Expansion Of Analytic Functions Research Articles

Solutions of Boundary-Value Problems for the Helmholtz Equation in Simply Connected Domains of the Complex Plane

The bases in the spaces of functions analytic in simply connected domains are constructed with the help of conformal mappings of these domains onto a circle. The obtained basis functions are biorthogonal to the Faber polynomials. By using the expansions of analytic functions in series in systems of basis functions, we determine the solutions of boundary-value problems for the Helmholtz equation whose boundary values coincide with the boundary values of these functions.

On the expansions of analytic functions on convex locally closed sets in exponential series

Let Qbe a bounded, convex, locally closed subset of CN with nonempty interior. For N>1 sufficient conditions are obtained that an operator of the representation of analytic functions on Q by exponential series has a continuous linear right inverse. For N=1 the criterions for the existence of a continuous linear right inverse for the representation operator are proved.

Expansion of analytic functions in q-orthogonal polynomials

A classical result on the expansion of an analytic function in a series of Jacobi polynomials is extended to a class of q-orthogonal polynomials containing the fundamental Askey–Wilson polynomials and their special cases. The function to be expanded has to be analytic inside an ellipse in the complex plane with foci at ±1. Some examples of explicit expansions are discussed.

Unified Formulation of Nodal Expansion Method and Analytic Nodal Method Solutions to Two-Group Diffusion Equations

This paper demonstrates that the analytic nodal method (ANM) solution to two-group (2-G) diffusion equations can be formulated in the same way as the nodal expansion method (NEM) solution, and thereby, the two most popular transverse integrated nodal method formulations can be integrated into a unified nodal method (UNM) formulation. For this purpose, the analytic solution, i.e., the combined homogeneous and particular solution, of transverse-integrated one-dimensional, 2-G diffusion equations is represented by an expansion of analytic basis functions while the expansion coefficients are obtained in the same way as the NEM. The advantages of the UNM formulation are then discussed. It is a stable method in itself so that it does not require approximate schemes to avoid the instability at the near-critical nodes. Because it does not introduce any approximate scheme in conjunction with the stability questions at the near-critical nodes, it is more accurate than the conventional ANM formulation in the case where the latter needs to introduce approximations. It is readily incorporated into a number of existing NEM production codes. These advantages are demonstrated in terms of numerical solutions of Nuclear Energy Agency Committee on Reactor Physics pressurized water reactor benchmark problems.

Joseph L. Walsh: Selected papers, edited by Theodore J. Rivlin and Edward B. Saff. Pp. 682. £86. 2000. ISBN 0 387 98782 7 (Springer Verlag).

Joseph L. Walsh: Selected papers, edited by Theodore J. Rivlin and Edward B. Saff. Pp. 682. £86. 2000. ISBN 0 387 98782 7 (Springer Verlag).

Expansions in Products of Heine—Stieltjes Polynomials

It is shown that products of polynomials introduced by Heine and Stieltjes form orthogonal bases in suitable function spaces. A theorem on the expansion of analytic function in these bases is proved.

Expansions of analytic functions in spherical harmonics

It is shown that every analytic function f(x1,…,xk) defined in the intersection of the complex sphere in C k with an open bail in C k can be expanded in spherical harmonics. An application to the theory of Lamé functions is given.

On Series Expansions for Meromorphic Functions

Series expansions for meromorphic functions obtained in the author′s earlier paper (Complex Variables Theory Appl. 25 (1994), 159-171) are derived in a different way-namely, from Cauchy′s theorem on partial fraction expansions-in the present paper. In addition to that, a certain result of Cauchy′s theorem just mentioned is specified in this paper (which allowed us to somewhat reformulate the theorem). Also, expansions of analytic functions on disks containing poles of the functions are studied, and examples of such expansions are given.

Expansions of analytic functions in products of Bessel functions

Expansions of analytic functions in products of Bessel functions

Expansion of Analytic Functions in Series of Floquet Solutions of First Order Differential Systems

