Imaginary Argument Research Articles

Bridging the p-Special Functions between the Generalized Hyperbolic and Trigonometric Families

Here, we study the extension of p-trigonometric functions sinp and cosp family in complex domains and p-hyperbolic functions sinhp and the coshp family in hyperbolic complex domains. These functions satisfy analogous relations as their classical counterparts with some unknown properties. We show the relationship of these two classes of special functions viz. p-trigonometric and p-hyperbolic functions with imaginary arguments. We also show many properties and identities related to the analogy between these two groups of functions. Further, we extend the research bridging the concepts of hyperbolic and elliptical complex numbers to show the properties of logarithmic functions with complex arguments.

Integrals Involving Jacobi Theta Functions and Applications in Quantum Systems

We study four expressions involving the integrals of Jacobi’s theta functions. From Poisson’s summation formula, we write the integrals of the functions θi, (i = 1,2,3,4) in terms of modified Bessel functions of the second kind. For the integrals of θ1 , θ2 and θ3, we get expressions with real arguments, but for the integral of θ4 , we find an expression with imaginary argument. In addition, we apply our results to the description of two kinds of interacting quantum systems: boson gas and fermion gas both under a thermal bath and an external magnetic field. PACS numbers:

Overlapping of Lévai’s and Milson’s e-Tangent-Polynomial Potentials along Symmetric Curves

The paper examines common elements between Lévai’s and Milson’s potentials obtained by Liouville transformations of two rational canonical Sturm–Liouville equations (RCSLEs) with even density functions which are exactly solvable via Jacobi polynomials in a real or accordingly imaginary argument. We refer to the polynomial numerators of the given rational density function as ‘tangent polynomial’ (TP) and thereby term the aforementioned potentials as ‘e-TP’. Special attention is given to the overlap between the two potentials along symmetric curves which represent two different rational forms of the Ginocchio potential exactly quantized via Gegenbauer and Masjed-Jamei polynomials, respectively. Our analysis reveals that the actual interconnection between Lévai’s parameters for these two rational realizations of the Ginocchio potential is much more complicated than one could expect based on the striking resemblance between two quartic equations derived by Lévai for ‘averaged’ Jacobi indexes.

An integrated model based on transient flow and the nanoconfinement effect of water transport through nanopores

The property of water confined in nanopores is significantly different from that of bulk water, resulting in a great influence on the rate of water development during the unsteady state. In this study, we focused on the unsteady behavior of water transport in nanopores. We established a transient equation and applied the Laplace transform and Bessel function of the imaginary argument according to the time-dependent momentum conservation equation of Poiseuille-flow. The apparent density, apparent viscosity, and slip of the confined water were modeled by considering the water-wall interactions and the pore size. Moreover, the transient transport equation was modified by these three models to obtain a transient analytical model, which considered the effect of the nanoconfinement effect. We verified the credibility of the proposed model compared with other models as well as through experiments and molecular dynamics simulations. The results indicated that the development rate of water confinement in the superhydrophobic nanopores was much greater than that in intermediate wettability and hydrophilic pores, and its velocity profile behaved as an approximately straight line throughout the unsteady state. Slip in the intermediate wettability pores was more likely to transmit its effects to the pore center than in hydrophilic pores. We found that small hydrophilic pores were dominated by the apparent density and apparent viscosity in the unsteady state, whereas large hydrophilic pores were dominated by slip in the initial state, which gradually changed to apparent density and apparent viscosity with time. For the superhydrophobic pores, regardless of pore size, slip was the main controlling factor in the unsteady state. Additionally, the sensitivity of confined water stabilization time to pore size was much greater in the superhydrophobic pores than in the hydrophilic pores.

