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asymptotic approximations and expansions for large |r|

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1: 33.21 Asymptotic Approximations for Large | r |
§33.21(i) Limiting Forms
§33.21(ii) Asymptotic Expansions
2: 2.11 Remainder Terms; Stokes Phenomenon
§2.11(i) Numerical Use of Asymptotic Expansions
§2.11(iii) Exponentially-Improved Expansions
For another approach see Paris (2001a, b).
§2.11(vi) Direct Numerical Transformations
3: 2.10 Sums and Sequences
§2.10 Sums and Sequences
This identity can be used to find asymptotic approximations for large n when the factor v j changes slowly with j , and u j is oscillatory; compare the approximation of Fourier integrals by integration by parts in §2.3(i). …
§2.10(iii) Asymptotic Expansions of Entire Functions
  • (c)

    The coefficients in the Laurent expansion

    2.10.27 g ( z ) = n = - g n z n , 0 < | z | < r ,

    have known asymptotic behavior as n ± .

  • 4: 8.11 Asymptotic Approximations and Expansions
    §8.11 Asymptotic Approximations and Expansions
    where δ denotes an arbitrary small positive constant. … For an exponentially-improved asymptotic expansion2.11(iii)) see Olver (1991a). … This expansion is absolutely convergent for all finite z , and it can also be regarded as a generalized asymptotic expansion2.1(v)) of γ ( a , z ) as a in | ph a | π - δ . …
    5: Bibliography W
  • R. Wong (1973a) An asymptotic expansion of W k , m ( z ) with large variable and parameters. Math. Comp. 27 (122), pp. 429–436.
  • R. Wong (1973b) On uniform asymptotic expansion of definite integrals. J. Approximation Theory 7 (1), pp. 76–86.
  • R. Wong (1981) Asymptotic expansions of the Kontorovich-Lebedev transform. Appl. Anal. 12 (3), pp. 161–172.
  • R. Wong (1983) Applications of some recent results in asymptotic expansions. Congr. Numer. 37, pp. 145–182.
  • R. Wong (1995) Error bounds for asymptotic approximations of special functions. Ann. Numer. Math. 2 (1-4), pp. 181–197.
  • 6: 2.6 Distributional Methods
    §2.6 Distributional Methods
    §2.6(ii) Stieltjes Transform
    To derive an asymptotic expansion of 𝒮 f ( z ) for large values of | z | , with | ph z | < π , we assume that f ( t ) possesses an asymptotic expansion of the form … An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). … We now derive an asymptotic expansion of I μ f ( x ) for large positive values of x . …
    7: 13.9 Zeros
    where n is a large positive integer. … where n is a large positive integer. For fixed a and z in , U ( a , b , z ) has two infinite strings of b -zeros that are asymptotic to the imaginary axis as | b | .
    8: 18.26 Wilson Class: Continued
    18.26.9 lim β R n ( x ; - N - 1 , β , γ , δ ) = R n ( x ; γ , δ , N ) .
    18.26.12 r ( x ; β , c , N ) = x ( x + β + c - 1 ( 1 - c ) N ) ,
    18.26.13 lim N R n ( r ( x ; β , c , N ) ; β - 1 , c - 1 ( 1 - c ) N , N ) = M n ( x ; β , c ) .
    §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). …
    9: 25.11 Hurwitz Zeta Function
    The function ζ ( s , a ) was introduced in Hurwitz (1882) and defined by the series expansionFor other series expansions similar to (25.11.10) see Coffey (2008). …
    §25.11(xii) a -Asymptotic Behavior
    As a in the sector | ph a | π - δ ( < π ) , with s ( 1 ) and δ fixed, we have the asymptotic expansionSimilarly, as a in the sector | ph a | 1 2 π - δ ( < 1 2 π ) , …
    10: 2.7 Differential Equations
    Note that the coefficients in the expansions (2.7.12), (2.7.13) for the “late” coefficients, that is, a s , 1 , a s , 2 with s large, are the “early” coefficients a j , 2 , a j , 1 with j small. …
    §2.7(iii) Liouville–Green (WKBJ) Approximation
    For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: … By approximatingThe first of these references includes extensions to complex variables and reversions for zeros. …