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1: 14.26 Uniform Asymptotic Expansions
§14.26 Uniform Asymptotic Expansions
2: 10.57 Uniform Asymptotic Expansions for Large Order
§10.57 Uniform Asymptotic Expansions for Large Order
3: 10.69 Uniform Asymptotic Expansions for Large Order
§10.69 Uniform Asymptotic Expansions for Large Order
4: 10.19 Asymptotic Expansions for Large Order
§10.19 Asymptotic Expansions for Large Order
§10.19(i) Asymptotic Forms
§10.19(ii) Debye’s Expansions
§10.19(iii) Transition Region
See also §10.20(i).
5: 10.41 Asymptotic Expansions for Large Order
§10.41 Asymptotic Expansions for Large Order
§10.41(i) Asymptotic Forms
§10.41(ii) Uniform Expansions for Real Variable
6: 10.20 Uniform Asymptotic Expansions for Large Order
§10.20 Uniform Asymptotic Expansions for Large Order
10.20.5 Y ν ( ν z ) - ( 4 ζ 1 - z 2 ) 1 4 ( Bi ( ν 2 3 ζ ) ν 1 3 k = 0 A k ( ζ ) ν 2 k + Bi ( ν 2 3 ζ ) ν 5 3 k = 0 B 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).
7: Bibliography O
  • F. W. J. Olver (1951) A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order. Proc. Cambridge Philos. Soc. 47, pp. 699–712.
  • F. W. J. Olver (1952) Some new asymptotic expansions for Bessel functions of large orders. Proc. Cambridge Philos. Soc. 48 (3), pp. 414–427.
  • F. W. J. Olver (1954) The asymptotic expansion of Bessel functions of large order. Philos. Trans. Roy. Soc. London. Ser. A. 247, pp. 328–368.
  • F. W. J. Olver (1959) Uniform asymptotic expansions for Weber parabolic cylinder functions of large orders. J. Res. Nat. Bur. Standards Sect. B 63B, pp. 131–169.
  • F. W. J. Olver (1962) Tables for Bessel Functions of Moderate or Large Orders. National Physical Laboratory Mathematical Tables, Vol. 6. Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.
  • 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: 10.24 Functions of Imaginary Order
    In consequence of (10.24.6), when x is large J ~ ν ( x ) and Y ~ ν ( x ) comprise a numerically satisfactory pair of solutions of (10.24.1); compare §2.7(iv). … For mathematical properties and applications of J ~ ν ( x ) and Y ~ ν ( x ) , including zeros and uniform asymptotic expansions for large ν , see Dunster (1990a). …
    10: Bibliography H
  • B. A. Hargrave and B. D. Sleeman (1977) Lamé polynomials of large order. SIAM J. Math. Anal. 8 (5), pp. 800–842.
  • C. J. Howls and A. B. Olde Daalhuis (1999) On the resurgence properties of the uniform asymptotic expansion of Bessel functions of large order. Proc. Roy. Soc. London Ser. A 455, pp. 3917–3930.