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Sunday, May 10, 2020 | History

2 edition of Expansion problems in connection with the hypergeometric differential equation found in the catalog.

Expansion problems in connection with the hypergeometric differential equation

Bernhard Paul Reinsch

# Expansion problems in connection with the hypergeometric differential equation

## by Bernhard Paul Reinsch

Published in [Baltimore, Md .
Written in English

Subjects:
• Hypergeometric functions.

• Edition Notes

Classifications The Physical Object Statement by Bernhard Paul Reinsch ... LC Classifications QA351 .R4 1924 Pagination 1 p. l., p. 45-70, 1 l. Number of Pages 70 Open Library OL6675967M LC Control Number 25006973 OCLC/WorldCa 8868686

A supplement for elementary and intermediate courses in differential equations, this text features more than problems and answers. Suitable for undergraduate students of mathematics, engineering, and physics, this volume also represents a helpful tool for professionals wishing /5(2). In mathematics, the Gaussian or ordinary hypergeometric function ${}_2F_1(a,b;c;z)$ is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE).

12 HYPERGEOMETRIC DISTRIBUTION Examples: 1. Five cards are chosen from a well shuﬄed deck. X = the number of diamonds selected. 2. An audio ampliﬁer contains six transistors. It has been ascertained that three of the transistors are faulty but it is not known which three. Amy removes three tran-sistors at random, and inspects Size: 66KB. Now we consider second order linear di erential equations of the form y00+ p(z)y0+ q(z)y= 0 or d2y dz2 + p(z) dy dz + q(z)y= 0: (1) Then we have the following de nition: De nition 2. A point z 0 2C is called a regular point of the di erential equation (1) if both p(z) and q(z) are analytic in z 0. Otherwise z 0 is called a singular point of (1).File Size: KB.

An Expansion involving Confluent Hypergeometric Functions. by Ragab, the Airy equation, the generalized Airy equation, and the confluent hypergeometric equation. Seller Inventory # mon are now used in many engineering and physical problems. This book is intended to further this development. The important practical. Hyperbolic Schwarz map of the confluent hypergeometric differential equation SAJI, Kentaro, SASAKI, Takeshi, and YOSHIDA, Masaaki, Journal of the Mathematical Society of Japan, ; Monodromy groups of hypergeometric functions satisfying algebraic equations Kato, Mitsuo and Noumi, Masatoshi, Tohoku Mathematical Journal, ; A System of Linear Differential Equations for the Distribution of.

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### Expansion problems in connection with the hypergeometric differential equation by Bernhard Paul Reinsch Download PDF EPUB FB2

In the following we solve the second-order differential equation called the hypergeometric differential equation using Frobenius method, named after Ferdinand Georg is a method that uses the series solution for a differential equation, where we assume the solution takes the form of a series.

This is usually the method we use for complicated ordinary differential equations. In mathematics, the Gaussian or ordinary hypergeometric function 2 F 1 (a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting 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. SOLUTION OF DIFFERENTIAL EQUATIONS OF HYPERGEOMETRIC TYPE J. PALMER Abstract. We present a method for solving the classical linear ordinary dif-ferential equations of hypergeometric type , including Bessel’s equation, Le-gendre’s equation, and others with polynomial coeﬃcients of File Size: KB.

For instance, Legendre's differential equation can be shown to be a special case of the hypergeometric differential equation. Hence, by solving the hypergeometric differential equation, one may directly compare its solutions to get the solutions of Legendre's differential equation, after.

The expansion of the Riemann function, which is defined by Eq. (), with respect to the Bessel functions when the toroidal beam has been cut from the flow (), can be obtained by processing the hypergeometric equation () using the same algorithm (Syrovoy, ) as for Eq. erential equation by means of a suitable change of variables.

e solutions of hypergeometric di erential equation include many of the most interesting special functions of mathematical physics. Solutions to the hypergeometric dif-ferential equation are built out of the hypergeometric series.

e solution of Euler s hypergeometric di erential equation. The solutions of hypergeometric differential equation include many of the most interesting special functions of mathematical physics. Solutions to the hypergeometric differential equation are built out of the hypergeometric series.

Definition 1. The Pochhammer -symbol is defined as Cited by: 8. Hypergeometric Functions Reading Problems Introduction The hypergeometric function F(a, b; c; x) is deﬁned as F(a; b; c; x)= 2F 1(a, b; c; x)=F(b, a; c; x) =1+ ab c x + a(a +1)b(b +1) c(c +1) x2 2.

