Integrals Of Elementary Functions Research Articles

Bifurcations Associated with Three-Phase Polynomial Dynamical Systems and Complete Description of Symmetry Relations Using Discriminant Criterion

The behavior and bifurcations of solutions to three-dimensional (three-phase) quadratic polynomial dynamical systems (DSs) are considered. The integrability in elementary functions is proved for a class of autonomous polynomial DSs. The occurrence of bifurcations of the type-twisted fold is discovered on the basis and within the frames of the elements of the developed DS qualitative theory. The discriminant criterion applied originally to two-phase quadratic polynomial DSs is extended to three-phase DSs investigated in terms of their coefficient matrices. Specific classes of D- and S-vectors are introduced and a complete description of the symmetry relations inherent to the DS coefficient matrices is performed using the discriminant criterion.

Derivation of Some Entries in the Tables of David Bierens De Haan and Anatolii Prudnikov: An Exercise in Integration Theory

It is always useful to improve the catalogue of definite integrals available in tables. In this paper we use our previous work on Lobachevsky integrals to derive entries in the tables by Bierens De Haan and Anatolli Prudnikov featuring errata results and new integral formula for interested readers. In this work we derive a definite integral given by (1) in terms of the Lerch function. The importance of this work lies in the derivation of known and new results not presently found in current literature. We used our contour integral method and applied it to an integral in Prudnikov and used it to derive a closed form solution in terms of a special function. The advantage of using a special function is the added benefit of analytic continuation which widens the range of computation of the parameters. Special functions have significance in mathematical analysis, functional analysis, geometry, physics, and other applications. Special functions are used in the solutions of differential equations or integrals of elementary functions. Special functions are linked to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics.

Effective free-fermionic form factors and the XY spin chain

We introduce effective form factors for one-dimensional lattice fermions with arbitrary phase shifts. We study tau functions defined as series of these form factors. On the one hand we perform the exact summation and present tau functions as Fredholm determinants in the thermodynamic limit. On the other hand simple expressions of form factors allow us to present the corresponding series as integrals of elementary functions. Using this approach we re-derive the asymptotics of static correlation functions of the XY quantum chain at finite temperature.

Representations of $$\zeta (2n + 1)$$ and Related Numbers in the Form of Definite Integrals and Rapidly Convergent Series

Let $$\zeta (s)$$ and $$\beta (s)$$ be the Riemann zeta function and the Dirichlet beta function. The formulas for calculating the values of $$\zeta (2m)$$ and $$\beta (2m - 1)$$ ( $$m = 1,\;2,\; \ldots $$ ) are classical and well known. Our aim is to represent $$\zeta (2m + 1)$$ , $$\beta (2m)$$ , and related numbers in the form of definite integrals of elementary functions and rapidly converging numerical series containing $$\zeta (2m)$$ . By applying the method of this work, on the one hand, both classical formulas and ones relatively recently obtained by others researchers are proved in a uniform manner, and on the other hand, numerous new results are derived.

Euler’s formula for the zeta function at the positive even integers

We give a new proof of Euler’s formula for the values of the Riemann zeta function at the positive even integers. The proof involves estimating a certain integral of elementary functions two different ways and using a recurrence relation for the Bernoulli polynomials evaluated at 12.

New cases of integrable systems with dissipation on tangent bundles of two- and three-dimensional spheres

Integrability in elementary functions is demonstrated for some classes of dynamic systems on tangent bundles of two- and three-dimensional spheres. The force fields possess the so-called variable dissipation with a zero mean and generalize those considered earlier.

A multidimensional pendulum in a nonconservative force field under the presence of linear damping

A nonconservative force field in the dynamics of a multidimensional solid is constructed according to the results from the dynamics of real solids occurring in the force field of the action of the medium. In this case, it becomes possible to generalize the equations of motion of a multidimensional solid in a similarly constructed field of forces and to obtain a complete list of, generally speaking, transcendental first integrals expressed through a finite combination of elementary functions. In the study, the integrability in elementary functions is shown for the simultaneous equations of motion of a dynamically symmetric fixed multidimensional solid under the action of a nonconservative pair of forces in the presence of the linear damping moment (the additional dependence of the force field on the tensor of angular velocity of the solid).

