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11—20 of 37 matching pages
11: 28.8 Asymptotic Expansions for Large
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28.8.1
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12: 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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On one-parameter families of Painlevé III.
Stud. Appl. Math. 101 (3), pp. 321–341.
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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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Tables of the functions of the parabolic cylinder for negative integer parameters.
Zastos. Mat. 13, pp. 261–273.
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13: Bibliography O
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Uniform asymptotic expansions for hypergeometric functions with large parameters. I.
Analysis and Applications (Singapore) 1 (1), pp. 111–120.
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Uniform asymptotic expansions for hypergeometric functions with large parameters. II.
Analysis and Applications (Singapore) 1 (1), pp. 121–128.
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Uniform asymptotic expansions for hypergeometric functions with large parameters. III.
Analysis and Applications (Singapore) 8 (2), pp. 199–210.
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Legendre functions with both parameters large.
Philos. Trans. Roy. Soc. London Ser. A 278, pp. 175–185.
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Whittaker functions with both parameters large: Uniform approximations in terms of parabolic cylinder functions.
Proc. Roy. Soc. Edinburgh Sect. A 86 (3-4), pp. 213–234.
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14: 11.6 Asymptotic Expansions
15: 36.5 Stokes Sets
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►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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►The Stokes set is itself a cusped curve, connected to the cusp of the bifurcation set:
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►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4).
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►The first sheet corresponds to and is generated as a solution of Equations (36.5.6)–(36.5.9).
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►This consists of a cusp-edged sheet connected to the cusp-edged sheet of the bifurcation set and intersecting the smooth sheet of the bifurcation set.
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16: 26.10 Integer Partitions: Other Restrictions
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►The set is denoted by .
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26.10.5
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►It is known that for , , with strict inequality for sufficiently large, provided that , or ; see Yee (2004).
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►where is the modified Bessel function (§10.25(ii)), and
…The quantity is real-valued.
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17: 12.10 Uniform Asymptotic Expansions for Large Parameter
§12.10 Uniform Asymptotic Expansions for Large Parameter
… ►In this section we give asymptotic expansions of PCFs for large values of the parameter that are uniform with respect to the variable , when both and are real. … ►§12.10(ii) Negative ,
… ►§12.10(vii) Negative , . Expansions in Terms of Airy Functions
… ►18: 8.17 Incomplete Beta Functions
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►Throughout §§8.17 and 8.18 we assume that , , and .
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►Addendum: For a companion equation see (8.17.24).
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►For a historical profile of see Dutka (1981).
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►With , , and ,
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►The expansion (8.17.22) converges rapidly for .
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19: 3.8 Nonlinear Equations
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►Suppose also depends on a parameter
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Then the sensitivity of a simple zero to changes in is given by
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►Consider and .
We have and .
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►It is called a Julia set.
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20: 18.40 Methods of Computation
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