inhomogeneous Airy functions
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11: 2.8 Differential Equations with a Parameter
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►These are elementary functions in Case I, and Airy functions (§9.2) in Case II.
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►For error bounds, extensions to pure imaginary or complex , an extension to inhomogeneous differential equations, and examples, see Olver (1997b, Chapter 10).
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►For and see §9.2.
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►of smallest absolute value, and define the envelopes of and by
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►For other examples of uniform asymptotic approximations and expansions of special functions in terms of Airy functions see especially §10.20 and §§12.10(vii), 12.10(viii); also §§12.14(ix), 13.20(v), 13.21(iii), 13.21(iv), 15.12(iii), 18.15(iv), 30.9(i), 30.9(ii), 32.11(ii), 32.11(iii), 33.12(i), 33.12(ii), 33.20(iv), 36.12(ii), 36.13.
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12: Bibliography O
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Uniform Airy-type expansions of integrals.
SIAM J. Math. Anal. 25 (2), pp. 304–321.
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On higher-order Stokes phenomena of an inhomogeneous linear ordinary differential equation.
J. Comput. Appl. Math. 169 (1), pp. 235–246.
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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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Distribution of the partition function modulo
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Ann. of Math. (2) 151 (1), pp. 293–307.
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Modified quotients of cylinder functions.
Math. Tables Aids Comput. 10, pp. 27–28.
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