Beam Models and Quantum Systems with Linear Potentials Based on Airy Functions

According to legend, Leo Szilard's baths were ruined by his conversion to biology. He had enjoyed soaking for hours while thinking about physics. But as a convert he found this pleasure punctuated by the frequent need to leap out and search for a fact. In physics—particularly theoretical physics—we can get by with a few basic principles without knowing many facts; that is why the subject attracts those of us cursed with poor memory.But there is a corpus of mathematical information that we do need. Much of this consists of formulas for the "special" functions. How many of us remember the expansion of cos 5x in terms of cos x and sin x, or whether an integral obtained in the course of a calculation can be identified as one of the many representations of a Bessel function, or whether the asymptotic expression for the gamma function involves (n + 12) or (n − 12)? For such knowledge, we theorists have traditionally relied on compilations of formulas. When I started research, my peers were using Jahnke and Emde's Tables of Functions with Formulae and Curves (J&E) 1 1. E. Jahnke, F. Emde, Tables of Functions with Formulae and Curves, Dover Publications, New York (1945). or Erdélyi and coauthors' Higher Transcendental Functions. 2 2. A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions, 5 vols., Krieger Publishing, Melbourne, Fla. (1981) [first published in 1953]. Then in 1964 came Abramowitz and Stegun's Handbook of Mathematical Functions (A&S), 3 3. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, vol. 55, US Government Printing Office, Washington, DC (1964). perhaps the most successful work of mathematical reference ever published. It has been on the desk of every theoretical physicist. Over the years, I have worn out three copies. Several years ago, I was invited to contemplate being marooned on the proverbial desert island. What book would I most wish to have there, in addition to the Bible and the complete works of Shakespeare? My immediate answer was: A&S. If I could substitute for the Bible, I would choose Gradsteyn and Ryzhik's Table of Integrals, Series and Products. 4 4. I. S. Gradsteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (translated from Russian by Scripta Technika), Academic Press, New York (2000) [first published in 1965]. Compounding the impiety, I would give up Shakespeare in favor of Prudnikov, Brychkov and Marichev's of Integrals and Series. 5 5. A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, 5 vols. (translated from Russian by N. M. Queen), Gordon and Breach, New York (1986–1992). On the island, there would be much time to think about physics and much physics to think about: waves on the water that carve ridges on the sand beneath and focus sunlight there; shapes of clouds; subtle tints in the sky. … With the arrogance that keeps us theorists going, I harbor the delusion that it would be not too difficult to guess the underlying physics and formulate the governing equations. It is when contemplating how to solve these equations—to convert formulations into explanations—that humility sets in. Then, compendia of formulas become indispensable.Nowadays the emphasis is shifting away from books towards computers. With a few keystrokes, the expansion of cos 5x, the numerical values of Bessel functions, and many analytical integrals can all be obtained easily using software such as Mathematica and Maple. (In the spirit of the times, I must be even handed and refer to both the competing religions.) A variety of resources is available online. The most ambitious initiative in this direction is being prepared by NIST, the descendant of the US National Bureau of Standards, which published A&S. NIST's forthcoming Digital Library of Mathematical Functions (DLMF) will be a free Web-based collection of formulas (dlmf.nist.gov), cross-linked and with live graphics that can be magnified and rotated. (Stripped-down versions of the project will be issued as a book and a CD-ROM for people who prefer those media.)The DLMF will reflect a substantial increase in our knowledge of special functions since 1964, and will also include new families of functions. Some of these functions were (with one class of exceptions) known to mathematicians in 1964, but they were not well known to scientists, and had rarely been applied in physics. They are new in the sense that, in the years since 1964, they have been found useful in several branches of physics. For example, string theory and quantum chaology now make use of automorphic functions and zeta functions; in the theory of solitons and integrable dynamical systems, Painlevé transcendents are widely employed; and in optics and quantum mechanics, a central role is played by "diffraction catastrophe" integrals, generated by the polynomials of singularity theory—my own favorite, and the subject of a chapter I am writing with Christopher Howls for the DLMF.This continuing and indeed increasing reliance on special functions is a surprising development in the sociology of our profession. One of the principal applications of these functions was in the compact expression of approximations to physical problems for which explicit analytical solutions could not be found. But since the 1960s, when scientific computing became widespread, direct and "exact" numerical solution of the equations of physics has become available in many cases. It was often claimed that this would make the special functions redundant. Similar skepticism came from some pure mathematicians, whose ignorance about special functions, and lack of interest in them, was almost total. I remember that when singularity theory