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1: 2.11 Remainder Terms; Stokes Phenomenon
The choice of α and β is facilitated by a knowledge of the relevant Stokes lines; see §2.11(iv) below. … Rays (or curves) on which one contribution in a compound asymptotic expansion achieves maximum dominance over another are called Stokes lines ( θ = π in the present example). … The relevant Stokes lines are ph z = ± π for w 1 ( z ) , and ph z = 0 , 2 π for w 2 ( z ) . … For example, extrapolated values may converge to an accurate value on one side of a Stokes line2.11(iv)), and converge to a quite inaccurate value on the other.
2: 36.5 Stokes Sets
§36.5 Stokes Sets
§36.5(i) Definitions
§36.5(ii) Cuspoids
Elliptic Umbilic Stokes Set (Codimension three)
§36.5(iv) Visualizations
3: 8.22 Mathematical Applications
8.22.1 F p ( z ) = Γ ( p ) 2 π z 1 - p E p ( z ) = Γ ( p ) 2 π Γ ( 1 - p , z ) ,
plays a fundamental role in re-expansions of remainder terms in asymptotic expansions, including exponentially-improved expansions and a smooth interpretation of the Stokes phenomenon. … The function Γ ( a , z ) , with | ph a | 1 2 π and ph z = 1 2 π , has an intimate connection with the Riemann zeta function ζ ( s ) 25.2(i)) on the critical line s = 1 2 . …
4: Bibliography D
  • T. M. Dunster (1990a) Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.
  • T. M. Dunster (1996b) Asymptotics of the generalized exponential integral, and error bounds in the uniform asymptotic smoothing of its Stokes discontinuities. Proc. Roy. Soc. London Ser. A 452, pp. 1351–1367.