The calculation of the coefficients of certain asymptotic series by means of linear recurrent relations

1952 ◽  
Vol 2 (1) ◽  
pp. 77-84 ◽  
Author(s):  
H. A. Lauwerier
Keyword(s):  
Asymptotic Series ◽  
Recurrent Relations

An asymptotic series for the mean value of DirichletL-functions

1981 ◽  
Vol 56 (1) ◽  
pp. 148-161 ◽  
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Asymptotic Series ◽  
Mean Value ◽  
The Mean

A property of the asymptotic series for a class of Titchmarsh–Weyl m-functions

1986 ◽  
Vol 102 (3-4) ◽  
pp. 253-257 ◽  
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Linear Differential Equation ◽  
Asymptotic Series ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
The Real ◽  
Linear Differential ◽  
Image Position

SynopsisIn an earlier paper [6] we showed that if q ϵ CN[0, ε) for some ε > 0, then the Titchmarsh–Weyl m(λ) function associated with the second order linear differential equationhas the asymptotic expansionas |A| →∞ in a sector of the form 0 < δ < arg λ < π – δ.We show that if the real valued function q admits the expansionin a neighbourhood of 0, then

Derivation of the Explicit Solution of the Inverse Involute Function and its Applications

1992 ◽  
Author(s):  
Harry Hui Cheng
Keyword(s):  
Series Solution ◽  
Asymptotic Series ◽  
Explicit Solutions ◽  
Series Solutions ◽  
Perturbation Techniques ◽  
Pocket Calculator ◽  
Practical Applications ◽  
Involute Gears ◽  
Simple Expansion ◽  
Explicit Series Solutions

Abstract The involute function ε = tanϕ – ϕ or ε = invϕ, and the inverse involute function ϕ = inv−1(ε) arise in the tooth geometry calculations of involute gears, involute splines, and involute serrations. In this paper, the explicit series solutions of the inverse involute function are derived by perturbation techniques in the ranges of |ε| &lt; 1.8, 1.8 &lt; |ε| &lt; 5, and |ε| &gt; 5. These explicit solutions are compared with the exact solutions, and the expressions for estimated errors are also developed. Of particular interest in the applications are the simple expansion ϕ = inv−1(ε) = (3ε)1/3 – 2ε/5 which gives the angle ϕ (&lt; 45°) with error less than 1.0% in the range of ε &lt; 0.215, and the economized asymptotic series expansion ϕ = inv−1 (ε) = 1.440859ε1/3 – 0.3660584ε which gives ϕ with error less than 0.17% in the range of ε &lt; 0.215. The four, seven, and nine term series solutions of ϕ = inv−1 (ε) are shown to have error less than 0.0018%, 4.89 * 10−6%, and 2.01 * 10−7% in the range of ε &lt; 0.215, respectively. The computation of the series solution of the inverse involute function can be easily performed by using a pocket calculator, which should lead to its practical applications in the design and analysis of involute gears, splines, and serrations.

A distributional theory for asymptotic expansions

1990 ◽  
Vol 428 (1875) ◽  
pp. 399-430 ◽  
Keyword(s):  
Asymptotic Expansions ◽  
Asymptotic Series ◽  
Generalized Functions ◽  
The Moment ◽  
The One ◽  
Topological Tensor Products ◽  
Multidimensional Integrals ◽  
Vector Valued ◽  
Selection Of ◽  
Mathematics And Physics

We present various techniques for the asymptotic expansions of generalized functions. We show that the moment asymptotic expansions hold for a very wide variety of kernels such as generalized functions of rapid decay and rapid oscillations. We do not use Mellin transform techniques as done by previous authors in the field. Instead, we introduce a direct approach that not only solves the one-dimensional problems but also applies to various multidimensional integrals and oscillatory kernels as well. This approach also helps in the development of various asymptotic series arising in diverse fields of mathematics and physics. We find that the asymptotic expansions of generalized functions depend on the selection of suitable spaces of test functions. Accordingly, we have exercised special care in classifying the spaces and the distributions defined on them. Furthermore, we use the theory of topological tensor products to obtain the expansions of vector-valued distributions. We present several examples to illustrate that many classical results follow in a simple manner. For instance, we derive from our results the asymptotic expansions of certain series considered by Ramanujan.

scholarly journals Closed-Form Asymptotic Sampling Distributions under the Coalescent with Recombination for an Arbitrary Number of Loci