was being applied to optics in the 1970s, and I was seeking a graduate student to pursue these investigations, a mathematician recommended somebody as being very bright, very knowledgeable, and interested in applications. But this student had never heard of Bessel functions (nor could he carry out the simplest integrations, but that is another story).The persistence of special functions is puzzling as well as surprising. What are they, other than just names for mathematical objects that are useful only in situations of contrived simplicity? Why are we so pleased when a complicated calculation "comes out" as a Bessel function, or a Laguerre polynomial? What determines which functions are "special"? These are slippery and subtle questions to which I do not have clear answers. Instead, I offer the following observations.There are mathematical theories in which some classes of special functions appear naturally. A familiar classification is by increasing complexity, starting with polynomials and algebraic functions and progressing through the "elementary" or "lower" transcendental functions (logarithms, exponentials, sines and cosines, and so on) to the "higher" transcendental functions (Bessel, parabolic cylinder, and so on). Functions of hypergeometric type can be ordered by the behavior of singular points of the differential equations representing them, or by a group-theoretical analysis of their symmetries. But all these classifications are incomplete, in the sense of omitting whole classes that we find useful. For example, Mathieu functions fall outside the hypergeometric class, and gamma and zeta functions are not the solutions of simple differential equations. Moreover, even when the classifications do apply, the connections they provide often appear remote and unhelpful in our applications.One reason for the continuing popularity of special functions could be that they enshrine sets of recognizable and communicable patterns and so constitute a common currency. Compilations like A&S and the DLMF assist the process of standardization, much as a dictionary enshrines the words in common use at a given time. Formal grammar, while interesting for its own sake, is rarely useful to those who use natural language to communicate. Arguing by analogy, I wonder if that is why the formal classifications of special functions have not proved very useful in applications.Sometimes the patterns embodying special functions are conjured up in the form of pictures. I wonder how useful sines and cosines would be without the images, which we all share, of how they oscillate. In 1960, the publication in J&E of a three-dimensional graph showing the poles of the gamma function in the complex plane acquired an almost iconic status. With the more sophisticated graphics available now, the far more complicated behavior of functions of several variables can be explored in a variety of two-dimensional sections and three-dimensional plots, generating a large class of new and shared insights."New" is important here. Just as new words come into the language, so the set of special functions increases. The increase is driven by more sophisticated applications, and by new technology that enables more functions to be depicted in forms that can be readily assimilated.Sometimes the patterns are associated with the asymptotic behavior of the functions, or of their singularities. Of the two Airy functions, Ai is the one that decays towards infinity, while Bi grows; the J Bessel functions are regular at the origin, the Y Bessel functions have a pole or a branch point.Perhaps standardization is simply a matter of establishing uniformity of definition and notation. Although simple, this is far from trivial. To emphasize the importance of notation, Robert Dingle in his graduate lectures in theoretical physics at the University of St. Andrews in Scotland would occasionally replace the letters representing variables by nameless invented squiggles, thereby inducing instant incomprehensibility. Extending this one level higher, to the names of functions, just imagine how much confusion the physicist John Doe would cause if he insisted on replacing sin x by doe(x), even with a definition helpfully provided at the start of each paper.To paraphrase an aphorism attributed to the biochemist Albert Szent-Györgyi, perhaps special functions provide an economical and shared culture analogous to books: places to keep our knowledge in, so that we can use our heads for better things. Kelvin'S ship-wave pattern, calculated with the Airy function, the simplest special function in the hierarchy of diffraction catastrophes.PPT|High resolution A cross section of the elliptic umbilic, a member of the hierarchy of diffraction catastrophes.PPT|High resolution The cusp, a member of the hierarchy of diffraction catastrophes.PPT|High resolutionREFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. E. Jahnke, F. Emde, Tables of Functions with Formulae and Curves, Dover Publications, New York (1945). Google Scholar2. A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions, 5 vols., Krieger Publishing, Melbourne, Fla. (1981) [first published in 1953]. Google Scholar3. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, vol. 55, US Government Printing Office, Washington, DC (1964). Google Scholar4. I. S. Gradsteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (translated from Russian by Scripta Technika), Academic Press, New York (2000) [first published in 1965]. Google Scholar5. A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, 5 vols. (translated from Russian by N. M. Queen), Gordon and Breach, New York (1986–1992). Google Scholar© 2001 American Institute of Physics.