In this paper we study boundary eigenvalue problems for first order systems of ordinary differential equations of the form \[zy'\left( z \right) = \left( {\lambda A_1 \left( z \right) + A_0 \left( z \right)} \right)y\left( z \right),\,\,y\left( {ze^{2\pi i} } \right) = e^{2\pi iv} y\left( z \right)\] for z ϵ Slog, where S is a ring region around zero, Slog denotes the Riemann surface of the logarithm over S, the coefficient matrix functions A1(z) and A0(z) are holomorphic on S, and v is a complex number. The eigenfunctions of this eigenvalue problem are the Floquet solutions of the differential system with v as characteristic exponent. For an open subset S0 of S, the notion of A1-convexity of the pair (S0, S) is introduced. For A1-convex pairs (S0, S) it is shown that the expansion into eigenfunctions and associated functions of holomorphic functions on Slog, satisfying the monodromy condition y(ze2πi) = e2πivy(z), converges regularly on Slog0 and is unique. If S is a pointed neighbourhood of 0 and A1(z) is holomorphic in SU{0}, it is shown that there is a pointed neighbourhood S0 of 0 such that (S0, S) is A1-convex. It follows from the results of this paper that many expansions of analytic functions in terms of special functions can be considered as eigenfunction expansions of this kind.

The coefficients of the Laurent series expansions of real analytic functions

Letf be a real analytic function of a real variable such that 0 is an isolated (possibly essential) singularity off. In the existing literature the coefficients of the Laurent series expansion off around 0 are obtained by applying Cauchy's integral formula to the analytic continuation off on the complex plane. Here by means of a conformal mapping we derive a formula which determines the Laurent coefficients off solely in terms of the values off and the derivatives off at a real point of analyticity off. Using a more complicated mapping, we similarly determine the coefficients of the Laurent expansion off around 0 where now 0 is a singularity off which is not necessarily isolated.

Some elements of a theory of multidimensional complex variables: part II. expansions of analytic functions and application to fluid flows

The three-dimensional (3-D) analytic-function theory whose foundation was developed in Part I is further developed and applied to representing ideal flow solutions in fluid dynamics. A 3-D complex velocity and 3-D complex potential are defined in terms of analytic functions of a 3-d variable. Components of the complex potential relate to a 3-D flow velocity potential and 3-D stream functions. It was shown in Part I that the parts of every analytic function of one 3-D variable provide the components of a solenoidal and irrotational ( S and I) harmonic vector. In this paper, S and I harmonic vectors are shown also to be provided by the vector terms of a certain trigonometric-series expansion of every elementary 3-D analytic function. (“Elementary” here refers to an analytic function extended from the set of ordinary complex analytic functions.) The “primary part” (defined as the first real vector term in such an expansion) of every elementary analytic function of one 3-D variable produces an axisymmetric-flow solution. Specific simple examples are shown, including the functions whose respective primary parts give the velocity potential and Stokes stream function of elementary flows including a uniform stream, stagnation flow, source flow, flow due to a doublet (dipole) and flow over a sphere.

Expansion of analytic functions in series of classical orthogonal polynomials

Expansion of analytic functions in series of classical orthogonal polynomials

Expansion of analytic functions in series of rational functions

Expansion of analytic functions in series of rational functions

The expansion of analytic functions in Dirichlet series

The expansion of analytic functions in Dirichlet series

Expansion of Analytic Functions in Jacobi Series

A classical theorem on expansion of an analytic function in a series of Jacobi polynomials $P_n^{(\alpha ,\beta )} $ is extended so that $\alpha $ and $\beta $, instead of being assumed real, may have any complex values provided $\alpha + \beta \ne - 2, - 3, - 4, \cdots $. The function to be expanded may be analytic inside an arbitrary ellipse in the complex plane, since we use notation which does not fix the foci at $ -1 $ and 1. If the ellipse is a circle, Jacobi’s series reduces to Taylor’s series as a special case, not a limiting case. Cauchy’s inequality for the coefficients of Taylor’s series has an analogue for Jacobi's series. The coefficients are determined uniquely by the analytic function. Some special series are listed.

Expansion of Analytic Functions in Exponential Polynomials

Expansion of Analytic Functions in Exponential Polynomials

Expansion of analytic functions in exponential polynomials

The expansion of analytic functions in interpolation series by the method of kernel expansion has been treated in [3], [4] and the detailed application of the method to expansions in the classical polynomials has culminated in the well-known monograph of Boas and Buck [2]. In the present paper we use the kernel expansion technique to obtain a new expansion in which the terms are not polynomials in the usual sense of powers but instead are the exponential polynomials

Expansion of Analytic Functions in Infinite Series and Infinite Products with Application to Multiple Valued Functions

(1965). Expansion of Analytic Functions in Infinite Series and Infinite Products with Application to Multiple Valued Functions. The American Mathematical Monthly: Vol. 72, No. 8, pp. 861-864.

Expansion of Analytic Functions in Infinite Series and Infinite Products with Application to Multiple Valued Functions

Expansion of Analytic Functions in Infinite Series and Infinite Products with Application to Multiple Valued Functions