analytically continued
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11—20 of 42 matching pages
11: Bibliography W
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Analytic Theory of Continued Fractions.
D. Van Nostrand Company, Inc., New York.
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12: 8.17 Incomplete Beta Functions
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►However, in the case of §8.17 it is straightforward to continue most results analytically to other real values of , , and , and also to complex values.
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13: 1.12 Continued Fractions
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►For analytical and numerical applications of continued fractions to special functions see §3.10.
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14: 3.8 Nonlinear Equations
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►This is an iterative method for real twice-continuously differentiable, or complex analytic, functions:
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15: 18.40 Methods of Computation
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►The question is then: how is this possible given only , rather than itself? often converges to smooth results for off the real axis for at a distance greater than the pole spacing of the , this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to and evaluating these on the real axis in regions of higher pole density that those of the approximating function.
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16: 33.23 Methods of Computation
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►Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions (§3.11(iv)) for the analytic continuations of Coulomb functions.
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17: 4.37 Inverse Hyperbolic Functions
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4.37.6
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►Elsewhere on the integration paths in (4.37.1) and (4.37.2) the branches are determined by continuity.
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►These functions are analytic in the cut plane depicted in Figure 4.37.1(iv), (v), (vi), respectively.
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►It should be noted that the imaginary axis is not a cut; the function defined by (4.37.19) and (4.37.20) is analytic everywhere except on .
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18: 4.12 Generalized Logarithms and Exponentials
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►Both and are continuously differentiable.
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►For analytic generalized logarithms, see Kneser (1950).
19: Frank W. J. Olver
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►Olver continued to maintain a connection to NIST after moving to the university.
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►He continued his editing work until the time of his death on April 22, 2013 at age 88.
20: 10.72 Mathematical Applications
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►In regions in which (10.72.1) has a simple turning point , that is, and are analytic (or with weaker conditions if is a real variable) and is a simple zero of , asymptotic expansions of the solutions for large can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order (§9.6(i)).
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►The number can also be replaced by any real constant
in the sense that
is analytic and nonvanishing at ; moreover, is permitted to have a single or double pole at .
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►In regions in which the function has a simple pole at and is analytic at (the case in §10.72(i)), asymptotic expansions of the solutions of (10.72.1) for large can be constructed in terms of Bessel functions and modified Bessel functions of order , where is the limiting value of as .
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►In (10.72.1) assume and depend continuously on a real parameter , has a simple zero and a double pole , except for a critical value , where .
Assume that whether or not , is analytic at .
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