Study of the Hypergeometric Equation via Data Driven Koopman-EDMD Theory

Introduction. Birkhofft has developed an expansion theory associated with linear differential equations on an interval of the real axis containing no singular points of the equation. The purpose of this paper is to develop a similar theory for the domain of complex variables in a region containing a regular singular point of the equation. To take the place of the two-point boundary conditions of the real variable case, we have corresponding relations between the values of the solution and its derivatives at a given point, and the values obtained by analytic continuation about the regular singular point back to the given point. The boundary conditions, n in number, are assumed to be linear, homogeneous and linearly independent, the order of the equation being n. An expansion theory associated with the linear equation of the second order with polynomial coefficients has been developed by 0. Volk.$ The method used in this paper is different from that employed by Volk and the coefficients are not assumed to be polynomials. The results of this paper and those of Volk overlap each other for the second-order equation, but neither includes the whole of the other. The NeumannGegenbauer expansions in Bessel's functions are obtained as special cases. The Legendre and hypergeometric differential equations give rise to expansion theories if infinity is considered as the regular singular point and the parameter is suitably chosen as a function of the arbitrary constants of the respective equations. The expansion theory for the hypergeometric equation as developed by Reinsch? is not included as a special case, since the parameter does not enter in the same way. In another paper it is hoped to extend the present theory so as to include the latter as a special case of a more general theory. The Laurent expansion is included as a special case arising from differential equations of every order. A solution of a differential equation with a regular singular point is, in

The Gaussian or ordinary hypergeometric function is a special function represented by the hypergeometric series that includes many other special functions as speci fic or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation. This paper claims that the natural logarithm can be represented by the Gaussian hypergeometric function.

Exact solutions of the Klein–Gordon–Fock (KGF) general relativistic equation that describe the dynamics of a massive, electrically charged scalar particle in the curved spacetime geometry of an electrically charged, rotating Kerr–Newman–(anti) de Sitter black hole are investigated. In the general case of a rotating, charged, cosmological black hole the solution of the KGF equation with the method of separation of variables results in Fuchsian differential equations for the radial and angular parts which for most of the parameter space contain more than three finite singularities and thereby generalise the Heun differential equations. For particular values of the physical parameters (i.e. mass of the scalar particle) these Fuchsian equations reduce to the case of the Heun equation and the closed form analytic solutions we derive are expressed in terms of Heun functions. For other values of the parameters some of the extra singular points are false singular points. We derive the conditions on the coefficients of the generalised Fuchsian equation such that a singular point is a false point. In such a case the exact solution of the Fuchsian equation can in principle be simplified and expressed in terms of Heun functions. This is the generalisation of the case of a Heun equation with a false singular point in which the exact solution of Heun’s differential equation is expressed in terms of Gauß hypergeometric function. We also derive the exact solutions of the radial and angular equations for a charged massive scalar particle in the Kerr–Newman spacetime. The analytic solutions are expressed in terms of confluent Heun functions. Moreover, we derived the constraints on the parameters of the theory such that the solution simplifies and expressed in terms of confluent Kummer hypergeometric functions. We also investigate the radial solutions in the KN case in the regions near the event horizon and far from the black hole. Finally, we construct several expansions of the solutions of the Heun equation in terms of generalised hypergeometric functions of Lauricella–Appell.

The solutions to the hypergeometric differential equation ― two regular singular points in the finite complex plane (at z = 0 and z = 1) and a regular singular point at infinity ― were analyzed in detail in Chapter 5. In this chapter, the solutions of the differential equation with four regular singular points are investigated. We restrict ourselves to the situation where there are three regular singular points in the finite complex plane, and a regular singular point at infinity.

On global solutions and monodromy groups of differential equations having at least two singularities on the Riemann sphere

One of the oldest, and still one of the most interesting, applications of group theory arises in the study of the transformations of an ordinary differential equation. If we know that a given differential equation admits a group of transformations, then we know that the solution set must admit that same group of transformations, and we can deduce properties of all the solutions from the properties of any one of them. A case in point is offered by the celebrated hypergeometric equation (See Eq. (1) below), whose solutions include many of the most interesting special functions of mathematical physics. In his book [3], Einar Hille notes that this equation has a venerable history associated with such names as Gauss, Euler, Riemann, and Kummer. The hypergeometric equation is in fact a prototype: every ordinary differential equation of second order with at most three regular singular points can be brought to the hypergeometric equation by means of suitably chosen changes of variable [S]. In 1836 Kummer published a set of 6 distinct solutions of the hypergeometric equation. These include the hypergeometric function of Gauss, and all of them could be expressed in terms of Gauss's function (See Table 1 below). A useful summary of their basic properties is found in [1, p. 105ff.]. A glance at the list of these Kummer solutions reveals a rather complicated set of relationships which pleads for some simple explanation. We show here that the Kummer solutions are related by a finite group of transformations which serve to explain their relationships and to exemplify the use of transformation groups in the study of differential equations.

