asymptotic expansions as ϵ→0
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1: 2.2 Transcendental Equations
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2.2.7
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2.2.8
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►where and () is the coefficient of in the asymptotic expansion of (Lagrange’s formula for the reversion of
series).
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2: 2.1 Definitions and Elementary Properties
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2.1.13
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2.1.14
in .
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2.1.16
in ,
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►As an example, in the sector () each of the functions , and (principal value) has the null asymptotic expansion
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2.1.18
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3: 33.20 Expansions for Small
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§33.20(iii) Asymptotic Expansion for the Irregular Solution
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33.20.7
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§33.20(iv) Uniform Asymptotic Expansions
►For a comprehensive collection of asymptotic expansions that cover and as and are uniform in , including unbounded values, see Curtis (1964a, §7). …4: 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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►If has a double zero , or more generally is a zero of order , , then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order .
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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 .
These asymptotic expansions are uniform with respect to , including cut neighborhoods of , and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation.
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5: 33.11 Asymptotic Expansions for Large
6: 4.13 Lambert -Function
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4.13.11
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7: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
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►For asymptotic expansions of as in various sectors of the complex -plane for fixed real values of and fixed real or complex values of , see Wright (1935) when , and Wright (1940b) when .
For exponentially-improved asymptotic expansions in the same circumstances, together with smooth interpretations of the corresponding Stokes phenomenon (§§2.11(iii)–2.11(v)) see Wong and Zhao (1999b) when , and Wong and Zhao (1999a) when .
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8: 7.17 Inverse Error Functions
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§7.17(ii) Power Series
… ►where and the other coefficients follow from the recursion … ►§7.17(iii) Asymptotic Expansion of for Small
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7.17.3
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9: 2.3 Integrals of a Real Variable
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(b)
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(b)
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2.3.2
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2.3.7
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As
2.3.14
and the expansion for is differentiable. Again and are positive constants. Also (consistent with (a)).
2.3.16
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As the asymptotic expansions (2.3.14) apply, and each is infinitely differentiable. Again , , and are positive.