ELASTIC TRANSVERSAL-ISOTROPIC SPACE WITH TWO UNIAXIAL PARALLEL CIRCULAR CRACKS AND ASSOCIATED PROBLEMS OF BASICITY

In the paper a problem of the stress state in a transversely isotropic space with two parallel circular cracks, the centers of which are located on the axis of anisotropy of the space, is investigated. A constant normal load acts on the crack planes. The problem is solved by the generalized Fourier method. For this purpose, systems of compressed spheroidal coordinates are introduced, the origins of which are connected to the centers of cracks. The general solution of the problem is constructed in the form of series based on axisymmetric variants of the general vector solutions of the system of equations of equilibrium of a transversely isotropic body in spheroidal coordinates, which were previously constructed by one of the authors of the paper. To implement the method, it is further generalized to compressed spheroidal coordinate systems with origins shifted along the axis. For this purpose, new addition theorems for basic vector displacements of transversally isotropic bodies in the above-mentioned coordinate systems are obtained. After applying the generalized Fourier method, the problem is reduced to an infinite system of linear algebraic equations. It is proved that under certain geometrical and mechanical conditions the operator of the system is a Fredholm operator. The reduction method is used for the numerical solution of the system. Graphs of normal stresses in the plane of one of the cracks outside its boundaries, as well as values of stress intensity factor at the top of the crack for different geometric parameters of the cracks, are obtained. The obtained results agree with the known value of the stress intensity factor in the problem with one crack. The practical convergence of the reduction method is studied.
 As an important related problem, the problem of proving the basicity of a general axisymmetric set of external solutions of the system of equilibrium equations of a transversally isotropic body whose boundary is described in compressed spheroidal coordinates is considered. The key problem here is obtaining subtle estimates from below of the modulus of the determinant of the first boundary value problem. As a corollary of the obtained result, several important estimates from the theory of special functions are derived, in which Legendre functions of the second kind from a purely imaginary argument appear.

Two general series identities involving modified Bessel functions and a class of arithmetical functions

AbstractWe consider two sequences $a(n)$ and $b(n)$ , $1\leq n<\infty $ , generated by Dirichlet series $$ \begin{align*}\sum_{n=1}^{\infty}\frac{a(n)}{\lambda_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{\mu_n^{s}},\end{align*} $$ satisfying a familiar functional equation involving the gamma function $\Gamma (s)$ . Two general identities are established. The first involves the modified Bessel function $K_{\mu }(z)$ , and can be thought of as a ‘modular’ or ‘theta’ relation wherein modified Bessel functions, instead of exponential functions, appear. Appearing in the second identity are $K_{\mu }(z)$ , the Bessel functions of imaginary argument $I_{\mu }(z)$ , and ordinary hypergeometric functions ${_2F_1}(a,b;c;z)$ . Although certain special cases appear in the literature, the general identities are new. The arithmetical functions appearing in the identities include Ramanujan’s arithmetical function $\tau (n)$ , the number of representations of n as a sum of k squares $r_k(n)$ , and primitive Dirichlet characters $\chi (n)$ .

On the Highly Accurate Evaluation of the Voigt/Complex Error Function with Small Imaginary Argument

A rapidly convergent series, based on Taylor expansion of the imaginary part of the complex error function, is presented for highly accurate approximation of the Voigt/complex error function with small imaginary argument y ≤ 0.1. Error analysis and run-time tests in double-precision arithmetic reveals that in the real and imaginary parts, the proposed algorithm provides an average accuracy exceeding 10−15 and 10−16, respectively, and the calculation speed is as fast as that reported in recent publications. An optimized MATLAB code providing rapid computation with high accuracy is presented.

