On some solutions to generalized spheroidal wave equations and applications

Previous article Next article A Series of Hypergeometric FunctionsH. M. SrivastavaH. M. Srivastavahttps://doi.org/10.1137/1017075PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout"A Series of Hypergeometric Functions." SIAM Review, 17(4), pp. 685–686 Previous article Next article FiguresRelatedReferencesCited byDetails Volume 17, Issue 4| 1975SIAM Review History Published online:17 February 2012 InformationCopyright © 1975 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1017075Article page range:pp. 685-686ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

Consider the differential equation d2y/dz2 + k2Qy = 0, where Q = Q0 + Q1/k + Q2/k2 + . . .,

The Gauss hypergeometric function ${}_2F_1(a,b,c;z)$ can be computed by using the power series in powers of $z, z/(z-1), 1-z, 1/z, 1/(1-z),(z-1)/z$. With these expansions ${}_2F_1(a,b,c;z)$ is not completely computable for all complex values of $z$. As pointed out in Gil, {\it et al.} [2007, \S2.3], the points $z=e^{\pm i\pi/3}$ are always excluded from the domains of convergence of these expansions. B\"uhring [1987] has given a power series expansion that allows computation at and near these points. But, when $b-a$ is an integer, the coefficients of that expansion become indeterminate and its computation requires a nontrivial limiting process. Moreover, the convergence becomes slower and slower in that case. In this paper we obtain new expansions of the Gauss hypergeometric function in terms of rational functions of $z$ for which the points $z=e^{\pm i\pi/3}$ are well inside their domains of convergence . In addition, these expansion are well defined when $b-a$ is an integer and no limits are needed in that case. Numerical computations show that these expansions converge faster than B\"uhring's expansion for $z$ in the neighborhood of the points $e^{\pm i\pi/3}$, especially when $b-a$ is close to an integer number.

New solutions for the elliptic Darboux equation are obtained as particular cases of solutions constructed for Heun's general equation. We consider two groups of power series expansions and two new groups of expansions in series of Gauss hypergeometric functions. The convergence of one group in power series is determined by means of ratio tests for infinite series, while the other groups are designed to solve problems which admit finite‐series solutions. Actually, we envisage periodic quasi‐exactly solvable potentials for which the stationary one‐dimensional Schrödinger equation is reduced to the Darboux equation. In general, finite‐ and infinite‐series solutions are obtained from power series expansions for Heun's equation. However, we show that the Schrödinger equation admits additional finite‐series expansions in terms of hypergeometric functions for a family of associated Lamé potentials used in band theory of solids.For each finite‐series solution, we find as well four infinite‐series expansions which are bounded and convergent for all values of the independent variable.

Expansions in series of functions are one of the most important tools of the applied mathematician, particularly expansions in series of the classical orthogonal polynomials, e.g. Laguerre, Jacobi and Hermite polynomials. In applied problems, the uniqueness of the particular expansion is usually intrinsic to the analysis, and often implicitly assumed. Indeed, in those cases where the functions in the series are orthogonal, uniqueness can often be proved by an argument that runs as follows. Let {φn(x)} (n = 0, 1, 2, …) be a sequence of functions orthogonal with respect to the weight function ρ(x) over the interval [0, 1], and suppose thatthe series being boundedly convergent for 0 ≤ x ≤ 1.

In the past, high order series expansion techniques have been used to study the nonlinear equations that govern the form of periodic Stokes waves moving steadily on the surface of an inviscid fluid. In the present study, two such series solutions are recomputed using exact arithmetic, eliminating any loss of accuracy due to accumulation of round-off error, allowing a much greater number of terms to be found with confidence. It is shown that a higher order behavior of the series generated by the solution casts doubt over arguments that rely on estimating the series’ radius of convergence. Further, the exact nature of the series is used to shed light on the unusual nature of convergence of higher order Padé approximants near the highest wave. Finally, it is concluded that, provided exact values are used in the series, these Padé approximants prove very effective in successfully predicting three turning points in both the dispersion relation and the total energy.

Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl–Teller–Ginocchio potential wave functions

An analytical method to solve a general class of nonlinear reactive transport models

In this paper, a method is proposed for improving the accuracy of the stability criterion by enlarging the estimation of the stability region determined by means of Lyapunov's direct method using series expansion, and by putting it closer to the actual stability region. The Lyapunov function obtained by energy integration is used as the first estimation. The method is applied to a single-machine infinite bus power system, and the stability domain in the δ-ω plane is shown together with the actual stability domain obtained by numerical integration. It is shown that the application of the proposed method results in a considerable improvement in the stability boundary estimation over that given by the original Lyapunov function. Zubov's method, which obtains an accurate stability boundary using the series expansion, is related to the method proposed in this paper. The last section is, therefore, devoted to comparing the two methods and examining the possibility of their application to multi-machine power systems.

