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11—20 of 30 matching pages
11: 36.4 Bifurcation Sets
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36.4.1
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►This is the codimension-one surface in space where critical points coalesce, satisfying (36.4.1) and
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►This is the codimension-one surface in space where critical points coalesce, satisfying (36.4.2) and
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12: Bibliography G
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Algorithm 259: Legendre functions for arguments larger than one.
Comm. ACM 8 (8), pp. 488–492.
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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules.
ACM Trans. Math. Software 20 (1), pp. 21–62.
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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Mutual integrability, quadratic algebras, and dynamical symmetry.
Ann. Phys. 217 (1), pp. 1–20.
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On Mill’s ratio.
Proc. Cambridge Philos. Soc. 67, pp. 363–364.
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13: 20.11 Generalizations and Analogs
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►In the case identities for theta functions become identities in the complex variable
, with , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7).
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►However, in this case is no longer regarded as an independent complex variable within the unit circle, because is related to the variable
of the theta functions via (20.9.2).
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►For specialization to the one-dimensional theta functions treated in the present chapter, see Rauch and Lebowitz (1973) and §21.7(iii).
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►A further development on the lines of Neville’s notation (§20.1) is as follows.
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►Such sets of twelve equations include derivatives, differential equations, bisection relations, duplication relations, addition formulas (including new ones for theta functions), and pseudo-addition formulas.
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14: 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 .
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►In the following subsections, only Stokes sets involving at least one real saddle are included unless stated otherwise.
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36.5.4
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36.5.7
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►One of the sheets is symmetrical under reflection in the plane , and is given by
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15: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
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Complex Analysis: An Introduction of the Theory of Analytic Functions of One Complex Variable.
2nd edition, McGraw-Hill Book Co., New York.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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Commentary on the Paper “On certain special sets of orthogonal polynomials”.
Contemporary Mathematicians, Birkhäuser, Boston, Mass..
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16: Bibliography D
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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire
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Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
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On the real roots of Euler polynomials.
Monatsh. Math. 106 (2), pp. 115–138.
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On the computation of Mathieu functions.
J. Engrg. Math. 7, pp. 39–61.
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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17: 26.3 Lattice Paths: Binomial Coefficients
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►The number of lattice paths from to , , that stay on or above the line is
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26.3.3
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26.3.4
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18: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
19: 18.39 Applications in the Physical Sciences
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Introduction and One-Dimensional (1D) Systems
… ►As in classical dynamics this sum is the total energy of the one particle system. … ►1D Quantum Systems with Analytically Known Stationary States
… ►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. … ►The discrete variable representations (DVR) analysis is simplest when based on the classical OP’s with their analytically known recursion coefficients (Table 3.5.17_5), or those non-classical OP’s which have analytically known recursion coefficients, making stable computation of the and , from the J-matrix as in §3.5(vi), straightforward. …20: Bibliography K
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Orthogonal Polynomials on
-spheres: Gegenbauer, Jacobi and Heun.
In Topics in Polynomials of One and Several Variables and their
Applications,
pp. 299–322.
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
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The Askey scheme as a four-manifold with corners.
Ramanujan J. 20 (3), pp. 409–439.
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