+ = ∞ n=0 (a)n(b) n (c)n xn n. |x| File Size: 90KB. hypergeometric solutions of di erential equations. A linear homogeneous di erential equation with rational function coe cients corresponds to a di erential operator L2 C(x)[@] where @ = d dx. For example, if L= a [email protected] + a [email protected]+ a 0 is a di erential operator with a 2;a 1;a 0 2C(x), then the corresponding di erential equation L(y) = 0 is a 2y00+a.

Chapter 1 Ordinary linear diﬀerential equations Diﬀerential equations and systems of equa-tions A diﬀerential ﬁeld K is a ﬁeld equipped with a derivation, that is, a map ∂: K → K which has the following properties, For all a,b ∈ K we have ∂(a+b) = ∂a+∂b.

For all a,b ∈ K we have ∂(ab) = a∂b+b∂a. The subset C:= {a ∈ K|∂a = 0} is a subﬁeld of K and is File Size: KB. General Solution: If, and are all non-integers, the general solution for the hypergeometric differential equation is: which is valid for.

Gamma Function: A hypergeometric function can be expressed in terms of gamma functions. Solutions of -Hypergeometric Differential Equations Article (PDF Available) in Journal of Applied Mathematics April with 3, Reads How we measure 'reads'.

us all m-hypergeometric right factors of operator L from (3), and hence all primitive m-hypergeometric so-lutions of recurrence Ly = O. 2 Basic Algorithm In this section we apply algorithm HYPER — the al-gorithm of the previous section with m = 1 — to find hypergeometric solutions of.

hypergeometric functions of Gauss, Horn, Appell, and Lauricella. We will emphasize the alge-braic methods of Saito, Sturmfels, and Takayama to construct hypergeometric series and the connection with deformation techniques in commutative algebra. We end with a brief discussion of the classiﬁcation problem for rational hypergeometric functions.

$\begingroup$ I should point out that the connection between hypergeometric functions and algebraic equations is more than a pure coincidence of series inversion. Felix Klein's 'Lectures on the Icosahedron' offers a very nice derivation of the roots from a geometric viewpoint, using the symmetry group of the icosahedron and a Galois resolvent.

into the form of a Kummer's or confluent hypergeometric differential equation: $$y w''(y) + [f - y] w'(y) + g w(y) = 0$$ I know it may have something to do with merging two of the singularities of the original equation, and maybe doing something with y, making it $\frac{y}{b}$ and taking b to infinity, but I don't know and can't find the.

The book is well written and easy to read In the first chapter, the author shows how to solve linear second order differential equation by power series and obtain directly the power serie expansion of the solution. Second Chapter is devoted to the differential equation defining hypergeometric function and its solution in power by: A similar generalized Coulomb problem for a class of general Natanzon confluent potentials is exactly solved in  by reducing the corresponding system to confluent hypergeometric differential recently, in , the authors succeeded to solve the eigenvalue wave equation for an electron in the field of a molecule with an electric dipole moment by expanding the solutions of a.

Sturm Liouville differential equation and hypergeometric functions I'm trying to understand how to solve this differential equation: $[z^2(1-z)\dfrac{d^2}{dz} - z^2 \dfrac{d}{dz} - \lambda] f(z) = 0$ I know the solution is related to the hypergeometric function.

The Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems, is an exceptional and complete reference for scientists and engineers as it contains over 7, ordinary. In the present paper, starting from the second-order difference hypergeometric equation on the non-uniform lattice x(s) satisfied by the set of discrete hypergeometric orthogonal q-polynomials {p n}, we find analytical expressions of the expansion coefficients of any q-polynomial r m (x(s)) on x(s) and of the product r m (x(s))q j (x(s)) in series of the set {p n}.Cited by: The confluent hypergeometric function is useful in many problems in theoretical physics, in particular as the solution of the differential equation for the velocity distribution function of electrons in a high frequency gas discharge.

This report presents some of the properties of this function together with six-figure tables and charts for the.Abstract. In a previous paper  one of us developed an expansion for the con-fluent hypergeometric function in series of Bessel functions. A different expansion of the same kind given by Buchholz  was also studied.

Since publication of , it was found that Rice  has also developed an expansion of .