Asymptotic expansions of the oblate spheroidal eigenvalues and wave functions for large parameter c

2001 ◽  
Vol 79 (5) ◽  
pp. 813-831 ◽  
Author(s):  
Tam Do-Nhat
Keyword(s):  
Asymptotic Expansion ◽  
Power Series ◽  
Asymptotic Expansions ◽  
Wave Functions ◽  
Laguerre Functions ◽  
Large Parameter ◽  
Oblate Spheroidal ◽  
Asymptotic Eigenvalue ◽  
Expansion Coefficients ◽  
Analytical Expressions

The asymptotic expansion of the oblate spheroidal eigenfunctions can be expanded in terms of the Laguerre functions of the first and second kinds, from which their asymptotic eigenvalue can be expressed in an inverse power series of c, where the parameter c is proportional to the operating frequency. Analytical expressions of the eigenvalue coefficients, as well as those of the expansion coefficients of the eigenfunctions, are derived and verified with numerical results. PACS Nos.: 02.30Gp, 03.65ge

Hyperasymptotics for integrals with saddles

1991 ◽  
Vol 434 (1892) ◽  
pp. 657-675 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Saddle Point ◽  
Multiple Scattering ◽  
Positive Constant ◽  
Asymptotic Expansions ◽  
Steepest Descent ◽  
The Other ◽  
Large Parameter ◽  
Original Series ◽  
Method Of Steepest Descent

Integrals involving exp { – k f ( z )}, where | k | is a large parameter and the contour passes through a saddle of f ( z ), are approximated by refining the method of steepest descent to include exponentially small contributions from the other saddles, through which the contour does not pass. These contributions are responsible for the divergence of the asymptotic expansion generated by the method of steepest descent. The refinement is achieved by means of an exact ‘resurgence relation', expressing the original integral as its truncated saddle-point asymptotic expansion plus a remainder involving the integrals through certain ‘adjacent’ saddles, determined by a topological rule. Iteration of the resurgence relation, and choice of truncation near the least term of the original series, leads to a representation of the integral as a sum of contributions associated with ‘multiple scattering paths’ among the saddles. No resummation of divergent series is involved. Each path gives a ‘hyperseries’, depending on the terms in the asymptotic expansions for each saddle (these depend on the particular integral being studied and so are non-universal), and certain ‘hyperterminant’ functions defined by integrals (these are always the same and hence universal). Successive hyperseries get shorter, so the scheme naturally halts. For two saddles, the ultimate error is approximately ∊ 2.386 , where ∊ (proportional to exp (— A │ k │) where A is a positive constant), is the error in optimal truncation of the original series. As a numerical example, an integral with three saddles is computed hyperasymptotically.

Asymptotics of the Titchmarsh-Weyl m-coefficient for integrable potentials

1986 ◽  
Vol 103 (3-4) ◽  
pp. 347-358 ◽  
Author(s):  
Hans G. Kaper ◽  
Man Kam Kwong
Keyword(s):  
Asymptotic Expansion ◽  
Power Series ◽  
Asymptotic Behaviour ◽  
Potential Function ◽  
Error Bound ◽  
Power Series Expansion ◽  
Smoothness Condition ◽  
Asymptotic Power ◽  
Simple Method ◽  
Expansion Coefficients

This article is concerned with the asymptotic behaviour of m(λ), the Titchmarsh-Weyl m-coefficient, for the singular eigenvalue equation y“ + (λ − q(x))y = 0 on [0, ∞), as λ →∞ in a sector in the upper half of the complex plane. It is assumed that the potential function q is integrable near 0. A simplified proof is given of a result of Atkinson [7], who derived the first two terms in the asymptotic expansion of m(λ), and a sharper error bound is obtained. Theproof is then generalised to derive subsequent terms in the asymptotic expansion. It is shown that the Titchmarsh-Weyl m-coefficient admits an asymptotic power series expansion if the potential function satisfies some smoothness condition. A simple method to compute the expansion coefficients is presented. The results for the first few coefficients agree with those given by Harris [9].