Non-global logarithms (NGLs) are the leading manifestation of correlations between distinct phase space regions in QCD and gauge theories and have proven a challenge to understand using traditional resummation techniques. Recently, the dressed gluon expansion was introduced that enables an expansion of the NGL series in terms of a "dressed gluon" building block, defined by an all-orders factorization theorem. Here, we clarify the nature of the dressed gluon expansion, and prove that it has an infinite radius of convergence as a solution to the leading logarithmic and large-$N_c$ master equation for NGLs, the Banfi-Marchesini-Smye (BMS) equation. The dressed gluon expansion therefore provides an expansion of the NGL series that can be truncated at any order, with reliable uncertainty estimates. In contrast, manifest in the results of the fixed-order expansion of the BMS equation up to 12-loops is a breakdown of convergence at a finite value of $\alpha_s$log. We explain this finite radius of convergence using the dressed gluon expansion, showing how the dynamics of the buffer region, a region of phase space near the boundary of the jet that was identified in early studies of NGLs, leads to large contributions to the fixed order expansion. We also use the dressed gluon expansion to discuss the convergence of the next-to-leading NGL series, and the role of collinear logarithms that appear at this order. Finally, we show how an understanding of the analytic behavior obtained from the dressed gluon expansion allows us to improve the fixed order NGL series using conformal transformations to extend the domain of analyticity. This allows us to calculate the NGL distribution for all values of $\alpha_s$log from the coefficients of the fixed order expansion.

We present the Olsson.wl Mathematica package which aims to find transformations for some classes of multivariable hypergeometric functions. It is based on a well-known method developed by P. O. M. Olsson in J. Math. Phys. 5, 420 (1964) to derive the analytic continuations of the Appell F_1 double hypergeometric series using the linear transformations of the Gauss _2F_1 hypergeometric function. We provide a brief description of the method of Olsson and demonstrate the use of the commands of the Olsson.wl package using some examples. In particular, we reproduce various results of the literature on multivariable hypergeometric functions and show practical applications of this package in the derivation of novel formulas. In the context of high energy physics, we also demonstrate how it can be used to disentangle some known results about the analytic continuation of some series representations of the one-loop pentagon in multi-Regge kinematics and D=6-2ϵ. We also present companion package, called ROC2.wl, which is dedicated to the derivation of the regions of convergence of double hypergeometric series. This package can be used independently of Olsson.wl.

provided that d(X) #0. Heref and g are members of the function class C[[a, b]. The function d(X) is an entire analytic function of the complex parameter X and D(s, t; X), which is a continuous function of (s, t), is also an entire analytic function of X. The display of analyticity in the solution of the Fredholm equation of the second kind is an early signal of the important role which analyticity was destined to play in spectral theory. Anticipations of spectral theory itself can be traced at least as far back as 1836, which was the date of publication of the work of Sturm [38], [39] and Liouville [28], [29] on second-order ordinary differential equations and the associated eigenvalue and expansion theorems. In 1904 Hilbert [20] showed how to transform a differential boundary value problem into a problem of solving an integral equation. Moreover, the Green's function of the differential problem turned out, under certain conditions, to be a kernel of Hilbert-Schmidt type, from which followed a great many things of interest. The utility of analytic functions and the calculus of residues in connection with expansions in series of special functions has been known since the work of Cauchy. In 1894 Poincare [34] dealt with the problem of the vibrating membrane and related the eigenvalues and eigenfunction expansions to the meromorphic character of the Green's function's dependence on the parameter. In 1908 Birkhoff made explicit the use of the meromorphic behavior of the Green's

For Lauricella's hypergeometric function F(n)D of n variables, we prove two formulas exhibiting its behaviour near the boundaries of the n-dimensional region of convergence of the multiple series defining it. Each of these results can be applied to deduce the corresponding properties of several simpler hypergeometric functions of one, two, and more variables.

In this paper, we present novel advancements in the theory of bicomplex hypergeometric functions and their applications. We extend the hypergeometric function to bicomplex parameters, analyse its convergence region, and define its integral and derivative representations. Furthermore, we delve into the k-Riemann–Liouville fractional integral and derivative within a bicomplex operator, proving several significant theorems. The developed bicomplex hypergeometric functions and bicomplex fractional operators are demonstrated to have practical applications in various fields. This paper also highlights the essential concepts and properties of bicomplex numbers, special functions, and fractional calculus. Our results enhance the overall comprehension and possible applications of bicomplex numbers in mathematical analysis and applied sciences, providing a solid foundation for future research in this field.

We consider a class of assets whose risk-neutral pricing dynamics are described by an exponential Lévy-type process subject to default. The class of processes we consider features locally dependent drift, diffusion, and default intensity as well as a locally dependent Lévy measure. Using techniques from regular perturbation theory and Fourier analysis, we derive a series expansion for the price of a European-style option. We also provide precise conditions under which this series expansion converges to the exact price. Additionally, for a certain subclass of assets in our modeling framework, we derive an expansion for the implied volatility induced by our option pricing formula. The implied volatility expansion is exact within its radius of convergence. As an example of our framework, we propose a class of CEV-like Lévy-type models. Within this class, approximate option prices can be computed by a single Fourier integral and approximate implied volatilities are explicit (i.e., no integration is required). Furthermore, the class of CEV-like Lévy-type models is shown to provide a tight fit to the implied volatility surface of S&P500 index options.