approximations for large parameters
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21: 15.12 Asymptotic Approximations
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§15.12(i) Large Variable
… ►§15.12(ii) Large
… ►§15.12(iii) Other Large Parameters
… ►As , … ►For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).22: 2.8 Differential Equations with a Parameter
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►in which is a real or complex parameter, and asymptotic solutions are needed for large
that are uniform with respect to in a point set in or .
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►In Case III the approximating equation is
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►For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv).
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►For further examples of uniform asymptotic approximations in terms of parabolic cylinder functions see §§13.20(iii), 13.20(iv), 14.15(v), 15.12(iii), 18.24.
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►For examples of uniform asymptotic approximations in terms of Whittaker functions with fixed second parameter see §18.15(i) and §28.8(iv).
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23: 18.26 Wilson Class: Continued
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§18.26(v) Asymptotic Approximations
►For asymptotic expansions of Wilson polynomials of large degree see Wilson (1991), and for asymptotic approximations to their largest zeros see Chen and Ismail (1998). ►Koornwinder (2009) rescales and reparametrizes Racah polynomials and Wilson polynomials in such a way that they are continuous in their four parameters, provided that these parameters are nonnegative. Moreover, if one or more of the new parameters becomes zero, then the polynomial descends to a lower family in the Askey scheme.24: Bibliography W
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Asymptotic approximations for certain - and - symbols.
J. Phys. A 32 (39), pp. 6901–6902.
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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 Approximations to Truncation Errors of Series Representations for Special Functions.
In Algorithms for Approximation, A. Iske and J. Levesley (Eds.),
pp. 331–348.
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Wave functions for large arguments by the amplitude-phase method.
Phys. Rev. 52, pp. 1123–1127.
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An asymptotic expansion of with large variable and parameters.
Math. Comp. 27 (122), pp. 429–436.
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25: 28.20 Definitions and Basic Properties
26: 28.25 Asymptotic Expansions for Large
27: 10.72 Mathematical Applications
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►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter.
…where is a real or complex variable and is a large real or complex parameter.
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►The order of the approximating Bessel functions, or modified Bessel functions, is , except in the case when has a double pole at .
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►Then for large
asymptotic approximations of the solutions can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on and ).
These approximations are uniform with respect to both and , including , the cut neighborhood of , and .
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28: 12.16 Mathematical Applications
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►PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi).
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►PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs.
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29: 8.13 Zeros
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8.13.1
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►For asymptotic approximations for and as see Tricomi (1950b), with corrections by Kölbig (1972b).
For more accurate asymptotic approximations see Thompson (2012).
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►For information on the distribution and computation of zeros of and in the complex -plane for large values of the positive real parameter
see Temme (1995a).
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►Approximations to , for large
can be found in Kölbig (1970).
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