Special Functions Of Mathematical Physics Research Articles

A Method of Generating Integral Relations by the Simultaneous Separability of Generalized Schrödinger Equations

One of the most important methods in the theory of special functions of mathematical physics is that of generating integral relations for these functions by the simultaneous separability of the 3-dimensional wave equation in different orthogonal coordinate systems. In the present paper it will be shown that a consequent application of this principle of simultaneous separability to more general partial differential equations and higher dimensions yields various types of integral relations for the solutions of a wide class of ordinary differential equations which especially contains all second-order equations of Fuchsian type.

An addition formula for Green's functions

Abstract : A formula yielding an expression for the Green's function of a linear separable elliptic partial differential equation in terms of the Green's function for two simpler equations, one of which is elliptic and the other hyperbolic, is derived utilizing separation of variables, transform techniques for solving ordinary differential equations, Parseval's identity from Fourier transform theory, and general properties of Green's and Riemann functions. The formula is applied successfully to specific problems, some of which are believed to be previously unsolved, and Green's functions are represented using the special functions of mathematical physics. A more detailed overview of the thesis may be found in Appendix A, which is the recommended starting point for reading this thesis. (Author)

Monotonic Properties of Analytic Functions

A theorem is proved which enables one to obtain monotonic properties of the real part, imaginary part, modules and phase of an arbitrary analytic function in the complex plane. The monotonic properties are established from the behavior of the analytic function and its derivative on the boundary of the domain in which the monotonic properties are required. As an applicationsome monotonic results are derived for the Bessel functions $J_\nu (z)$ and $H_\nu ^{(1)} (z)$, where z is complex and $\nu$ is real and positive. The theorem and a corollary can be used to obtain monotonic properties of many other special functions of mathematical physics.

Table errata: Formulas and theorems for the special functions of mathematical physics (Springer, New York, 1966) by W. Magnus, F. Oberhettinger and R. P. Soni

Table errata: Formulas and theorems for the special functions of mathematical physics (Springer, New York, 1966) by W. Magnus, F. Oberhettinger and R. P. Soni

Table erratum: Formulas and theorems for the special functions of mathematical physics (third edition of English translation, Springer, New York, 1966) by W. Magnus, F. Oberhettinger and R. P. Soni

Table erratum: Formulas and theorems for the special functions of mathematical physics (third edition of English translation, Springer, New York, 1966) by W. Magnus, F. Oberhettinger and R. P. Soni

A Further Extension of the Leibniz Rule to Fractional Derivatives and Its Relation to Parseval’s Formula

The familiar Leibniz rule for the Nth derivative of the product of two functions is $D^N uv = \sum {\left( {\begin{array}{*{20}c} N n \end{array} } \right)D^{N - n} uD^n v}$. A generalization of this formula for fractional derivatives is given as $D^\alpha uv = \sum a {\left( {\begin{array}{*{20}c} \alpha {an + \gamma } \end{array} } \right)D^{\alpha - an - \gamma } uD^{an + \gamma } v} $, where $\alpha $ need not be a natural number and $0 < a \leqq 1$. (The special case, $a = 1$, appeared previously.) Further generalizations of the Leibniz rule are also given and are derived from a generalization of Taylor’s series given previously by the author. It is shown that these new series are generalizations of Parseval's formula from the study of Fourier series. Finally, new series expansions relating the special functions of mathematical physics are derived as special cases of the generalizations of the Leibniz rule. These series include a generalized Dougall's formula, several series of the Cardin...

On Lie Algebras of Difference Operators and the Special Functions of Mathematical Physics

A method is given whereby special functions of hypergeometric type and their generalizations can be derived in a systematic fashion from Lie algebras of difference and differential operators in one complex variable. The method automatically yields contour integral representations, generatingfunctions, recurrence formulas, integral identities, and infinite sum identities for each function treated. This approach unites Vilenkin’s integral transform method for special functions and the older factorization method into a single flexible tool.

Table errata: Formulas and theorems for the special functions of mathematical physics (third edition of English translation, Springer, New York, 1966) by W. Magnus, F. Oberhettinger and R. P. Soni

Table errata: Formulas and theorems for the special functions of mathematical physics (third edition of English translation, Springer, New York, 1966) by W. Magnus, F. Oberhettinger and R. P. Soni

Basic Equations and Special Functions of Mathematical Physics

<i>Basic Equations and Special Functions of Mathematical Physics</i>

Table errata: Formulas and theorems for the special functions of mathematical physics (third edition of the English translation, Springer, New York, 1966) by W. Magnus, F. Oberhettinger and R. P. Soni

Table errata: Formulas and theorems for the special functions of mathematical physics (third edition of the English translation, Springer, New York, 1966) by W. Magnus, F. Oberhettinger and R. P. Soni

Table errata: Formulas and theorems for the special functions of mathematical physics

Table errata: Formulas and theorems for the special functions of mathematical physics

Basic equations and special functions of mathematical physics, by V. Ya. Arsenin (Translated by S. Chomet). Iliffe Books Ltd., London, 1968. 361 pages.

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Table errata: Formulas and theorems for the special functions of mathematical physics (third edition of the English translation, Springer, New York, 1966) by W. Magnus, F. Oberhettinger and R. P. Soni

Table errata: Formulas and theorems for the special functions of mathematical physics (third edition of the English translation, Springer, New York, 1966) by W. Magnus, F. Oberhettinger and R. P. Soni

Special Functions and the Complex Euclidean Group in 3-Space. I

It is shown that the general addition theorems of Gegenbauer, relating Bessel functions and Gegenbauer polynomials, are special cases of identities for special functions obtained from a study of certain local irreducible representations of the complex Euclidean group in 3-space. Among the physically interesting results generalized by this analysis are the expansion for a plane wave as a sum of spherical waves and the addition theorem for spherical waves. This paper is one of a series attempting to derive the special functions of mathematical physics and their basic properties from the representation theory of Lie symmetry groups.

Formulas and Theorems for the Special Functions of Mathematical Physics

Formulas and Theorems for the Special Functions of Mathematical Physics

Formulas and Theorems for the Special Functions of Mathematical Physics

<i>Formulas and Theorems for the Special Functions of Mathematical Physics</i>

Formulas and Theorems for the Special Functions of Mathematical Physics

First Page

Special Functions of Mathematical Physics from the Viewpoint of Lie Algebra

Families of special functions, known from mathematical physics, are defined here by their recursion relations. The operators which raise and lower indices in these functions are considered as generators of a Lie algebra. The general element of the corresponding Lie group thus operates on the function in two ways: on the one hand it shifts the argument of the function; on the other hand it produces an infinite sum of functions (at the unchanged argument) with shifted indices. Equating the two results of the operation gives us ``addition theorems,'' hitherto derived by analytical methods. The present paper restricts itself to the study of 2- and 3-parameter Lie groups.

On Lie algebras and some special functions of mathematical physics

On Lie algebras and some special functions of mathematical physics

Special Functions of Mathematical Physics.

Special Functions of Mathematical Physics.