plane curves
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11—20 of 23 matching pages
11: 1.9 Calculus of a Complex Variable
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Jordan Curve Theorem
…12: 36.5 Stokes Sets
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►In Figures 36.5.1–36.5.6 the plane is divided into regions by the dashed curves (Stokes sets) and the continuous curves (bifurcation sets).
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13: 10.20 Uniform Asymptotic Expansions for Large Order
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►The equations of the curved boundaries and in the -plane are given parametrically by
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►The curves
and in the -plane are the inverse maps of the line segments
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14: 36.7 Zeros
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►The zeros in Table 36.7.1 are points in the
plane, where is undetermined.
…Inside the cusp, that is, for , the zeros form pairs lying in curved rows.
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►Deep inside the bifurcation set, that is, inside the three-cusped astroid (36.4.10) and close to the part of the -axis that is far from the origin, the zero contours form an array of rings close to the planes
…In the symmetry planes (e.
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►The zeros of these functions are curves in space; see Nye (2007) for and Nye (2006) for .
15: 28.33 Physical Applications
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►As runs from to , with and fixed, the point moves from to along the ray given by the part of the line that lies in the first quadrant of the -plane.
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►For points that are at intersections of with the characteristic curves
or , a periodic solution is possible.
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16: 19.25 Relations to Other Functions
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19.25.36
►The sign on the right-hand side of (19.25.35) will change whenever one crosses a curve on which , for some .
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►The sign on the right-hand side of (19.25.40) will change whenever one crosses a curve on which , for some .
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17: 2.11 Remainder Terms; Stokes Phenomenon
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►Thus if (), then lies in the right half-plane.
…On the other hand, when , is in the left half-plane and differs from 2 by an exponentially-small quantity.
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►Rays (or curves) on which one contribution in a compound asymptotic expansion achieves maximum dominance over another are called Stokes lines ( in the present example).
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18: Bibliography
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Algorithm 39: Areas under the normal curve.
The Computer Journal 12 (2), pp. 197–198.
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High-order interior caustics produced in scattering of a diagonally incident plane wave by a circular cylinder.
J. Opt. Soc. Amer. A 14 (6), pp. 1305–1315.
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Szegő Type Asymptotics for the Reproducing Kernel in Spaces of Full-Plane Weighted Polynomials.
Comm. Math. Phys. 398 (3), pp. 1291–1348.
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Plane partitions. III. The weak Macdonald conjecture.
Invent. Math. 53 (3), pp. 193–225.
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19: 25.11 Hurwitz Zeta Function
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has a meromorphic continuation in the -plane, its only singularity in being a simple pole at with residue .
As a function of , with () fixed, is analytic in the half-plane
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25.11.30
, ,
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