envelope functions
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9 matching pages ♦
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9 matching pages
1: 2.8 Differential Equations with a Parameter
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►of smallest absolute value, and define the envelopes of and by
…These envelopes are continuous functions of , and as
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►Again, an alternative way of representing the error terms in (2.8.29) and (2.8.30) is by means of envelope functions.
…Define
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►(For envelope functions for parabolic cylinder functions see §14.15(v)).
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2: 13.21 Uniform Asymptotic Approximations for Large
3: 33.3 Graphics
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33.3.1
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4: 13.20 Uniform Asymptotic Approximations for Large
5: 14.15 Uniform Asymptotic Approximations
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14.15.11
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14.15.12
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14.15.15
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►Here we introduce the envelopes of the parabolic cylinder functions
, , which are defined in §12.2.
For or , with and nonnegative,
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6: 13.8 Asymptotic Approximations for Large Parameters
7: 18.15 Asymptotic Approximations
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18.15.19
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►Here denotes the Bessel function (§10.2(ii)), denotes its envelope (§2.8(iv)), and is again an arbitrary small positive constant.
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18.15.22
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►Here denotes the Airy function (§9.2), denotes its derivative, and denotes its envelope (§2.8(iii)).
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8: Errata
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Equation (14.15.23)
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Originally used to represent both and . This has been replaced by two equations giving explicit definitions for the two envelope functions. Some slight changes in wording were needed to make this clear to readers.