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1: 10.17 Asymptotic Expansions for Large Argument
§10.17(i) Hankel’s Expansions
§10.17(iii) Error Bounds for Real Argument and Order
§10.17(v) Exponentially-Improved Expansions
For higher re-expansions of the remainder terms see Olde Daalhuis and Olver (1995a) and Olde Daalhuis (1995, 1996).
2: 10.40 Asymptotic Expansions for Large Argument
§10.40(i) Hankel’s Expansions
3: 10.20 Uniform Asymptotic Expansions for Large Order
§10.20 Uniform Asymptotic Expansions for Large Order
10.20.6 H ν ( 1 ) ( ν z ) H ν ( 2 ) ( ν z ) } 2 e π i / 3 ( 4 ζ 1 - z 2 ) 1 4 ( Ai ( e ± 2 π i / 3 ν 2 3 ζ ) ν 1 3 k = 0 A k ( ζ ) ν 2 k + e ± 2 π i / 3 Ai ( e ± 2 π i / 3 ν 2 3 ζ ) ν 5 3 k = 0 B k ( ζ ) ν 2 k ) ,
10.20.9 H ν ( 1 ) ( ν z ) H ν ( 2 ) ( ν z ) } 4 e 2 π i / 3 z ( 1 - z 2 4 ζ ) 1 4 ( e 2 π i / 3 Ai ( e ± 2 π i / 3 ν 2 3 ζ ) ν 4 3 k = 0 C k ( ζ ) ν 2 k + Ai ( e ± 2 π i / 3 ν 2 3 ζ ) ν 2 3 k = 0 D k ( ζ ) ν 2 k ) ,
§10.20(iii) Double Asymptotic Properties
For asymptotic properties of the expansions (10.20.4)–(10.20.6) with respect to large values of z see §10.41(v).
4: 10.19 Asymptotic Expansions for Large Order
§10.19 Asymptotic Expansions for Large Order
§10.19(i) Asymptotic Forms
§10.19(iii) Transition Region
10.19.13 H ν ( 1 ) ( ν + a ν 1 3 ) H ν ( 2 ) ( ν + a ν 1 3 ) } - 2 5 3 ν 2 3 e ± π i / 3 Ai ( e π i / 3 2 1 3 a ) k = 0 R k ( a ) ν 2 k / 3 + 2 4 3 ν 4 3 e π i / 3 Ai ( e π i / 3 2 1 3 a ) k = 0 S k ( a ) ν 2 k / 3 ,
See also §10.20(i).
5: Bibliography W
  • R. Wong (1976) Error bounds for asymptotic expansions of Hankel transforms. SIAM J. Math. Anal. 7 (6), pp. 799–808.
  • R. Wong (1977) Asymptotic expansions of Hankel transforms of functions with logarithmic singularities. Comput. Math. Appl. 3 (4), pp. 271–286.
  • 6: 10.41 Asymptotic Expansions for Large Order
    §10.41(v) Double Asymptotic Properties (Continued)
    We first prove that for the expansions (10.20.6) for the Hankel functions H ν ( 1 ) ( ν z ) and H ν ( 2 ) ( ν z ) the z -asymptotic property applies when z ± i , respectively. …
    7: Bibliography N
  • G. Nemes (2017b) Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions. Acta Appl. Math. 150, pp. 141–177.
  • 8: 10.1 Special Notation
    For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
    9: Bibliography G
  • E. A. Galapon and K. M. L. Martinez (2014) Exactification of the Poincaré asymptotic expansion of the Hankel integral: spectacularly accurate asymptotic expansions and non-asymptotic scales. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2162), pp. 20130529, 16.
  • 10: 10.22 Integrals
    For asymptotic expansions of Hankel transforms see Wong (1976, 1977), Frenzen and Wong (1985a) and Galapon and Martinez (2014). …