scholarly journals An Asymptotic Expansion for the Number of Permutations with a Certain Number of Inversions

10.37236/1528 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Lane Clark
Keyword(s):  
Asymptotic Expansion ◽  
Complete Asymptotic Expansion ◽  
Trigonometric Integral

Let $b(n,k)$ denote the number of permutations of $\{1,\ldots,n\}$ with precisely $k$ inversions. We represent $b(n,k)$ as a real trigonometric integral and then use the method of Laplace to give a complete asymptotic expansion of the integral. Among the consequences, we have a complete asymptotic expansion for $b(n,k)/n!$ for a range of $k$ including the maximum of the $b(n,k)/n!$.

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.

scholarly journals Data Processing Method of Well Testing

2021 ◽  
Vol 134 (3) ◽  
pp. 35-38
Author(s):  
A. M. Svalov ◽  
Keyword(s):  
Boundary Condition ◽  
Asymptotic Expansion ◽  
Test Data ◽  
Traditional Method ◽  
Processing Method ◽  
Well Testing ◽  
Well Test ◽  
Conductivity Equation ◽  
Complete Asymptotic Expansion ◽  
Data Processing Method

Horner’s traditional method of processing well test data can be improved by a special transformation of the pressure curves, which reduces the time the converted curves reach the asymptotic regimes necessary for processing these data. In this case, to take into account the action of the «skin factor» and the effect of the wellbore, it is necessary to use a more complete asymptotic expansion of the exact solution of the conductivity equation at large values of time. At the same time, this method does not allow to completely eliminate the influence of the wellbore, since the used asymptotic expansion of the solution for small values of time is limited by the existence of a singular point, in the vicinity of which the asymptotic expansion ceases to be valid. To solve this problem, a new method of processing well test data is proposed, which allows completely eliminating the influence of the wellbore. The method is based on the introduction of a modified inflow function to the well, which includes a component of the boundary condition corresponding to the influence of the wellbore.

Landau constants and their q-analogues

2015 ◽  
Vol 13 (02) ◽  
pp. 217-231 ◽  
Author(s):  
Mourad E. H. Ismail ◽  
Xin Li ◽  
M. Rahman
Keyword(s):  
Asymptotic Expansion ◽  
Large Degree ◽  
Complete Asymptotic Expansion

We derive inequalities and a complete asymptotic expansion for the Landau constants Gn, as n → ∞ using the asymptotic sequence n!/(n + k)!. We also introduce a q-analogue of the Landau constants and calculate their large degree asymptotics. In the process, we also establish q-analogues of identities due to Ramanujan and Bailey.

scholarly journals A REMARK ON LONG RANGE SCATTERING FOR THE NONLINEAR KLEIN–GORDON EQUATION

2005 ◽  
Vol 02 (01) ◽  
pp. 77-89 ◽  
Author(s):  
HANS LINDBLAD ◽  
AVY SOFFER
Keyword(s):  
Asymptotic Expansion ◽  
Initial Data ◽  
Long Range ◽  
Gordon Equation ◽  
Scattering Problem ◽  
One Dimension ◽  
Complete Asymptotic Expansion ◽  
Arbitrary Size ◽  
Klein Gordon ◽  
Klein Gordon Equation

We consider the scattering problem for the nonlinear Klein–Gordon Equation with long range nonlinearity in one dimension. We prove that for all prescribed asymptotic solutions there is a solution of the equation with such behavior, for some choice of initial data. In the case the nonlinearity has the good sign (repulsive) the result hold for arbitrary size asymptotic data. The method of proof is based on reducing the long range phase effects to an ODE; this is done via an appropriate ansatz. We also find the complete asymptotic expansion of the solutions.

Hyperasymptotics for integrals with finite endpoints

1992 ◽  
Vol 439 (1906) ◽  
pp. 373-396 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Complex Plane ◽  
Airy Function ◽  
Error Function ◽  
Steepest Descent ◽  
Similar Form ◽  
Complete Asymptotic Expansion ◽  
Complementary Error Function ◽  
Plane Passing ◽  
Method Of Steepest Descent

Berry & Howls (1991) (hereinafter called BH) refined the method of steepest descent to study exponentially accurate asymptotics of a general class of integrals involving exp {– kf ( z )} along doubly infinite contours in the complex plane passing over saddlepoints of f ( z ). Here we derive analogous results for integrals with integrands of a similar form, but whose local expansions in powers of 1/ k are made about the finite endpoints of semi-infinite contours of integration. We treat endpoints where f ( z ) behaves locally linearly or quadratically. Generically, local endpoint expansions made by the method of steepest descent diverge because of the presence of saddles of f ( z ). We derive ‘resurgence relations’ which express the original integral exactly as a truncated endpoint expansion plus a remainder, involving the global saddle structure of f ( z ) via integrals through certain ‘adjacent’ saddles. The saddles adjacent to the endpoint are determined by a topological rule. If the least term of the endpoint expansion is the N 0 ( k ) th, summing to here calculates the endpoint integral up to an error of approximately exp ( – N 0 ( k )). We develop a scheme, involving iteration of the new resurgence relations with a similar one derived in BH, which can reduce this error down to exp( – 2.386 N 0 ( k )). This ‘hyperasymptotic’ formalism parallels that of BH and incorporates automatically any change in the complete asymptotic expansion as the endpoint moves in the complex plane, provided that it does not coincide with other saddles. We illustrate the analytical and numerical use of endpoint hyperasymptotics by application to the complementary error function erfc( x ) and a constructed ‘incomplete’ Airy function.

scholarly journals Large-time asymptotics for the density of a branching Wiener process

2005 ◽  
Vol 42 (4) ◽  
pp. 1081-1094 ◽  
Author(s):  
Pál Révész ◽  
Jay Rosen ◽  
Zhan Shi
Keyword(s):  
Asymptotic Expansion ◽  
Wiener Process ◽  
Large Time ◽  
Large Time Asymptotics ◽  
Complete Asymptotic Expansion ◽  
Time Asymptotics ◽  
Number Of Particles

Given an ℝd-valued supercritical branching Wiener process, let ψ(A, T) be the number of particles in A ⊂ ℝd at time T (T = 0, 1, 2, …). We provide a complete asymptotic expansion of ψ(A, T) as T → ∞, generalizing the work of X. Chen.

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