Hyperasymptotic expansions of confluent hypergeometric functions

Daalhuis, A. B. Olde

doi:10.1093/imamat/49.3.203

Cited by 21 publications

(33 citation statements)

References 5 publications

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“…The expansion will be in terms of the function Fa+N,,(z), and functions of the form U(l -b, c, Ape;<e ·"l), where A is a positive constant. Notice that for confluent hypergeometric functions with the present z-domain, we have already derived a hyperasymptotic expansion in Section 2 and [5].…”

Section: Hyperasymptotic Expansions Of U(a C Z) Near the Stokes Limentioning

confidence: 91%

“…Again, in [5] it is explained how to compute the integrals in (3.13 Next we want to expand g2(r, z) in a Taylor series at r = 12· By wntmg gz(r,z)={g1(r,z)-L;;';,,(/bn,1(r-11)n}(r-y1)-N 1 , it follows that the Taylor series of g2(r, z) at r = y2 can be computed with the Taylor series of g1(r, z) at r = 'Y2· We write gi(r, z) = bo,2 + b1,2(r -12) + ... + bN 2 -1,2(r -12)N 2 -1 + (r -'Y2)N 2 g3(r, z), (3.16) and we obtain…”

Section: Hyperasymptotic Expansions Of U(a C Z) Near the Stokes Limentioning

confidence: 96%

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Hyperasymptotics and the Stokes' phenomenon

Daalhuis

1993

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Synopsis!"lyperas~mptotic expansions were recently introduced by Berry and Howls, and yield refined information by expanding remainders in asymptotic expansions. In a recent paper of Olde Daalhuis, a method was given for obtaining hyperasymptotic expansions of integrals that represent the confluent hypergeometric V-function. This paper gives an extension of that method to neighbourhoods of the so-called Stokes lines. At each level, the remainder is exponentially small compared with the previous remainders. Two numerical illustrations confirm these exponential improvements.

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Section: Hyperasymptotic Expansions Of U(a C Z) Near the Stokes Limentioning

confidence: 91%

Section: Hyperasymptotic Expansions Of U(a C Z) Near the Stokes Limentioning

confidence: 96%

Hyperasymptotics and the Stokes' phenomenon

Daalhuis

1993

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…Stokes' phenomenon for the Stirling expansion for log Γ(z). (For an earlier discussion, see [34,35,36,4,20,26,6] and [37] 6 .) The Stirling function P (z) given by (9) can be represented as…”

Section: Improved Error Bounds In the Regionsmentioning

confidence: 99%

“…Let the spaces B(R, A, ε) and W(R, A, ε) be given by (37) and (38), respectively. Assume that F (t) ∈ B(R, A, ε).…”

Section: Duality Theoremmentioning

confidence: 99%

Error bounds, duality, and the Stokes phenomenon. I

Gurarii¹

2010

St. Petersburg Math. J.

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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].

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“…We follow existing terminology [1,2,5,9] for these new types of asymptotic expansions. Thus re-expansions in terms of generalized exponential integrals are called 1 to e~2l z l|z|~2.…”

Section: Introductionmentioning

confidence: 99%

Hyperasymptotic solutions of second-order linear differential equations I

Daalhuis¹,

Olver²

1995

Methods and Applications of Analysis

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ABSTRACT. A sequence of re-expansions is developed for the remainder terms in the well-known Poincare series expansions of the solutions of homogeneous linear differential equations of the second order in the neighborhood of an irregular singularity of rank one. These re-expansions are in series of repeated integrals of the generalized exponential integral, and the coefficients in these series are the same as those of the original Poincare expansions. Each step of the process reduces the estimate of the error term by the same exponentially-small factor, while increasing the region of validity.It also is shown how to ensure that the process is numerically stable. A numerical example is included.

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Hyperasymptotic expansions of confluent hypergeometric functions

Cited by 21 publications

References 5 publications

Hyperasymptotics and the Stokes' phenomenon

Hyperasymptotics and the Stokes' phenomenon

Error bounds, duality, and the Stokes phenomenon. I

Hyperasymptotic solutions of second-order linear differential equations I

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