ХАРАКТЕРИСТИКИ КОЛЬЦЕВОГО РЕБРА ПРЯМОУГОЛЬНОГО ПРОФИЛЯ С ЭНЕРГОВЫДЕЛЕНИЕМ

The heat conduction equation for an annular fin with an arbitrary profile in the presence of energy release in the fin is obtained in the article. The resulting equation differs from the approximate equation given in the literature by the presence of energy release and a more accurate determination of the length of the arc element. As boundary conditions, the temperature of the base of the fin is set, and at the end of the fin, heat exchange occurs according to the Newton - Richmann law with the environment. The equation for the fin of a rectangular profile is an inhomogeneous modified Bessel equation. Its solution contains the Bessel functions of the imaginary argument of the first and second kind of zero order. The efficiency of the fin and the heat flow through the base of the fin are determined. The energy release in the fin increases its efficiency compared to the efficiency of the fin in the absence of energy release, and also reduces the heat flow. The restriction by the values of energy release in the fin is found as condition for the applicability of the finning. The fin efficiency must be less than one. If the efficiency exceeds one, the fin plays the opposite role: the flow is directed in the reverse side. In the article, an expression is obtained for the surface build-up coefficient kh. When calculating the heating (cooling) of a body with a finned surface, the heat transfer coefficient should be increased by kh times.

The Dirichlet Problem for an Elliptic Equation with Singular Coefficients in a Semi-Cylindrical Domain

We consider a three-dimensional equation of elliptic type with two singular coefficients, for which the Dirichlet problem is studied in a half-cylinder. The study of the problem is carried out using the method of spectral analysis. For the problem posed, using the Fourier method, one-dimensional spectral problems are obtained. Based on the completeness property of systems of eigenfunctions of these problems, the uniqueness theorem is proved. The solution of the problem under study is constructed in the form of the sum of the Fourier–Bessel series. In substantiating the uniform convergence of the constructed series, we used asymptotic estimates of the Bessel functions of the real and imaginary argument. Based on them, estimates are obtained for each member of the series, which made it possible to prove the convergence of the resulting series and its derivatives to the second order inclusive, as well as the existence theorem in the class of regular solutions.

ENGLISH-THE UPSHOTS AND SOCIO-CULTURAL CHARACTER FOR MUSLIM WOMEN IN ISLAMIC SOCIETY

Social advances and legal-based socio-cultural scholars have argued that “character legislative issues” weaken social advances and undermine their effectiveness in an open approach. I put forward the imaginary argument that some kind of “character assembly issues” could have the opposite impact. In particular, when composing underestimated clusters around the characteristics of a "network area", they master the local crowd, create feelings associated with the growth of strong members, and do alert work programs as well as create modules for comprehensive alliances. The U.S. In revealing, personal business material through colorful women strengthens the growth of women and does not legally enhance the reaction of experts to a shadow of a shadow against women and brutality against women. This article will be a focus on how and why the enemy usually spreads sex to seize power in the public arena, rather than acting out government issues in the cultural and religion. The article focuses mainly on the penances of women and their area as different personalities around in the Islamic Society.

The asymptotic zero distribution of Lommel polynomials as functions of their order with a variable complex argument

We study the asymptotic distribution of roots of Lommel polynomials as polynomials of the order with a variable and purely imaginary argument. The roots are complex and accumulate on certain curves in the complex plane. We prove existence of the weak limit of corresponding root-counting measures and deduce formulas for the supporting curves and density. The obtained result represents a solvable example of a more general problem which is still open. Numerical illustrations of the main result are also involved.

Analysis of free oscillations of circular plates with variable thickness based on the symmetry method

