Elliptic basis for the Zernike system: Heun function solutions

2018 ◽  
Vol 59 (7) ◽  
pp. 073503 ◽  
Author(s):  
Natig M. Atakishiyev ◽  
George S. Pogosyan ◽  
Kurt Bernardo Wolf ◽  
Alexander Yakhno
Keyword(s):  
Heun Function ◽  
Elliptic Basis

Duffin–Kemmer–Petiau oscillator in the presence of a cosmic screw dislocation

2020 ◽  
Vol 35 (30) ◽  
pp. 2050195
Author(s):  
Soroush Zare ◽  
Hassan Hassanabadi ◽  
Marc de Montigny
Keyword(s):  
Elastic Medium ◽  
Screw Dislocation ◽  
Topological Defect ◽  
Potential Functions ◽  
Wave Functions ◽  
Screw Dislocations ◽  
Energy Eigenvalues ◽  
Heun Function ◽  
Uvarov Method ◽  
The Impact

We examine the behavior of spin-zero bosons in an elastic medium which possesses a screw dislocation, which is a type of topological defect. Therefore, we solve analytically the Duffin–Kemmer–Petiau (DKP) oscillator for bosons in the presence of a screw dislocation with two types of potential functions: Cornell and linear-plus-cubic potential functions. For each of these functions, we analyze the impact of screw dislocations by determining the wave functions and the energy eigenvalues with the help of the Nikiforov–Uvarov method and Heun function.

scholarly journals Expansions of the Solutions of the General Heun Equation Governed by Two-Term Recurrence Relations for Coefficients

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
T. A. Ishkhanyan ◽  
T. A. Shahverdyan ◽  
A. M. Ishkhanyan
Keyword(s):  
Power Series ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Recurrence Relations ◽  
Power Series Expansion ◽  
Heun Equation ◽  
Generalized Hypergeometric Function ◽  
Gamma Functions ◽  
Heun Function ◽  
Finite Sum

We examine the expansions of the solutions of the general Heun equation in terms of the Gauss hypergeometric functions. We present several expansions using functions, the forms of which differ from those applied before. In general, the coefficients of the expansions obey three-term recurrence relations. However, there exist certain choices of the parameters for which the recurrence relations become two-term. The coefficients of the expansions are then explicitly expressed in terms of the gamma functions. Discussing the termination of the presented series, we show that the finite-sum solutions of the general Heun equation in terms of generally irreducible hypergeometric functions have a representation through a single generalized hypergeometric function. Consequently, the power-series expansion of the Heun function for any such case is governed by a two-term recurrence relation.

Elliptic basis of circular oscillator

1985 ◽  
Vol 88 (1) ◽  
pp. 43-56 ◽  
Author(s):  
L. G. Mardoyan ◽  
G. S. Pogosyan ◽  
A. N. Sissaklan ◽  
V. M. Ter-Antonyan
Keyword(s):  
Elliptic Basis

scholarly journals EMG-based facial gesture recognition through versatile elliptic basis function neural network

2013 ◽  
Vol 12 (1) ◽  
pp. 73 ◽  
Author(s):  
Mahyar Hamedi ◽  
Sh-Hussain Salleh ◽  
Mehdi Astaraki ◽  
Alias Noor
Keyword(s):  
Neural Network ◽  
Gesture Recognition ◽  
Basis Function ◽  
Elliptic Basis

scholarly journals Quantized black hole and Heun function

2012 ◽  
Vol 90 (9) ◽  
pp. 877-881 ◽  
Author(s):  
D. Momeni ◽  
Koblandy Yerzhanov ◽  
Ratbay Myrzakulov
Keyword(s):  
Differential Equation ◽  
Black Hole ◽  
Wave Equation ◽  
Physical Properties ◽  
Natural Generalization ◽  
Expectation Value ◽  
Hypergeometric Differential Equation ◽  
Heun Function ◽  
New Applications ◽  
Feynman Theorem

In this paper, following the simple proposal by He and Ma for quantization of a black hole (BH) using Bohr’s method, we discuss the solvability of the wave equation for such a BH. We consequentially solve the associated Schrödinger equation. The eigenfunction problem reduces to the HeunB, H(α, β, γ, δ; z), differential equation, which is a natural generalization of the hypergeometric differential equation. We investigate some physical properties of the wavefunction. We then obtain the expectation value of the kinetic and the potential energies, using Hellmann–Feynman theorem. Our work introduces some new applications of the Heun function.

On the cubic lattice Green functions

1994 ◽  
Vol 445 (1924) ◽  
pp. 463-477 ◽  
Keyword(s):  
Basic Result ◽  
Honeycomb Lattice ◽  
Lattice Statistics ◽  
Green Functions ◽  
Cubic Lattice ◽  
Heun Function ◽  
Hypercubic Lattice ◽  
The Face ◽  
Complete Elliptic Integrals ◽  
The Green Function

It is proved that K ( k + ) = [(4 – η ) 1/2 – (1 – η ) 1/2 ] K ( k _), where η is a complex variable which lies in a certain region R 2 of the η plane, and K ( k ± ) are complete elliptic integrals of the first kind with moduli k ± which are given by k 2 ± Ξ k 2 ± ( η ) = 1/2 + 1/4 η (4 – η ) 1/2 – 1/4 (2 – η ) (1 – η ) 1/2 . This basic result is then used to express the face-centred cubic and simple cubic lattice Green functions at the origin in terms of the square of a complete ellip­tic integral of the first kind. Several new identities involving the Heun function are also derived. F(a, b; α,β, γ, δ; η) are also derived. Next it is shown that the three cubic lat­tice Green functions all have parametric representations which involve the Green function for the two-dimensional honeycomb lattice. Finally, the results are ap­plied to a variety of problems in lattice statistics. In particular, a new simplified formula for the generating function of staircase polygons on a four-dimensional hypercubic lattice is derived.

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