asymptotic behavior for large variable
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11: 8.12 Uniform Asymptotic Expansions for Large Parameter
§8.12 Uniform Asymptotic Expansions for Large Parameter
… ►For the asymptotic behavior of as see Dunster et al. (1998) and Olde Daalhuis (1998c). … ►Lastly, a uniform approximation for for large , with error bounds, can be found in Dunster (1996a). ►For other uniform asymptotic approximations of the incomplete gamma functions in terms of the function see Paris (2002b) and Dunster (1996a). ►Inverse Function
…12: Bibliography W
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Hill’s equation with a large potential.
SIAM J. Appl. Math. 45 (2), pp. 200–214.
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Asymptotic behavior of stability regions for Hill’s equation.
SIAM J. Appl. Math. 47 (5), pp. 941–958.
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Wave functions for large arguments by the amplitude-phase method.
Phys. Rev. 52, pp. 1123–1127.
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On the asymptotic behavior of the Fourier coefficients of Mathieu functions.
J. Res. Nat. Inst. Standards Tech. 113 (1), pp. 11–15.
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An asymptotic expansion of with large variable and parameters.
Math. Comp. 27 (122), pp. 429–436.
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13: 18.2 General Orthogonal Polynomials
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►It is assumed throughout this chapter that for each polynomial that is orthogonal on an open interval the variable
is confined to the closure of
unless indicated otherwise. (However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.)
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►For OP’s with weight function in the class there are asymptotic formulas as , respectively for outside and for , see Szegő (1975, Theorems 12.1.2, 12.1.4).
…This says roughly that the series (18.2.25) has the same pointwise convergence behavior as the same series with , a Chebyshev polynomial of the first kind, see Table 18.3.1.
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►If these satisfy then Szegő type asymptotics outside can be given for the corresponding OP’s, see Simon (2011, Corollary 3.7.2 and following).
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►For a large class of OP’s there exist pairs of differentiation formulas
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