Integrals with a large parameter. Several nearly coincident saddle points

1972 ◽  
Vol 72 (1) ◽  
pp. 49-65 ◽  
Author(s):  
F. Ursell
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Analytic Functions ◽  
Special Functions ◽  
Saddle Points ◽  
Positive Parameter ◽  
Large Parameter ◽  
Airy Functions ◽  
Ordinary Method ◽  
Image Position

AbstractIn the contour integralthe functions g( ) and f( ) are analytic functions of their arguments, and N is a large positive parameter. When N tends to ∞, asymptotic expansions can usually be found by the method of steepest descents, which shows that the principal contributions arise from the saddle-points, i.e. the values of z at which ∂f/∂z = 0. The position of the saddle-points varies with the l parameters, denoted by α = (α1,α2,…,αl), and it may happen that two or more saddle-points coincide at a certain value of α. The asymptotic expansions given by the ordinary method of steepest descents are then non-uniform. The case of a single parameter and two nearly coincident saddle- points was studied earlier and leads to Airy functions. Here we are concerned with the case of l parameters and of m saddle-points which are nearly coincident when α is near 0. It is then known that f(z,α) can be transformed locally into a polynomial of degree m + 1, and accordingly we here restrict f(z,α) to be such a polynomial. The form of the resulting asymptotic expansion had already been conjectured (correctly as we now see), and involves certain special functions of m–1 variables. To establish the validity of the expansion we must show that the remainder after any number of terms is of smaller magnitude than the last term retained in the expansion. In other words, we must establish an inequality between integrals which is to hold for sufficiently small α and sufficiently large N, and which involves 1 parameters and m nearly coincident saddle-points.

scholarly journals Asymptotic expansions and positivity of coefficients for large powers of analytic functions

2003 ◽  
Vol 2003 (16) ◽  
pp. 1003-1025 ◽  
Author(s):  
Valerio De Angelis
Keyword(s):  
Asymptotic Expansion ◽  
Power Series ◽  
Asymptotic Expansions ◽  
Analytic Functions ◽  
Large Range ◽  
Consecutive Integers ◽  
Positive Coefficients

We derive an asymptotic expansion asn→∞for a large range of coefficients of(f(z))n, wheref(z)is a power series satisfying|f(z)|<f(|z|)forz∈ℂ,z∉ℝ+. Whenfis a polynomial and the two smallest and the two largest exponents appearing infare consecutive integers, we use the expansion to generalize results of Odlyzko and Richmond (1985) on log concavity of polynomials, and we prove that a power offhas only positive coefficients.

Asymptotic expansion of the Mathieu and prolate spheroidal eigenvalues for large parameter c

1999 ◽  
Vol 77 (8) ◽  
pp. 635-652 ◽  
Author(s):  
T Do-Nhat
Keyword(s):  
Wave Number ◽  
Parabolic Cylinder ◽  
Prolate Spheroidal ◽  
Spheroidal Wave Functions ◽  
Prolate Spheroidal Wave Functions ◽  
Asymptotic Eigenvalue ◽  
Parabolic Cylinder Functions ◽  
Expansion Coefficients ◽  
Analytical Expressions ◽  
Operating Wave

The asymptotic expansions of the Mathieu eigenfunctions and the prolate spheroidal wave functions can be expanded in terms of the parabolic cylinder functions, from which their asymptotic eigenvalue can be expressed in an inverse power series of c, where the parameter c is proportional to the operating wave number. Analytical expressions of the eigenvalues, as well as those of the expansion coefficients of the eigenfunctions, are derived and verified with numerical results.PACS. No.: 2.30 MV

scholarly journals Generating Asymptotics for Factorially Divergent Sequences

10.37236/5999 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Michael Borinsky
Keyword(s):  
Asymptotic Expansion ◽  
Power Series ◽  
Formal Power Series ◽  
Asymptotic Expansions ◽  
Series Form ◽  
Chord Diagrams ◽  
Algebraic Properties ◽  
Chain Rules ◽  
Formal Power ◽  
And Inversion

