scholarly journals Asymptotics of some generalized Mathieu series

2020 ◽  
Vol 126 (3) ◽  
pp. 424-450
Author(s):  
Stefan Gerhold ◽  
Friedrich Hubalek ◽  
Živorad Tomovski
Keyword(s):  
Dirichlet Series ◽  
Gamma Function ◽  
Mellin Transform ◽  
Asymptotic Estimates ◽  
Precise Asymptotics ◽  
Singular Behavior ◽  
Natural Boundary ◽  
First Order ◽  
Elementary Estimate ◽  
Mathieu Series

We establish asymptotic estimates of Mathieu-type series defined by sequences with power-logarithmic or factorial behavior. By taking the Mellin transform, the problem is mapped to the singular behavior of certain Dirichlet series, which is then translated into asymptotics for the original series. In the case of power-logarithmic sequences, we obtain precise first order asymptotics. For factorial sequences, a natural boundary of the Mellin transform makes the problem more challenging, but a direct elementary estimate gives reasonably precise asymptotics. As a byproduct, we prove an expansion of the functional inverse of the gamma function at infinity.

scholarly journals A note on a functional equation

1973 ◽  
Vol 15 (4) ◽  
pp. 385-388
Author(s):  
Chung-Ming An
Keyword(s):  
Functional Equation ◽  
Dirichlet Series ◽  
Gamma Function ◽  
Generalized Gamma ◽  
Special Cases ◽  
Generalized Gamma Function

The object of this note is to give an aspect to the problem of the functional equation of the generalized gamma function and Dirichlet series which are defined in [1]. In general, we cannot answer the problem yet. But it is worthy to attack this problem for some special cases.

Analysis of Shells of Revolution Subjected to Symmetrical and Nonsymmetrical Loads

1964 ◽  
Vol 31 (3) ◽  
pp. 467-476 ◽  
Author(s):  
A. Kalnins
Keyword(s):  
Direct Integration ◽  
Shells Of Revolution ◽  
Natural Boundary ◽  
First Order ◽  
Rotationally Symmetric ◽  
Dependent Variables ◽  
Method Of Solution ◽  
Natural Boundary Conditions ◽  
Numerical Method Of Solution ◽  
Theory Of Shells

The boundary-value problem of deformation of a rotationally symmetric shell is stated in terms of a new system of first-order ordinary differential equations which can be derived for any consistent linear bending theory of shells. The dependent variables contained in this system of equations are those quantities which appear in the natural boundary conditions on a rotationally symmetric edge of a shell of revolution. A numerical method of solution which combines the advantages of both the direct integration and the finite-difference approach is developed for the analysis of rotationally symmetric shells. This method eliminates the loss of accuracy encountered in the usual application of the direct integration approach to the analysis of shells. For the purpose of illustration, stresses and displacements of a pressurized torus are calculated and detailed numerical results are presented.

INFINITE PRODUCTS OF CYCLOTOMIC POLYNOMIALS

2015 ◽  
Vol 91 (3) ◽  
pp. 400-411 ◽  
Author(s):  
WILLIAM DUKE ◽  
HA NAM NGUYEN
Keyword(s):  
Asymptotic Behaviour ◽  
Dirichlet Series ◽  
Unit Circle ◽  
Roots Of Unity ◽  
Cyclotomic Polynomials ◽  
Natural Boundary ◽  
Infinite Products

We study analytic properties of certain infinite products of cyclotomic polynomials that generalise some products introduced by Mahler. We characterise those that have the unit circle as a natural boundary and use associated Dirichlet series to obtain their asymptotic behaviour near roots of unity.

scholarly journals Instability of a vortex sheet leaving a right-angled wedge

2016 ◽  
Vol 803 ◽  
pp. 1-17
Author(s):  
Anthony M. J. Davis ◽  
Stefan G. Llewellyn Smith
Keyword(s):  
Mellin Transform ◽  
Infinite Plate ◽  
Vortex Sheet ◽  
Instability Wave ◽  
Helmholtz Instability ◽  
Order Difference ◽  
First Order ◽  
Complementary Function ◽  
Careful Handling ◽  
Order Difference Equation

