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asymptotic approximations and expansions for large |r|

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11: 10.21 Zeros
§10.21(vi) McMahon’s Asymptotic Expansions for Large Zeros
§10.21(vii) Asymptotic Expansions for Large Order
§10.21(viii) Uniform Asymptotic Approximations for Large Order
The asymptotic expansion of the large positive zeros (not necessarily the m th) of the function …
12: 30.16 Methods of Computation
For small | γ 2 | we can use the power-series expansion (30.3.8). …If | γ 2 | is large we can use the asymptotic expansions in §30.9. Approximations to eigenvalues can be improved by using the continued-fraction equations from §30.3(iii) and §30.8; see Bouwkamp (1947) and Meixner and Schäfke (1954, §3.93). … If | γ 2 | is large, then we can use the asymptotic expansions referred to in §30.9 to approximate 𝖯𝗌 n m ( x , γ 2 ) . … A fourth method, based on the expansion (30.8.1), is as follows. …
13: Errata
  • Equation (19.20.11)
    19.20.11 R J ( 0 , y , z , p ) = 3 2 p z ln ( 16 z y ) 3 p R C ( z , p ) + O ( y ln y ) ,

    as y 0 + , p ( 0 ) real, we have added the constant term 3 p R C ( z , p ) and the order term O ( y ln y ) , and hence was replaced by = .

  • Subsection 2.1(iii)

    A short paragraph dealing with asymptotic approximations that are expressed in terms of two or more Poincaré asymptotic expansions has been added below (2.1.16).

  • Equation (2.11.4)

    Because (2.11.4) is not an asymptotic expansion, the symbol that was used originally is incorrect and has been replaced with , together with a slight change of wording.

  • Equation (13.9.16)

    Originally was expressed in term of asymptotic symbol . As a consequence of the use of the O order symbol on the right-hand side, was replaced by = .

  • Subsection 13.8(iii)

    A new paragraph with several new equations and a new reference has been added at the end to provide asymptotic expansions for Kummer functions U ( a , b , z ) and 𝐌 ( a , b , z ) as a in | ph a | π δ and b and z fixed.

  • 14: 11.6 Asymptotic Expansions
    §11.6 Asymptotic Expansions
    §11.6(i) Large | z | , Fixed ν
    §11.6(ii) Large | ν | , Fixed z
    More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions2.1(v)). …