The algebraic properties of formal power series, whose coefficients show factorial growth and admit a certain well-behaved asymptotic expansion, are discussed. It is shown that these series form a subring of $\mathbb{R}[[x]]$. This subring is also closed under composition and inversion of power series. An `asymptotic derivation' is defined which maps a power series to the asymptotic expansion of its coefficients. Product and chain rules for this derivation are deduced. With these rules asymptotic expansions of the coefficients of implicitly defined power series can be obtained. The full asymptotic expansions of the number of connected chord diagrams and the number of simple permutations are given as examples.

Asymptotic expansions and converging factors VI. Application to physical prediction

1959 ◽  
Vol 249 (1257) ◽  
pp. 293-295 ◽  
Keyword(s):  
Magnetic Field ◽  
Asymptotic Expansion ◽  
Power Series ◽  
Asymptotic Expansions ◽  
Free Electron ◽  
Electron Gas ◽  
Perturbation Expansion ◽  
Exact Form ◽  
Precise Representation ◽  
The Magnetic Field

It is shown how methods developed in earlier papers of this series can be used to find a more precise representation of a physical quantity given as a power series (for instance, a perturbation expansion) and, moreover, to help in predicting the existence and magnitude of terms missed out in such an expansion because of its preconceived form as a power series. As an illustration, the exact form for the steady diamagnetism of a free-electron gas, and the oscillatory contribution (de Haas-van Alphen effect), are both predicted from a known asymptotic expansion in rising powers of the magnetic field.

scholarly journals Prime geodesics and averages of the Zagier L-series

2021 ◽  
pp. 1-24
Author(s):  
OLGA BALKANOVA ◽  
DMITRY FROLENKOV ◽  
MORTEN S. RISAGER
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Error Term ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Prime Geodesic Theorem ◽  
Error Terms ◽  
Average Size ◽  
Real Quadratic

Abstract The Zagier L-series encode data of real quadratic fields. We study the average size of these L-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem.

Asymptotic Expansions in the Exponent: a Compound Poisson Approach

1997 ◽  
Vol 29 (02) ◽  
pp. 374-387 ◽  
Author(s):  
V. Čekanavičius
Keyword(s):  
Asymptotic Expansion ◽  
Large Deviations ◽  
Poisson Distribution ◽  
Asymptotic Expansions ◽  
Compound Poisson ◽  
Poisson Measures ◽  
Poisson Approximations

The accuracy of the Normal or Poisson approximations can be significantly improved by adding part of an asymptotic expansion in the exponent. The signed-compound-Poisson measures obtained in this manner can be of the same structure as the Poisson distribution. For large deviations we prove that signed-compound-Poisson measures enlarge the zone of equivalence for tails.

scholarly journals Asymptotic expansions of Lambert series and related q-series

2017 ◽  
Vol 13 (08) ◽  
pp. 2097-2113 ◽  
Author(s):  
Shubho Banerjee ◽  
Blake Wilkerson
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Theta Functions ◽  
Lambert Series ◽  
Pochhammer Symbol ◽  
Complete Asymptotic Expansion ◽  
Polygamma Functions ◽  
Jacobi Theta Functions ◽  
Relative Errors ◽  
Asymptotic Forms

We study the Lambert series [Formula: see text], for all [Formula: see text]. We obtain the complete asymptotic expansion of [Formula: see text] near [Formula: see text]. Our analysis of the Lambert series yields the asymptotic forms for several related [Formula: see text]-series: the [Formula: see text]-gamma and [Formula: see text]-polygamma functions, the [Formula: see text]-Pochhammer symbol and the Jacobi theta functions. Some typical results include [Formula: see text] and [Formula: see text], with relative errors of order [Formula: see text] and [Formula: see text] respectively.

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