Computational Aspects of Three-Term Recurrence Relations

We study spherically-symmetric structures in Conformal Gravity and in a scalar-tensor extension and gain some more insight about these gravitational theories. In both cases we analyze solutions in two systems: perfect fluid solutions and boson stars of a self-interacting complex scalar field. In the purely tensorial (original) theory we find in a certain domain of parameter space finite mass solutions with a linear gravitational potential but without a Newtonian contribution. The scalar-tensor theory exhibits a very rich structure of solutions whose main properties are discussed. Among them, solutions with a finite radial extension, open solutions with a linear potential and logarithmic modifications and also a (scalar-tensor) gravitational soliton. This may also be viewed as a static self-gravitating boson star in purely tensorial Conformal Gravity.

7.1 Introduction The Bessel functions of the first and second kinds studied in Chap. 6 are often referred to as the standard Bessel or cylinder functions. In addition to these, there are a host of related functions also belonging to the general family of cylinder functions, the most notable of which are the modified Bessel functions of the first and second kinds. Although similar in definition to the standard Bessel functions, the modified Bessel functions are most clearly distinguished by their nonoscillatory behavior. For this reason, they often appear in applications that are different in nature from those for the standard functions. The general family of cylinder functions also include spherical Bessel functions, Hankel functions, Kelvin's functions, Lommel functions, Struve functions, Airy functions, and Anger and Weber functions. Of these, Hankel functions have special significance in that they enable us to obtain asymptotic formulas for large arguments for all the other types of Bessel functions.

The classical Euler–Bernoulli beam model and Kirchhoff plate model are very useful on the macroscopic scale. In the context of Eringen’s nonlocal elasticity theory, this article aims to develop the criteria of the applicability of nanoscale Euler–Bernoulli beam and Kirchhoff plate models based on the Timoshenko beam model and the Mindlin plate model via the wave propagation theory. The corresponding governing differential equations for the nanoscale Timoshenko beam and Mindlin plate are derived by the Hamilton’s principle, and the dispersion equations of wave are then obtained. By applying Taylor expansion to the corresponding solutions of the dispersion equations, new criteria are developed, simultaneously taking into account the effects of nonlocal parameters and material properties. When the nonlocal parameter is set to zero, the present criteria may be readily degenerated to their macroscopic counterparts. According to the present criteria, this article systematically evaluates the existing studies in literature. Various works in literature did not consider the effect of the nonlocal parameter, and hence, failed to satisfy the application conditions of the Euler–Bernoulli beam and Kirchhoff plate models on the nanoscale. The work in this article is of scientific significance to various studies on nanostructures.

Important modeling applications in marine engineering conduct us to a special class of solutions for difficult differential equations with variable coefficients. In order to be able to solve and implement such models (in wave theory, in acoustics, in hydrodynamics, in electromagnetic waves, but also in many other engineering fields), it is necessary to compute so called special functions: Bessel functions, modified Bessel functions, spherical Bessel functions, Hankel functions. The aim of this paper is to develop numerical solutions in Matlab for the above mentioned special functions. Taking into account the main properties for Bessel and modified Bessel functions, we shortly present analytically solutions (where possible) in the form of series. Especially it is studied the behavior of these special functions using Matlab facilities: numerical solutions and plotting. Finally, it will be compared the behavior of the special functions and point out other directions for investigating properties of Bessel and spherical Bessel functions. The asymptotic forms of Bessel functions and modified Bessel functions allow determination of important properties of these functions. The modified Bessel functions tend to look more like decaying and growing exponentials.

Exact solutions of an exponential type position dependent mass problem

We look for the solution to the eigenvalue problem, in a recently introduced phase-space representation, for the quantum particle in a linear potential. We find that the solution is not unique and that some of these functions correspond to the eigenfunctions in coordinate space and that others correspond to the classical limit. \textcopyright{} 1996 The American Physical Society.

