large variable and%2For large parameter
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11—20 of 73 matching pages
11: Bibliography W
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Asymptotische Darstellungen der hypergeometrischen Funktion für große Parameter unterschiedlicher Größenordnung.
Z. Anal. Anwendungen 5 (3), pp. 265–276 (German).
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The distribution of the zeros of Jacobian elliptic functions with respect to the parameter
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Comput. Methods Funct. Theory 9 (2), pp. 579–591.
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Hill’s equation with a large potential.
SIAM J. Appl. Math. 45 (2), pp. 200–214.
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Wave functions for large arguments by the amplitude-phase method.
Phys. Rev. 52, pp. 1123–1127.
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An asymptotic expansion of with large variable and parameters.
Math. Comp. 27 (122), pp. 429–436.
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12: 30.11 Radial Spheroidal Wave Functions
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►with , , , and as in §10.2(ii).
Then solutions of (30.2.1) with and are given by
…Here is defined by (30.8.2) and (30.8.6), and
…In (30.11.3) when , and when .
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§30.11(iii) Asymptotic Behavior
…13: 33.18 Limiting Forms for Large
14: 33.11 Asymptotic Expansions for Large
§33.11 Asymptotic Expansions for Large
►For large , with and fixed, ►
33.11.1
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33.11.7
►Here , , , , and for ,
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15: 10.72 Mathematical Applications
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►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter.
…where is a real or complex variable and is a large real or complex parameter.
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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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►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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►Then for large
asymptotic approximations of the solutions can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on and ).
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16: 8.27 Approximations
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DiDonato (1978) gives a simple approximation for the function (which is related to the incomplete gamma function by a change of variables) for real and large positive . This takes the form , approximately, where and is shown to produce an absolute error as .
17: 33.10 Limiting Forms for Large or Large
§33.10 Limiting Forms for Large or Large
►§33.10(i) Large
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33.10.2
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