transition%20points
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11—20 of 224 matching pages
11: Bibliography M
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Rational approximations, software and test methods for sine and cosine integrals.
Numer. Algorithms 12 (3-4), pp. 259–272.
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Calculation of the modified Bessel functions of the second kind with complex argument.
Math. Comp. 20 (95), pp. 407–412.
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Clothoid spline transition spirals.
Math. Comp. 59 (199), pp. 117–133.
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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems.
Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.
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The -analogue of the Laguerre polynomials.
J. Math. Anal. Appl. 81 (1), pp. 20–47.
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12: Bibliography T
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The numerical computation of special functions by use of quadrature rules for saddle point integrals. II. Gamma functions, modified Bessel functions and parabolic cylinder functions.
Report TW 183/78
Mathematisch Centrum, Amsterdam, Afdeling Toegepaste
Wiskunde.
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Asymptotic solution of a linear nonhomogeneous second order differential equation with a transition point and its application to the computations of toroidal shells and propeller blades.
J. Appl. Math. Mech. 23, pp. 1549–1565.
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13: 36.7 Zeros
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►The zeros in Table 36.7.1 are points in the plane, where is undetermined.
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►, ), the number of rings in the th row, measured from the origin and before the transition to hairpins, is given by
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14: 3.8 Nonlinear Equations
15: 20 Theta Functions
Chapter 20 Theta Functions
…16: 33.3 Graphics
17: 2.11 Remainder Terms; Stokes Phenomenon
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►When a rigorous bound or reliable estimate for the remainder term is unavailable, it is unsafe to judge the accuracy of an asymptotic expansion merely from the numerical rate of decrease of the terms at the point of truncation.
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►For large the integrand has a saddle point at .
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►In the transition through , changes very rapidly, but smoothly, from one form to the other; compare the graph of its modulus in Figure 2.11.1 in the case .
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►For example, using double precision is found to agree with (2.11.31) to 13D.
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18: 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.
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36.5.7
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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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19: 26.13 Permutations: Cycle Notation
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26.13.2
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►Cycles of length one are fixed points.
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►An element of with fixed points, cycles of length cycles of length , where , is said to have cycle type
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►A derangement is a permutation with no fixed points.
The derangement number, , is the number of elements of with no fixed points:
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20: 9.15 Mathematical Applications
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►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point.
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