at%20a%20point
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1: 26.12 Plane Partitions
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►A plane partition, , of a positive integer , is a partition of in which the parts have been arranged in a 2-dimensional array that is weakly decreasing (nonincreasing) across rows and down columns.
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►It is useful to be able to visualize a plane partition as a pile of blocks, one block at each lattice point
.
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►The plane partition in Figure 26.12.1 is an example of a cyclically symmetric plane partition.
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►A plane partition is totally symmetric if it is both symmetric and cyclically symmetric.
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►The example of a strict shifted plane partition also satisfies the conditions of a descending plane partition.
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2: 36.5 Stokes Sets
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►Stokes sets are surfaces (codimension one) in space, across which or acquires an exponentially-small asymptotic contribution (in ), associated with a complex critical point of or .
…where denotes a real critical point (36.4.1) or (36.4.2), and denotes a critical point with complex or , connected with by a steepest-descent path (that is, a path where ) in complex or space.
►In the following subsections, only Stokes sets involving at least one real saddle are included unless stated otherwise.
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►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.
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3: 12.11 Zeros
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►If , then has no real zeros.
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►If , then has no positive real zeros, and if , , then has a zero at
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►When the zeros are asymptotically given by and , where is a large positive integer and
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►For large negative values of the real zeros of , , , and can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii).
For example, let the th real zeros of and , counted in descending order away from the point
, be denoted by and , respectively.
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4: Bibliography P
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Sur les équations différentielles du second ordre à points critiques fixès.
C.R. Acad. Sc. Paris 143, pp. 1111–1117.
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Automatic computation of Bessel function integrals.
Comput. Phys. Comm. 25 (3), pp. 289–295.
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On the computation of zeros and turning points of Bessel functions.
Bull. Soc. Math. Grèce (N.S.) 31, pp. 117–122.
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Approximation for the turning points of Bessel functions.
J. Comput. Phys. 64 (1), pp. 253–257.
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5: 19.36 Methods of Computation
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►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.
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►where, in the notation of (19.19.7) with and ,
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►A summary for is given in Gautschi (1975, §3).
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►Near these points there will be loss of significant figures in the computation of or .
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►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).
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6: 3.4 Differentiation
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Two-Point Formula
… ►Three-Point Formula
… ►Four-Point Formula
… ►Five-Point Formula
… ►With the choice (which is crucial when is large because of numerical cancellation) the integrand equals at the dominant points , and in combination with the factor in front of the integral sign this gives a rough approximation to . …7: Bibliography B
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A program for computing the Riemann zeta function for complex argument.
Comput. Phys. Comm. 20 (3), pp. 441–445.
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Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities.
J. Math. Mech. 17, pp. 533–559.
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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.
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Asymptotic expansions for the coefficient functions that arise in turning-point problems.
Proc. Roy. Soc. London Ser. A 410, pp. 35–60.
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Occurrence of periodic Lamé functions at bifurcations in chaotic Hamiltonian systems.
J. Phys. A 34 (40), pp. 8199–8220.
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8: Bibliography L
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The central two-point connection problem for the Heun class of ODEs.
J. Phys. A 31 (18), pp. 4249–4261.
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A note on the uniform asymptotic expansion of integrals with coalescing endpoint and saddle points.
J. Phys. A 19 (3), pp. 329–335.
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A systematic “saddle point near a pole” asymptotic method with application to the Gauss hypergeometric function.
Stud. Appl. Math. 127 (1), pp. 24–37.
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On the points of inflection of Bessel functions of positive order. I.
Canad. J. Math. 42 (5), pp. 933–948.
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Uniform asymptotic expansions at a caustic.
Comm. Pure Appl. Math. 19, pp. 215–250.
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9: 25.12 Polylogarithms
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►The notation was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828):
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►In the complex plane has a branch point at
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►valid when , or , .
When and , (25.12.13) becomes (25.12.4).
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►For a uniform asymptotic approximation for see Temme and Olde Daalhuis (1990).