Based on symmetry and factorization methods, a general analytical solution to the fourth-order differential equation has been derived for a problem on the free axisymmetric oscillations of a circular plate of variable thickness. The law of the thickness change is the concave parabola h=H0(1–μρ)2, where µ is a constant coefficient that determines the degree of plate concaveness. The solution has been given by the Bessel functions of zero and the first order of the actual and imaginary argument. A circular ring plate has been considered whose inner contour is rigidly fixed and whose outer edge is free, for three values of the µ coefficient. We have determined the first three natural values for the problem (frequency numbers) and their natural functions (oscillation shapes). It has been shown that the natural frequencies of the first three shapes of oscillations decrease, with the increase in concaveness (increase in µ), to varying degrees, determined by the number of the frequency number λi (i=1, 2, 3). At µ=1.21417 and µ=1.39127, the frequencies decrease, compared to the case of µ=0.5985, by (1;1.3) %, (17.6;24) %, (22.85;30.35) %, respectively, for λ1, λ2, λ3. One can see a significant drop in frequency on the higher shapes of oscillations (λ2, λ3) and a slight drop in the basic shape (λ1). We have established the values and coordinates of extreme deflections (antinodes of oscillations) and the indicative coordinates of the nodal cross-sections. The reported numerical parameters, along with the frequency indicators, are a means of identifying the oscillational properties of a plate when it is studied in practice. We have built the graphic dependences for radial σr and tangential σθ cyclical stresses at the basic shape for each of the three variants of the concaveness of a parabolic plate. It has been established that the increase in the ratio of edge thickness, that is, concaveness, leads to an increase in σr in the cross-sections outside the end constraint. These stresses, which operate far from the free edge, for example at the end constraint or the area of the maximum σθ, are greater than σθ in varying degrees. Because of this, these stresses pose a major threat in terms of the cyclical strength of the plate when σr reaches destructive values. We have pointed to the possibility to provide, by increasing the concaveness of the parabolic plate, the optimal ratio between the value of σr at the end constraint and σr operating away from the fastening. This ratio, approximately equal to 1, is ensured at µ=1.39127 considered in the current work.

Keldysh problem for a three-dimensional equation of mixed type with three singular coefficients in a semi-infinite parallelepiped

This article studies the Keldysh problem for a three-dimensional equation of mixed type with three singular coefficients in a semi-infinite parallelepiped. Based on the completeness property of eigenfunction systems of two one-dimensional spectral problems, the uniqueness theorem is proved. To prove the existence of a solution to the problem, the Fourier spectral method based on the separation of variables is used. The solution to this problem is constructed in the form of a sum of a double Fourier-Bessel series. In substantiating the uniform convergence of the constructed series, we used asymptotic estimates of the Bessel functions of the real and imaginary argument. Based on them, estimates were obtained for each member of the series, which made it possible to prove the convergence of the series and its derivatives to the second order inclusive, as well as the existence theorem in the class of regular solutions.

Nonlocal Problem for an Elliptic Equation with Singular Coefficients in a Semi-infinite Parallelepiped

We consider a three-dimensional elliptic equation with two singular coefficients, for which a nonlocal problem is studied in a semi-infinite parallelepiped. The study of the problem is carried out using the method of separation of Fourier variables and spectral analysis. For the problem posed, using the Fourier method, two one-dimensional spectral problems are obtained. Based on the completeness property of the systems of eigenfunctions of these problems, the uniqueness theorem is proved. The solution to the problem is constructed in the form of the sum of a double Fourier series with respect to trigonometric and Bessel functions. In substantiating the uniform convergence of the constructed series, we used asymptotic estimates of the Bessel functions of the real and imaginary argument. Based on them, estimates are obtained for each member of the series, which made it possible to prove the convergence of the resulting series and its derivatives to the second order inclusive, as well as the existence theorem in the class of regular solutions.

The Unique Solvability of Boundary Value Problems for a 3D Elliptic Equation With Three Singular Coefficients

We state and study several boundary value problems for an elliptic equation with three singular coefficients in a rectangular parallelepiped. Using the method of energy integrals, we prove the uniqueness of solutions of the stated problems. For proving the existence of solutions, we use the spectral Fourier method based on separation of variables. We construct solutions to the stated problems in the form of the sum of the double Fourier—Bessel series. We justify the uniform convergence of the constructed series with the help of asymptotic bounds for Bessel functions of real and imaginary arguments. Using these bounds, we estimate each term of the series, which allows us to prove the convergence of the series and its derivatives up to the second order inclusive and to justify the existence theorem in the class of regular solutions.