2012 ◽  
Vol 44 (2) ◽  
pp. 391-407 ◽  
Author(s):  
Anand Bhaskar ◽  
Yun S. Song
Keyword(s):  
Asymptotic Expansion ◽  
Closed Form ◽  
Recombination Rate ◽  
Functional Form ◽  
Asymptotic Series ◽  
Sampling Distribution ◽  
Challenging Problem ◽  
Sampling Distributions ◽  
Asymptotic Sampling ◽  
New Framework

Obtaining a closed-form sampling distribution for the coalescent with recombination is a challenging problem. In the case of two loci, a new framework based on an asymptotic series has recently been developed to derive closed-form results when the recombination rate is moderate to large. In this paper, an arbitrary number of loci is considered and combinatorial approaches are employed to find closed-form expressions for the first couple of terms in an asymptotic expansion of the multi-locus sampling distribution. These expressions are universal in the sense that their functional form in terms of the marginal one-locus distributions applies to all finite- and infinite-alleles models of mutation.

Coordination sequences and layer-by-layer growth of periodic structures

2019 ◽  
Vol 234 (5) ◽  
pp. 291-299
Author(s):  
Anton Shutov ◽  
Andrey Maleev
Keyword(s):  
Generating Functions ◽  
Diophantine Equations ◽  
Periodic Structures ◽  
Layer Growth ◽  
Layer By Layer ◽  
Explicit Formulas ◽  
New Approach ◽  
Coordination Numbers ◽  
Linear Diophantine Equations ◽  
Recurrent Relations

Abstract A new approach to the problem of coordination sequences of periodic structures is proposed. It is based on the concept of layer-by-layer growth and on the study of geodesics in periodic graphs. We represent coordination numbers as sums of so called sector coordination numbers arising from the growth polygon of the graph. In each sector we obtain a canonical form of the geodesic chains and reduce the calculation of the sector coordination numbers to solution of the linear Diophantine equations. The approach is illustrated by the example of the 2-homogeneous kra graph. We obtain three alternative descriptions of the coordination sequences: explicit formulas, generating functions and recurrent relations.

scholarly journals Low Reynolds number flow in a heated tube of varying section

1983 ◽  
Vol 25 (2) ◽  
pp. 244-260 ◽  
Author(s):  
A. R. Bestman
Keyword(s):  
Reynolds Number ◽  
Free Convection ◽  
Asymptotic Series ◽  
Convection Heat Transfer ◽  
Axial Direction ◽  
Reynolds Number Flow ◽  
Free Convection Heat ◽  
Number Flow ◽  
Convection Parameter ◽  
Constricted Tube

AbstractIn this paper we study the effect of forced and free convection heat transfer on flow in an axisymmctric tube whose radius varies slowly in the axial direction. Asymptotic series expansions in terms of a small parameter ∈, which is a measure of the radius variation, are obtained for the velocity components, pressure and temperature on the assumption that the Reynolds number (R) is of order one. The effect of the free convection parameter or Grashof number (G) on the axial velocity, temperature distribution, shear stress and heat flux at the wall are discussed quantitatively for a locally constricted tube.

Laminar flow in symmetrical channels with slightly curved walls II. An asymptotic series for the stream function

1963 ◽  
Vol 272 (1350) ◽  
pp. 406-428 ◽  
Keyword(s):  
Laminar Flow ◽  
Asymptotic Series ◽  
Radius Of Curvature ◽  
Two Dimensional ◽  
The Third ◽  
Higher Order Terms ◽  
Curvature Parameter ◽  
Leading Term ◽  
Flow Through ◽  
Curved Walls

A class of two-dimensional channels, with walls whose radius of curvature is uniformly large relative to local channel width, is described, and the velocity field of laminar flow through these channels is obtained as a power series in the small curvature parameter. The leading term is the Jeffery-Hamel solution considered in part I, and it is shown here how the higher-order terms are found. Terms of the third approximation have been computed. The theory is applied to two examples, for one of which experimental results are available and confirm the theoretical values with fair accuracy.

scholarly journals Solution of the schödinger equation using Padé approximants to sum asymptotic series

1974 ◽  
Vol 4 (2) ◽  
pp. 375-376
Author(s):  
J.H. McGuire ◽  
J.E. Golden ◽  
J. Nuttall
Keyword(s):  
Asymptotic Series ◽  
Padé Approximants ◽  
Pade Approximants
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