Chapter 6 - Power Series Solutions of Ordinary Differential Equations

We use the method of regularization to construct two kinds of regularized asymptotic expansions (in a complex parameter) for a fundamental system of solutions of the Bessel equation. Expansions of the first kind are defined in the closed complex plane of the independent variable except for singular points of the spectral functions of the initial operator. We determine the domains of uniform and non-uniform convergence of the series involved. We study the resulting formulae on the positive real axis and prove that they yield Debye's familiar asymptotic expansions for Bessel functions on the interval (0,1), which lies in the domain of non-uniform convergence. The second kind of regularized uniform asymptotic expansions is constructed near a regular singular point in another domain of values of the parameter in the equations. Using these results, we get uniform asymptotic expansions of solutions of a boundary-value problem for the non-homogenous and homogeneous Bessel equations.

Chapter 4 - The hypergeometric function

For 1D acoustic wave propagation in the frequency domain, the complex nonlinear Riccati differential equation describes the dependence of the plane-wave reflection response in depth, defined as the upgoing part of the wavefield divided by the downgoing part of the wavefield at that depth. In the case that the model is a velocity gradient interface expressed in terms of a smooth Heaviside function defined by the Fermi-Dirac function, the Riccati equation is shown to have an analytical solution for the reflection response. The solution accounts for all wave phenomena in the gradient zone. The solution of the Riccati equation for the reflection response is obtained by introducing the Riccati equation for the reciprocal of the reflection response which can be related to a second-order self-adjoint linear differential equation. The two latter equations obey the same radiation condition. Furthermore, a connection between the second-order linear equation and the Heun equation is obtained, both being characterized by having four regular singular points. By a change of variables, the Heun equation can be transformed into Gauss’ hypergeometric equation, whereby the solution is expressed as the sum of two linearly independent functions containing hypergeometric functions. The advantage of applying the Heun-to-hypergeometric reduction to the nonlinear Riccati equation is that solutions to the hypergeometric equation are well known. The two functions allow us to analytically determine the Riccati solution because the ratio of the linearly independent functions at infinity is constant. In contrast to the classic frequency-independent reflection coefficient from two layers in welded contact, the reflection coefficient at the surface associated with the gradient interface depends on frequency. In the limit of zero frequency, the reflection coefficient approaches the classic one. We have implemented the iterative solution of the integral form of the Riccati equation and found that the solution builds up nonlinearly to the correct solution. The model is verified by a numerical example valid for 1D wave propagation.

1 - Introduction and Preliminaries

We present a multidomain spectral approach for Fuchsian ordinary differential equations in the particular case of the hypergeometric equation. Our hybrid approach uses Frobenius' method and Moebius transformations in the vicinity of each of the singular points of the hypergeometric equation, which leads to a natural decomposition of the real axis into domains. In each domain, solutions to the hypergeometric equation are constructed via the well-conditioned ultraspherical spectral method. The solutions are matched at the domain boundaries to lead to a solution which is analytic on the whole compactified real line $\mathbb{R}\cup {\infty}$, except for the singular points and cuts of the Riemann surface on which the solution is defined. The solution is further extended to the whole Riemann sphere by using the same approach for ellipses enclosing the singularities. The hypergeometric equation is solved on the ellipses with the boundary data from the real axis. This solution is continued as a harmonic function to the interior of the disk by solving the Laplace equation in polar coordinates with an optimal complexity Fourier--ultraspherical spectral method.

We study the integrability properties of nonlinearly perturbed Euler equations (linear ordinary differential equations with one regular singular point in the complex plane plus a nonlinear perturbation) near the singular point. We allow for first integrals with essential singularities and give sufficient conditions for the nonintegrability of the equations in the complex domain. We extend normal form theorems for singular equations and argue that equivalence to normal forms captures the spirit of the poly-Painlevé test and is a powerful tool for a rigorous approach to nonintegrability.

Starting from a second-order Fuchsian differential equation having five regular singular points, an equation obeyed by a function proportional to the first derivative of the solution of the Heun equation, we construct several expansions of the solutions of the general Heun equation in terms of Appell generalized hypergeometric functions of two variables of the first kind. Several cases when the expansions reduce to those written in terms of simpler mathematical functions such as the incomplete Beta function or the Gauss hypergeometric function are identified. The conditions for deriving finite-sum solutions via termination of the series are discussed. In general, the coefficients of the expansions obey four-term recurrence relations; however, there exist certain choices of parameters for which the recurrence relations involve only two terms, though not necessarily successive. For such cases, the coefficients of the expansions are explicitly calculated and the general solution of the Heun equation is constructed in terms of the Gauss hypergeometric functions.

We study the polariton problem for smooth boundaries, i.e. whether or not there exist some localized solutions of Maxwell’s equations for different types of smooth spatial variation of complex dielectric- and/or magnetic-permittivity tensors. For a particular dielectric permittivity profile varying according to the hyperbolic tangent law, the singular term is mathematically strongly taken into account in Maxwell’s equations, not in the boundary conditions. The problem is then reduced to the canonical form of Heun’s equation possessing four regular singular points. The solution to Heun’s equation as a power series is constructed, and an approximate solution involving a combination of two incomplete betafunctions is derived. Further, the exact eigenvalue solution to the polariton problem as a series in terms of incomplete beta-functions or, equivalently, Gauss hypergeometric functions is constructed. It is shown that the dispersion relation for the polariton wavenumber does not depend on the interface transition layer width, i.e. it is always exactly the same as the one derived in the limit of abrupt interface. We conjecture that the polariton wavenumber eigenvalue depends on either zeros of the dielectric permittivity variance profile or the poles of the logarithmic derivative of the latter.