for large ℜz
(0.004 seconds)
21—30 of 132 matching pages
21: Bibliography L
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The confluent hypergeometric functions and for large
and
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J. Comput. Appl. Math. 233 (6), pp. 1570–1576.
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22: 15.19 Methods of Computation
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►For example, in the half-plane we can use (15.12.2) or (15.12.3) to compute and , where is a large positive integer, and then apply (15.5.18) in the backward direction.
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23: 2.4 Contour Integrals
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►Except that is now permitted to be complex, with , we assume the same conditions on and also that the Laplace transform in (2.3.8) converges for all sufficiently large values of .
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►For large
, the asymptotic expansion of may be obtained from (2.4.3) by Haar’s method. This depends on the availability of a comparison function for that has an inverse transform
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►in which is a large real or complex parameter, and are analytic functions of and continuous in and a second parameter .
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►For large
, is approximated uniformly by the integral that corresponds to (2.4.19) when is replaced by a constant.
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24: 2.1 Definitions and Elementary Properties
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►For example, if is analytic for all sufficiently large
in a sector and as in , being real, then as in any closed sector properly interior to and with the same vertex (Ritt’s theorem).
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25: 28.20 Definitions and Basic Properties
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§28.20(iii) Solutions
…26: 10.72 Mathematical Applications
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►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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27: 2.7 Differential Equations
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►From the numerical standpoint, however, the pair and has the drawback that severe numerical cancellation can occur with certain combinations of and , for example if and are equal, or nearly equal, and , or , is large and negative.
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28: 2.11 Remainder Terms; Stokes Phenomenon
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►For large
, with (), the Whittaker function of the second kind has the asymptotic expansion (§13.19)
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29: 18.15 Asymptotic Approximations
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►For large
, fixed , and , Dunster (1999) gives asymptotic expansions of that are uniform in unbounded complex -domains containing .
…This reference also supplies asymptotic expansions of for large
, fixed , and .
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