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

By Boyd J.P.

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The term that changes dramatically in the neighborhood of the Stokes line is exponentially small compared to the sum of the asymptotic series. However, the smoothing does provide deep insights into the interlocking systems of caverns — interlocking systems of expansions about different saddle points and branch points — that lie beneath the surface of asymptotic approximations. The numerical smoothing of the discontinuity along a Stokes lines is based on the following ideas which will be explained below: 1.

3. Asymptotically solve the “inner” problem as | y |→ ∞, that is, compute the “outer limit of the inner solution”. 4. Sum the divergent outer limit of the inner problem by Borel summation or otherwise determine the connection formula, that is, the magnitude and phase of the discontinuity along the Stokes line radiating from the critical point to the real x-axis 5. 52 53 Exponential Asymptotics Real y-axis arg(y)=0 Im(t) Inner Matching Re(t) arg(y)= - π/2 arg(y)= -π Outer arg(y)= -(1/4) π Re(t) Im(t) Im(t) Real x-axis arg(y)= -(3/4) π Re(t) arg(y)= -(3/4) π Figure 12.

Thus, the error integral EN ( ) for one function, S( ), provides a hyperasymptotic approximation to an entire class of functions, namely all those of the form of Eq. (54) for which the convergence-limiting singularity of Φ(z) is a simple pole at z = −1. It might seem as if we would have to repeat the analysis for each different species of singularity — one family of error integrals when the singularity is a simple pole, another when the dominant singularity of Φ is a logarithm and so on. In reality, Dingle shows that for a very wide range of asymptotic expansions, both from integral representations, the WKB method, and so on, the coefficients are asymptotically of the form aj ∼ q(−1)j Γ(j + 1 − β) ρj+1−β (62) for some constants q, ρ and β.

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