piessens (1982) Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.
  • R. Piessens (1990) On the computation of zeros and turning points of Bessel functions. Bull. Soc. Math. Grèce (N.S.) 31, pp. 117–122.
  • R. Piessens and S. Ahmed (1986) Approximation for the turning points of Bessel functions. J. Comput. Phys. 64 (1), pp. 253–257.
  • 5: 19.36 Methods of Computation
    When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. … where, in the notation of (19.19.7) with a = 1 2 and n = 3 , … A summary for F ( ϕ , k ) is given in Gautschi (1975, §3). … Near these points there will be loss of significant figures in the computation of R J or R D . … For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …
    6: 3.4 Differentiation
    Two-Point Formula
    Three-Point Formula
    Four-Point Formula
    Five-Point Formula
    With the choice r = k (which is crucial when k is large because of numerical cancellation) the integrand equals e k at the dominant points θ = 0 , 2 π , and in combination with the factor k k in front of the integral sign this gives a rough approximation to 1 / k ! . …
    7: Bibliography B
  • A. Bañuelos and R. A. Depine (1980) A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.
  • N. Bleistein (1967) Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities. J. Math. Mech. 17, pp. 533–559.
  • A. A. Bogush and V. S. Otchik (1997) Problem of two Coulomb centres at large intercentre separation: Asymptotic expansions from analytical solutions of the Heun equation. J. Phys. A 30 (2), pp. 559–571.
  • W. G. C. Boyd (1987) Asymptotic expansions for the coefficient functions that arise in turning-point problems. Proc. Roy. Soc. London Ser. A 410, pp. 35–60.
  • M. Brack, M. Mehta, and K. Tanaka (2001) Occurrence of periodic Lamé functions at bifurcations in chaotic Hamiltonian systems. J. Phys. A 34 (40), pp. 8199–8220.
  • 8: 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.
  • C. Leubner and H. Ritsch (1986) A note on the uniform asymptotic expansion of integrals with coalescing endpoint and saddle points. J. Phys. A 19 (3), pp. 329–335.
  • J. L. López and P. J. Pagola (2011) A systematic “saddle point near a pole” asymptotic method with application to the Gauss hypergeometric function. Stud. Appl. Math. 127 (1), pp. 24–37.
  • L. Lorch and P. Szegő (1990) On the points of inflection of Bessel functions of positive order. I. Canad. J. Math. 42 (5), pp. 933–948.
  • D. Ludwig (1966) Uniform asymptotic expansions at a caustic. Comm. Pure Appl. Math. 19, pp. 215–250.
  • 9: 25.12 Polylogarithms
    The notation Li 2 ( z ) was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828): … In the complex plane Li 2 ( z ) has a branch point at z = 1 . … valid when s > 0 , a > 0 or s > 1 , a = 0 . When s = 2 and e 2 π i a = z , (25.12.13) becomes (25.12.4). … For a uniform asymptotic approximation for F s ( x ) see Temme and Olde Daalhuis (1990).
    10: 3.8 Nonlinear Equations
    Let z 1 , z 2 , be a sequence of approximations to a root, or fixed point, ζ . … 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. …