We examine the dynamics of a semi-infinite vortex sheet attached not to a semi-infinite plate but instead to a rigid right-angled wedge, with the sheet aligned along one of its edges. Our approach to this problem, which was suggested by David Crighton, accords well with the fundamental ethos of Crighton’s work, which was characterized by ‘the application of rigorous mathematical approximations to fluid mechanical idealizations of practically relevant problems’ (Ffowcs Williams, Annu. Rev. Fluid Mech., vol. 34, 2002, pp. 37–49). The resulting linearised unsteady potential flow is forced by an oscillatory dipole in the uniform stream passing along the top of the wedge, while there is stagnant fluid in the remaining quadrant. Spatial instability is considered according to well-established methods: causality is enforced by allowing the frequency to become temporarily complex. The essentially quadrant-type geometry replaces the usual Wiener–Hopf technique by the Mellin transform. The core difficulty is that a first-order difference equation of period 4 requires a solution of period unity. As a result, the complex fourth roots $(\pm 1\pm \text{i})$ of $-4$ appear in the complementary function. The Helmholtz instability wave is excited and requires careful handling to obtain explicit results for the amplitude of the instability wave.

Wetting a superomniphobic porous system

Soft Matter ◽  
2019 ◽  
Vol 15 (42) ◽  
pp. 8621-8626 ◽  
Author(s):  
J. Cimadoro ◽  
L. Ribba ◽  
S. Goyanes ◽  
E. Cerda
Keyword(s):  
Critical Pressure ◽  
Porous System ◽  
Aperture Size ◽  
Singular Behavior ◽  
Small Aperture ◽  
First Order ◽  
System P

We study experimentally and theoretically the critical pressure needed to move a liquid through a network of pores and show that, for small aperture size, wetting and leaking are typical first-order transitions, with a singular behavior at the omniphobic/omniphilic limit.

scholarly journals Dynamics of Single-City Influenza with Seasonal Forcing: From Regularity to Chaos

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
John H. M. Thornley ◽  
James France
Keyword(s):  
Avian Influenza ◽  
Gamma Function ◽  
Environmental Effects ◽  
Swine Influenza ◽  
Disease Status ◽  
Compartment Model ◽  
Control Measures ◽  
First Order ◽  
First Order Kinetics ◽  
Disease Mechanisms

Seasonal and epidemic influenza continue to cause concern, reinforced by connections between human and avian influenza, and H1N1 swine influenza. Models summarize ideas about disease mechanisms, help understand contributions of different processes, and explore interventions. A compartment model of single-city influenza is developed. It is mechanism-based on lower-level studies, rather than focussing on predictions. It is deterministic, without non-disease-status stratification. Categories represented are susceptible, infected, sick, hospitalized, asymptomatic, dead from flu, recovered, and one in which recovered individuals lose immunity. Most categories are represented with sequential pools with first-order kinetics, giving gamma-function progressions with realistic dynamics. A virus compartment allows representation of environmental effects on virus lifetime, thence affecting reproductive ratio. The model's behaviour is explored. It is validated without significant tuning against data on a school outbreak. Seasonal forcing causes a variety of regular and chaotic behaviours, some being typical of seasonal and epidemic flu. It is suggested that models use sequential stages for appropriate disease categories because this is biologically realistic, and authentic dynamics is required if predictions are to be credible. Seasonality is important indicating that control measures might usefully take account of expected weather.

Gamma function asymptotics by an extension of the method of steepest descents

1994 ◽  
Vol 447 (1931) ◽  
pp. 609-630 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Recent Work ◽  
Gamma Function ◽  
Error Bounds ◽  
Stokes Line ◽  
Asymptotic Estimates ◽  
Asymptotic Representations

Recent work of Berry & Howls, which reformulated the method of steepest de­scents, is exploited to derive a new representation for the gamma function. It is shown how this representation can be used to derive a number of properties of the asymptotic expansion of the gamma function, including explicit and realistic error bounds, the Berry transition between different asymptotic representations across a Stokes line, and asymptotic estimates for the late coefficients.

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