Elementary functions, Bessel functions, Legendre functions and many other special functions are included in the large family of mathematical functions known as generalized hypergeometric functions. A power series with coefficients that are rational functions of the index defines them. They are used in many disciplines, such as engineering, statistics and physics, because of their rich mathematical features and adaptability. The beauty and interdependence of mathematical ideas are demonstrated by the Generalized Hypergeometric Function. Researchers and practitioners from a wide range of disciplines find it to be an indispensable tool due to its unifying power, rich analytical features, and broad applications. Numerous unusual functions are included as particular examples of the generalized Hypergeometric function. Legendre polynomials, Bessel functions, the confluent Hypergeometric function, and numerous more noteworthy examples are also included. An order (q+1) linear homogeneous differential equation is satisfied by the generalized hypergeometric function. In many applications, but especially in mathematical physics, this differential equation is essential. It is possible to write the generalized hypergeometric function in terms of contour integrals, which offers different representations and makes it easier to evaluate some integrals. The generalized Hypergeometric function has a wealth of transformation formulas that allow one Hypergeometric function to be transformed into another with distinct parameters. These transformations are quite useful for examining relationships between various special functions and simplifying expressions.

We describe a new algorithm for computing special function solutions of the form y(x) = m(x)F(ξ(x)) of second order linear ordinary differential equations, where m(x) is an arbitrary Liouvillian function, ξ(x) is an arbitrary rational function, and F satisfies a given second order linear ordinary differential equation. Our algorithm, which is based on finding an appropriate point transformation between the equation defining F and the one to solve, is able to find all rational transformations for a large class of functions F, in particular (but not only) the 0F1 and 1F1 special functions of mathematical physics, such as Airy, Bessel, Kummer and Whittaker functions. It is also able to identify the values of the parameters entering those special functions, and can be generalized to equations of higher order.

The Wright function, which arises in the theory of the space-time fractional diffusion equation, is an interesting mathematical object which has diverse connections with other special and elementary functions. The Wright function provides a unified treatment of several classes of special functions, such as the Gaussian, Airy, Bessel, and Error functions, etc. The manuscript demonstrates an algorithm for symbolical representation in terms of finite sums of hypergeometric (HG) functions and polynomials. The HG functions are then represented by known elementary or other special functions, wherever possible. The algorithm is programmed in the open-source computer algebra system Maxima and can be used to for testing numerical algorithms for the evaluation of the Wright function.

We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions 0 F 1 , 1 F 1 , 2 F 1 , and 2 F 0 (or the Kummer U -function) are supported for unrestricted complex parameters and argument, and, by extension, we cover exponential and trigonometric integrals, error functions, Fresnel integrals, incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre functions, Jacobi polynomials, complete elliptic integrals, and other special functions. The output can be used directly for interval computations or to generate provably correct floating-point approximations in any format. Performance is competitive with earlier arbitrary-precision software and sometimes orders of magnitude faster. We also partially cover the generalized hypergeometric function p F q and computation of high-order parameter derivatives.

Uniform asymptotic expansions of integrals: a selection of problems

Eigenvalue problems arise in many areas of physics, from solving a classical electromagnetic problem to calculating the quantum bound states of the hydrogen atom. In textbooks, eigenvalue problems are defined for linear problems, particularly linear differential equations such as time-independent Schrödinger equations. Eigenfunctions of such problems exhibit several standard features independent of the form of the underlying equations. As discussed in Bender et al (2014 J. Phys. A: Math. Theor. 47 235204), separatrices of nonlinear differential equations share some of these features. In this sense, they can be considered eigenfunctions of nonlinear differential equations, and the quantized initial conditions that give rise to the separatrices can be interpreted as eigenvalues. We introduce a first-order nonlinear eigenvalue problem involving a general class of functions and obtain the large-eigenvalue limit by reducing it to a random walk problem on a half-line. The introduced general class of functions covers many special functions such as the Bessel and Airy functions, which are themselves solutions of second-order differential equations. For instance, in a special case involving the Bessel functions of the first kind, i.e. for y′(x) = J ν (xy), we show that the eigenvalues asymptotically grow as 241/42 n 1/4. We also introduce and discuss nonlinear eigenvalue problems involving the reciprocal gamma and the Riemann zeta functions, which are not solutions to simple differential equations. With the reciprocal gamma function, i.e. for y′(x) = 1/Γ(−xy), we show that the nth eigenvalue grows factorially fast as (1−2n)/Γ(r2n−1) , where r k is the kth root of the digamma function.

Uniform asymptotic methods for integrals