fixed points
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11: 2.4 Contour Integrals
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►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.
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12: 20.2 Definitions and Periodic Properties
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►For fixed
, each is an entire function of with period ; is odd in and the others are even.
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►The four points
are the vertices of the fundamental parallelogram in the -plane; see Figure 20.2.1.
The points
…are the lattice points.
The theta functions are quasi-periodic on the lattice:
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13: 2.8 Differential Equations with a Parameter
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►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.
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14: Bibliography U
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Integrals with a large parameter. Several nearly coincident saddle-points.
Proc. Cambridge Philos. Soc. 72, pp. 49–65.
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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.
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Integrals with a large parameter: Legendre functions of large degree and fixed order.
Math. Proc. Cambridge Philos. Soc. 95 (2), pp. 367–380.
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15: 31.15 Stieltjes Polynomials
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►The system (31.15.2) determines the as the points of equilibrium of movable (interacting) particles with unit charges in a field of particles with the charges
fixed at .
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16: 21.7 Riemann Surfaces
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►Consider the set of points in that satisfy the equation
…Equation (21.7.1) determines a plane algebraic curve in , 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 vanishes.
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►The zeros , of specify the finite branch points
, that is, points at which , on the Riemann surface.
Denote the set of all branch points by .
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17: 10.2 Definitions
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►This solution of (10.2.1) is an analytic function of , except for a branch point at when is not an integer.
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►For fixed
each branch of is entire in .
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►Whether or not is an integer has a branch point at .
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►For fixed
each branch of is entire in .
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►Each solution has a branch point at for all .
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18: 10.25 Definitions
19: 14.15 Uniform Asymptotic Approximations
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