Error bounds, duality, and the Stokes phenomenon. I

Abstract

Abstract. We consider classes of functions uniquely determined by coefficients of their divergent expansions. Approximating a function in such a class by partial sums of its expansion, we study how the accuracy changes when we move within a given region of the complex plane. Analysis of these changes allows us to propose a theory of divergent expansions, which includes a duality theorem and the Stokes phenomenon as essential parts. In its turn, this enables us to formulate necessary and sufficient conditions for a particular divergent expansion to encounter the Stokes phenomenon. We derive explicit expressions for the exponentially small terms that appear upon crossing Stokes lines and lead to an improvement in the accuracy of the expansion. §1. IntroductionThe Stokes phenomenon plays an exceptional role in complex analysis, in particular, in asymptotic analysis. It is very important for applications in theoretical and mathematical physics.This phenomenon was a stumbling block for Poincaré's asymptotic theory, which failed to explain it and which could not be used to evaluate the exponentially small Stokes' discontinuities.The stream of papers developing various ideas in the explanation of the phenomenon is vast and continues to grow. See, for example, [1,3,4,5,6,7,8], and the list of publications can be continued. The title of the paper "Stokes phenomenon demystified", see [9], is indicative of its extraordinary nature. It is difficult to think of another discovery of the middle part of the 19th century which still needs to be demystified 150 years after it was reported for the first time.In evaluating the position of fringes of supernumerary rainbows, Stokes [10] came upon the problem of approximation of an entire function (the Airy function) in the complex z-plane by multivalued functions (generated by formal solutions of Airy's differential equation). He discovered the existence of certain rays, Stokes rays, such that upon crossing these rays, one needs to add a term, exponentially small in z as z → ∞, to the approximation in order to retain the previous accuracy.Nearly one hundred years later, Pokrovskiȋ and Khalatnikov published the paper [11] in which they computed an exponentially small reflection which is not detected by the WKB method, and which would come to be called "asymptotics beyond-all-orders"; see [5], especially the papers by M. Berry and M. Kruskal et al.,and [12].

Error bounds and exponential improvement for Hermite's asymptotic expansion for the Gamma function

“…Using the integral representations (35) and (36), we deduce the estimate (7). To prove the bound (8), note that when z > 0 is real, we have 0 < , as can be seen from the representations (35) and (36).…”

Error bounds for the asymptotic expansion of the Hurwitz zeta function

“…The asymptotic expansion (4.3) is usually attributed to Stirling however, it was first discovered by De Moivre (for a detailed historical account, see [1, pp. 482-483] [20], and Guariȋ [9] can all be deduced as direct consequences of (4.4), (1.7) and the bounds given in Appendix A.…”

“…where N is any positive integer and R (Γ) N (z) including those of Lindelöf [12], F. W. Schäfke and A. Sattler [20], and Guariȋ [9] can all be deduced as direct consequences of (4.4), (1.7) and the bounds given in Appendix A.…”

Fractional Calculus in C and linear ODE's with Analytic Coefficients, Part 1

“…Remark 1. In paper [13], where our version of fractional calculus was first proposed, we denote by L the standard Laplace transform operator, see equations ( 44) and ( 46) with α = 0. We note that keeping in mind the application to differential equations with analytic coefficients, it is a little more convenient to use (5) instead of the standard definition.…”

“…Our main aim is to find a method of summation of the asymptotic series given by ( 9) with p k = p k (α) that would enable us to represent the results of summation of the series ∞ k=n p k (α) /ζ k in the form of an integral transformation of F (t, α) . The required function F (t, α) and the kernel of this transformation have been presented, with the appropriate change of notation, in Section 4 and Section 5 of [13], see equations ( 55), (100) and (105) where s = α + n. Later we show how to apply this technique to the asymptotic analysis of solutions of linear ODE's with analytic coefficients.…”