Hardy's Reduction for a Class of Liouville Integrals of Elementary Functions

This paper is concerned with a class of integrals whose integrands are the product of a rational function times the exponential of a nonconstant rational function. We call these Liouville integrals. For these integrals, we provide a student-friendly algorithm producing a two-term decomposition with minimum transcendental and maximum elementary components. This decomposition fulfills the conditions of Hardy's reduction theory, determines whether these integrals are elementary functions, and when in the affirmative, finds them. To achieve our goal, we use partial fraction decomposition, simple notions of linear algebra, and a special case of an 1835 theorem of Liouville that we refer to as Liouville's criterion on integration. There is in the literature a complete algorithm to decide if the integral of an elementary function is also elementary. Ours is a gentle alternative for the class of Liouville integrals.

A Multiple Scattering Theory Approach to Solving the Time-Dependent Schrödinger Equation with an Asymmetric Rectangular Potential

The exact expressions for an energy-dependent Green function (resolvent), space-time propagator and time-dependent solution for the wave function Ψ( r , t ) of a particle moving in the presence of an asymmetric rectangular well/barrier potential are obtained. It is done by applying to this problem the multiple scattering theory (MST), which is different from previous such approaches by using the localized at the potential jumps effective potentials responsible for transmission through and reflection from the considered rectangular potential. This approach (alternative to the path-integral one) enables considering these processes from a particle (rather than a wave) point of view. The solution for the wave function describes these quantum phenomena as a function of time and is related to the fundamental issues (such as measuring time) of quantum mechanics. It is presented in terms of integrals of elementary functions and is a sum of the forward- and backward-moving components of the wave packet. The relative contribution of these components and their interference as well as of the potential asymmetry to the probability density |Ψ( x, t )| 2 and particle dwell time is considered and numerically visualized for narrow and broad energy (momentum) distributions of the initial Gaussian wave packet. It is shown that in the case of a broad initial wave packet, the quantum mechanical counterintuitive effect of the influence of the backward-moving components on the considered quantities becomes significant.

Simulation of rigid body motion in a resisting medium and analogies with vortex streets

The author returns to constructing the nonlinear mathematical model of a plane-parallel impact of a medium on a rigid body with allowance for the dependence of the arm of force on the reduced angular velocity of the body (a kind of Strouhal number). In this case, the arm of the force is also a function of the angle of attack. As was shown by the processing of the experimental data on the motion of homogeneous circular cylinders in water, these circumstances should be taken into account in the simulation. The analysis of a plane model of the interaction of a rigid body with a medium revealed new cases of the complete integrability in elementary functions, which made it possible to find qualitative analogies between the motion of free bodies in a resisting medium and vibrations of partly fixed bodies in a flow of an oncoming medium. Phase patterns obtained in the analysis of the nonlinear model of the impact of the medium are compared with real Karman’s vortex streets.

CERTAIN CONDITIONS OF INTEGRABILITY OF DYNAMICAL SYSTEMS IN TRANSCENDENTAL FUNCTIONS

Certain general conditions of integrability in elementary functions for the systems on the tangent bundle of two-dimensional sphere are studied. At that an interesting example of three-dimensional phase pattern of pendulum-like system which describes the motion of spherical pendulum, placed in an over-run medium flow. Sufficient conditions of existence of the first integrals expressed through the finite combination of elementary functions, for multi-parametric third order systems are presented.

Improvement of Self-Inductance Calculations for Circular Coils of Rectangular Cross Section

In this paper, an improvement for calculating the self-inductance of circular coils of rectangular cross section is presented. An integral of elementary functions is applied to replace the Bessel and Struve functions in the integrand of the inductance expression, and the term of improper integral without the exponential factor is solved to a closed form which consists of the complete elliptic integrals of the first and second kind and generalized hypergeometric functions. Due to this method, the main difficulties of the numerical evaluations of the inductance expression are overcome. Excellent agreement with published method is confirmed by the numerical comparisons and the calculations can be accelerated to several tens to hundreds of times depending on the shape factors of the coils. Similar expressions of the special cases (thin-wall solenoids and disk coils) are also given and compared with previous methods. Furthermore, the asymptotic behaviors of the special coils which tend to be a circular ring are obtained and numerically validated.