To the Theory of C0-Operator Orthogonal Polynomials

Operator orthogonal polynomials are considered whose arguments are generators of strongly continuous semigroups of transformations of class C0 acting in a Banach space. Earlier such polynomials were considered by the authors in the case of the Chebyshev polynomials of the first and second kind. In this paper, more general classes of operator orthogonal polynomials are considered, which include the Jacobi and Aptekarev polynomials. Integral representations of operator fractional-rational functions and also of Bessel operator-valued functions of an imaginary argument are presented.

A sampling-based approximation of the complex error function and its implementation without poles

Recently we developed a new sampling methodology based on incomplete cosine expansion of the sinc function and applied it in numerical integration in order to obtain a rational approximation for the complex error function w(z)=e−z2(1+2iπ∫0zet2dt), where z=x+iy. As a further development, in this work we show how this sampling-based rational approximation can be transformed into alternative form for efficient computation of the complex error function w(z) at smaller values of the imaginary argument y=Im[z]. Such an approach enables us to avoid poles in implementation and to cover the entire complex plain with high accuracy in a rapid algorithm. An optimized Matlab code utilizing only three rapid approximations is presented.

A continued fraction of order twelve as a modular function

We study a continued fraction U ( τ ) U(\tau ) of order twelve using the modular function theory. We obtain the modular equations of U ( τ ) U(\tau ) by computing the affine models of modular curves X ( Γ ) X(\Gamma ) with Γ = Γ 1 ( 12 ) ∩ Γ 0 ( 12 n ) \Gamma = \Gamma _1 (12) \cap \Gamma _0(12n) for any positive integer n n ; this is a complete extension of the previous result of Mahadeva Naika et al. and Dharmendra et al. to every positive integer n n . We point out that we provide an explicit construction method for finding the modular equations of U ( τ ) U(\tau ) . We also prove that these modular equations satisfy the Kronecker congruence relations. Furthermore, we show that we can construct the ray class field modulo 12 12 over imaginary quadratic fields by using U ( τ ) U(\tau ) and the value U ( τ ) U(\tau ) at an imaginary quadratic argument is a unit. In addition, if U ( τ ) U(\tau ) is expressed in terms of radicals, then we can express U ( r τ ) U(r \tau ) in terms of radicals for a positive rational number r r .

Harmonic analysis associated with a discrete Laplacian

It is well-known that the fundamental solution of $$ u_t(n,t)= u(n+1,t)-2u(n,t)+u(n-1,t), \quad n\in\mathbb{Z}, $$ with $u(n,0) =\delta_{nm}$ for every fixed $m \in\mathbb{Z}$, is given by $u(n,t) = e^{-2t}I_{n-m}(2t)$, where $I_k(t)$ is the Bessel function of imaginary argument. In other words, the heat semigroup of the discrete Laplacian is described by the formal series $$ W_tf(n) = \sum_{m\in\mathbb{Z}} e^{-2t} I_{n-m}(2t) f(m). $$ By using semigroup theory, this formula allows us to analyze some operators associated with the discrete Laplacian. In particular, we obtain the maximum principle for the discrete fractional Laplacian, weighted $\ell^p(\mathbb{Z})$-boundedness of conjugate harmonic functions, Riesz transforms and square functions of Littlewood-Paley. Interestingly, it is shown that the Riesz transforms coincide essentially with the so called discrete Hilbert transform defined by D. Hilbert at the beginning of the XX century. We also see that these Riesz transforms are limits of the conjugate harmonic functions. The results rely on a careful use of several properties of Bessel functions.

Orthogonality of the Ferrers' associated Legendre functions of the second kind with imaginary argument

ABSTRACTIn this work we study the associated Legendre functions of the second kind with a purely imaginary argument . We derive the conditions under which they provide a set of square integrable functions when and we prove the relevant orthogonality relation that they satisfy.