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11: 8.12 Uniform Asymptotic Expansions for Large Parameter
Inverse Function
12: 28.31 Equations of Whittaker–Hill and Ince
Asymptotic Behavior
13: 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.
  • 14: 30.11 Radial Spheroidal Wave Functions
    §30.11(iii) Asymptotic Behavior
    15: 33.11 Asymptotic Expansions for Large ρ
    §33.11 Asymptotic Expansions for Large ρ
    16: 33.18 Limiting Forms for Large
    §33.18 Limiting Forms for Large
    17: 8.27 Approximations
  • DiDonato (1978) gives a simple approximation for the function F ( p , x ) = x - p e x 2 / 2 x e - t 2 / 2 t p d t (which is related to the incomplete gamma function by a change of variables) for real p and large positive x . This takes the form F ( p , x ) = 4 x / h ( p , x ) , approximately, where h ( p , x ) = 3 ( x 2 - p ) + ( x 2 - p ) 2 + 8 ( x 2 + p ) and is shown to produce an absolute error O ( x - 7 ) as x .

  • 18: 33.10 Limiting Forms for Large ρ or Large | η |
    §33.10(i) Large ρ
    §33.10(ii) Large Positive η
    §33.10(iii) Large Negative η
    19: 2.11 Remainder Terms; Stokes Phenomenon
    In order to guard against this kind of error remaining undetected, the wanted function may need to be computed by another method (preferably nonasymptotic) for the smallest value of the (large) asymptotic variable x that is intended to be used. …
    20: 33.12 Asymptotic Expansions for Large η
    §33.12 Asymptotic Expansions for Large η