LAGRANGIAN FEYNMAN FORMULAS FOR SECOND-ORDER PARABOLIC EQUATIONS IN BOUNDED AND UNBOUNDED DOMAINS

In this note a class of second-order parabolic equations with variable coefficients, depending on coordinate, is considered in bounded and unbounded domains. Solutions of the Cauchy–Dirichlet and the Cauchy problems are represented in the form of a limit of finite-dimensional integrals of elementary functions (such representations are called Feynman formulas). Finite-dimensional integrals in the Feynman formulas give approximations for functional integrals in the corresponding Feynman–Kac formulas, representing solutions of these problems. Hence, these Feynman formulas give an effective tool to calculate functional integrals with respect to probability measures generated by diffusion processes with a variable diffusion coefficient and absorption on the boundary.

On integrability in elementary functions of certain classes of complex nonautonomous equations

This work is devoted to the search for integrability cases of a certain class of complex, linear second-order, nonautonomous ordinary differential equations.

On integrability in elementary functions of certain classes of nonconservative dynamical systems

The results of the presented work are due to the study of the applied problem of the rigid body motion in a resisting medium; see [210, 211], where complete lists of transcendental first integrals expressed through a finite combination of elementary functions were obtained. This circumstance allowed the author to perform a complete analysis of all phase trajectories and highlight those properties of them which exhibit the roughness and preserve for systems of a more general form. The complete integrability of those systems is related to symmetries of a latent type. Therefore, it is of interest to study sufficiently wide classes of dynamical systems having analogous latent symmetries.

Complete integrability of the equations of motion of a spatial pendulum in a medium flow with rotational derivatives of the torque produced by the medium taken into account

We constructanonlinearmodelofthe mediumactiononarigidbodytakinginto account the dependence of the forcearm on the reducedangular velocityof the body.Inthis case, the moment of the action force itself is also a function of the angle of attack. As experimental data processing for the motion of homogeneous circular cylinders in water has shown, it is necessary to take these facts into account in modeling. Studying the model of interaction between the spatial pendulum and the medium, we found a new case of complete integrability in elementary functions. This allowed us to find several qualitative analogies between the motions of bodies that are free in the resisting environment and the oscilla- tions of bodies partially fixed in the homogeneous flow of incoming medium. DOI: 10.3103/S002565440703017X

On solving general algebraic equations by integrals of elementary functions

We obtain an integral formula for a solution to a general algebraic equation. In this formula the integrand is an elementary function and integration is carried out over an interval. The advantage of this formula over the well-known Mellin formula is that the integral has a broader convergence domain. This circumstance makes it possible to describe the monodromy of a solution for trinomial equations.

Expansion of the Hamiltonian of the Two-Planetary Problem into the Poisson Series in All Elements: Application of the Poisson Series Processor

This paper is the third in a series of articles devoted to one of the basic problems of celestial mechanics: the study of the evolution of solar-type planetary systems. In the previous papers a brief review of the history and current state of the problem was given; the plan of the study was outlined; the Jacobi coordinates and the related osculating elements were introduced; the form of the Poisson expansion of the Hamiltonian in all elements was given; and the expansion coefficients for the Hamiltonian of the two-planetary Sun–Jupiter–Saturn problem were obtained (though with impure accuracy) by a simple algorithm that is reduced to the calculation of multiple integrals of elementary functions. In the present paper the expansion of the Hamiltonian of the two-planetary Sun–Jupiter–Saturn problem into the Poisson series in all elements is constructed with the help of the PSP Poisson series processor, which is capable of required accuracy.

A simple solution of the bridged crack problem

A simple, yet effective solution is derived to the problem of a penny-shaped crack, whose faces are bridged by fibers, and subjected to normal loading. The mathematical apparatus developed by the author earlier (Fabrikant, 1989, 1991) is used to reduce the governing integral equation to a sequence of integrals of elementary functions, which can be easily solved by the method of collocations. The case of uniform loading is considered as an example. The accuracy of the obtained solution is rigorously verified by different criteria.

The Expansion of the Hamiltonian of the Two-Planetary Problem into a Poisson Series in All Elements: Estimation and Direct Calculation of Coefficients

This is the second paper in a series of articles devoted to one of the basic problems of celestial mechanics: the evolution of solar-type planetary systems. In the first paper (Kholshevnikov et al., 2001), we reviewed the history and the current state of the issue, outlined the scheme of the study, introduced Jacobi coordinates and related osculating elements, and indicated the form of the Hamiltonian expansion into a Poisson series in all elements. In this paper, the expansion coefficients are found according to a simple algorithm that is reduced to the calculation of multiple integrals of elementary functions. At the first stage, we restricted our analysis to the two-planetary problem (Sun–Jupiter–Saturn). The general case will be investigated in a forthcoming paper.