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1: 13.19 Asymptotic Expansions for Large Argument
§13.19 Asymptotic Expansions for Large Argument
2: 10.17 Asymptotic Expansions for Large Argument
§10.17 Asymptotic Expansions for Large Argument
§10.17(ii) Asymptotic Expansions of Derivatives
§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).
3: Bibliography Y
  • T. Yoshida (1995) Computation of Kummer functions U ( a , b , x ) for large argument x by using the τ -method. Trans. Inform. Process. Soc. Japan 36 (10), pp. 2335–2342 (Japanese).
  • 4: 10.40 Asymptotic Expansions for Large Argument
    §10.40 Asymptotic Expansions for Large Argument
    Products
    ν -Derivative
    §10.40(iv) Exponentially-Improved Expansions
    5: 13.7 Asymptotic Expansions for Large Argument
    §13.7 Asymptotic Expansions for Large Argument
    §13.7(ii) Error Bounds
    §13.7(iii) Exponentially-Improved Expansion
    For extensions to hyperasymptotic expansions see Olde Daalhuis and Olver (1995a).
    6: 10.67 Asymptotic Expansions for Large Argument
    §10.67 Asymptotic Expansions for Large Argument
    §10.67(i) ber ν x , bei ν x , ker ν x , kei ν x , and Derivatives
    §10.67(ii) Cross-Products and Sums of Squares in the Case ν = 0
    7: Bibliography N
  • M. Nardin, W. F. Perger, and A. Bhalla (1992a) Algorithm 707: CONHYP: A numerical evaluator of the confluent hypergeometric function for complex arguments of large magnitudes. ACM Trans. Math. Software 18 (3), pp. 345–349.
  • M. Nardin, W. F. Perger, and A. Bhalla (1992b) Numerical evaluation of the confluent hypergeometric function for complex arguments of large magnitudes. J. Comput. Appl. Math. 39 (2), pp. 193–200.
  • G. Nemes (2015b) On the large argument asymptotics of the Lommel function via Stieltjes transforms. Asymptot. Anal. 91 (3-4), pp. 265–281.
  • G. Nemes (2017) Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions. Acta Appl. Math. 150, pp. 141–177.
  • G. Nemes (2018) Error bounds for the large-argument asymptotic expansions of the Lommel and allied functions. Stud. Appl. Math. 140 (4), pp. 508–541.
  • 8: 11.13 Methods of Computation
    Then from the limiting forms for small argument (§§11.2(i), 10.7(i), 10.30(i)), limiting forms for large argument (§§11.6(i), 10.7(ii), 10.30(ii)), and the connection formulas (11.2.5) and (11.2.6), it is seen that H ν ( x ) and L ν ( x ) can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. …
    9: 10.18 Modulus and Phase Functions
    §10.18(iii) Asymptotic Expansions for Large Argument
    The remainder after k terms in (10.18.17) does not exceed the ( k + 1 ) th term in absolute value and is of the same sign, provided that k > ν - 1 2 .
    10: 10.68 Modulus and Phase Functions
    §10.68(iii) Asymptotic Expansions for Large Argument