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By Boyd J.P.

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The constant ws can be moved to the left side of the equation and the linear term is zero because dw/dt is zero at the saddle point. Taking the square root gives √ 1 d2 w w − ws = (t − ts ) (ts ) {1 + O(t − ts )} (71) 2 dt2 √ It follows that dt/dw is proportional to 1/ w − ws near the saddle point, which demonstrates the theorem. Denote the image-of-a-saddle-point of smallest absolute value by wmin . The coefficients bj of the power series of the integrand will then asymptote, for sufficiently high degree j, to those of a constant √ times 1/ w − wmin ; the contributions of the singularities that are more remote in the complex w-plane will decrease exponentially fast with j compared to the contribution of the square root branch point at w = wmin .

In other words, the calculation of the correction can be more formidable than that of the original asymptotic expansion. One is reminded of the dictum, sometimes asserted in physics, that getting an extra decimal place demands 100 times the effort expended on the previous one. ” — D. S. Jones (1990) [155] [pg. 261] Jones’ mildly pessimistic remarks are still true: hyperasymptotics is more work than superasymptotics and one does have to evaluate additional transcendentals. However, Dingle showed in a series of articles in the late fifties and early sixties, collected in his 1973 book, that there is a suprising universality to hyperasymptotics: a quartet of generic transcendentals suffices to cover almost all cases.

The integral representation for the Airy function is Ai(z) ≡ z 1/2 2π C exp z 3/2 i (t + t3 /3) dt (75) where C is a contour that originates at infinity at an angle arg(t) = (5/6)π − (1/2) arg(z) and returns to infinity at arg(t) = (1/6)π − (1/2) arg(z). Fig. 9 shows the steepest descent paths of integration for the Airy integral representation. As explained in the preceding section, the easiest way to generate the coefficients of the asymptotic series is to begin with a change-of-coordinate to a new integration variable w.

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Asymptotic, superasymptotic and hyperasymptotic series by Boyd J.P.

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