Morera’s Theorem. Sequences and Series of Functions. Uniform Convergence Inside a Domain. Power Series. Abel’s Theorem. Disk of Convergence. Radius of Convergence

2017 ◽  
pp. 87-95
Author(s):  
Alexander Isaev
Keyword(s):  
Power Series ◽  
Uniform Convergence ◽  
Radius Of Convergence ◽  
Convergence Radius ◽  
Series Of Functions

scholarly journals On Uniform Convergence of Sequences and Series of Fuzzy-Valued Functions

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Uğur Kadak ◽  
Hakan Efe
Keyword(s):  
Power Series ◽  
Uniform Convergence ◽  
Membership Function ◽  
Level Sets ◽  
Radius Of Convergence ◽  
Extension Principle ◽  
Membership Functions ◽  
Function Series ◽  
Convergence Of Sequences ◽  
The Relationship

The class of membership functions is restricted to trapezoidal one, as it is general enough and widely used. In the present paper since the utilization of Zadeh’s extension principle is quite difficult in practice, we prefer the idea of level sets in order to construct for a fuzzy-valued function via related trapezoidal membership function. We derive uniform convergence of fuzzy-valued function sequences and series with some illustrated examples. Also we study Hukuhara differentiation and Henstock integration of a fuzzy-valued function with some necessary inclusions. Furthermore, we introduce the power series with fuzzy coefficients and define the radius of convergence of power series. Finally, by using the notions of H-differentiation and radius of convergence we examine the relationship between term by term H-differentiation and uniform convergence of fuzzy-valued function series.

scholarly journals On the size of linearization domains

2008 ◽  
Vol 145 (2) ◽  
pp. 443-456
Author(s):  
XAVIER BUFF ◽  
CARSTEN L. PETERSEN
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Radius Of Convergence ◽  
Holomorphic Map ◽  
Root Of Unity ◽  
Rotation Numbers ◽  
Formal Power

AbstractAssume $f{:}\,U\subset \C\to \C$ is a holomorphic map fixing 0 with derivative λ, where 0 < |λ| ≤ 1. If λ is not a root of unity, there is a formal power series φf(z) = z + ${\cal O}$(z2) such that φf(λ z) = f(φf(z)). This power series is unique and we denote by Rconv(f) ∈ [0,+∞] its radius of convergence. We denote by Rgeom(f) the largest radius r ∈ [0, Rconv(f)] such that φf(D(0,r)) ⊂ U. In this paper, we present new elementary techniques for studying the maps f ↦ Rconv(f) and f ↦ Rgeom(f). Contrary to previous approaches, our techniques do not involve studying the arithmetical properties of rotation numbers.

scholarly journals Eigenelements of perturbed operators

1990 ◽  
Vol 49 (1) ◽  
pp. 138-148 ◽  
Author(s):  
B. V. Limaye ◽  
M. T. Nair
Keyword(s):  
Lower Bound ◽  
Power Series ◽  
Lower Bounds ◽  
Spectral Projection ◽  
Linear Perturbation ◽  
Convergence Radius

AbstractLet λ0 be a semisimple eigenvalue of an operator T0. Let Γ0 be a circle with centre λs0 containing no other spectral value of T0. Some lower bounds are obtained for the convergence radius of the power series for the spectral projection P(t) and for trace T(t)P(t) associated with linear perturbation family T(t) = T0 + tV0 and the circle Γ0. They are useful when T0 is a member of a sequence (Tn) which approximates an operator T in a collectively compact manner. These bounds result from a modification of Kato's method of majorizing series, based on an idea of Redont. I λ0 is simple, it is shown that the same lower bound are valid for the convergence radius of a power series yielding an eigenvector of T(t).

scholarly journals On (J, pn)-summability of Fourier series

1972 ◽  
Vol 18 (1) ◽  
pp. 13-17
Author(s):  
F. M. Khan
Keyword(s):  
Fourier Series ◽  
Power Series ◽  
Radius Of Convergence ◽  
Partial Sums ◽  
Summability Of Fourier Series ◽  
Image Position

Let pn>0 be such that pn diverges, and the radius of convergence of the power seriesis 1. Given any series σan with partial sums sn, we shall use the notationand

scholarly journals On composition of formal power series

2002 ◽  
Vol 30 (12) ◽  
pp. 761-770 ◽  
Author(s):  
Xiao-Xiong Gan ◽  
Nathaniel Knox
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Open Problem ◽  
Constant Term ◽  
Radius Of Convergence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Formal Power ◽  
Special Case

Given a formal power seriesg(x)=b0+b1x+b2x2+⋯and a nonunitf(x)=a1x+a2x2+⋯, it is well known that the composition ofgwithf,g(f(x)), is a formal power series. If the formal power seriesfabove is not a nonunit, that is, the constant term offis not zero, the existence of the compositiong(f(x))has been an open problem for many years. The recent development investigated the radius of convergence of a composed formal power series likefabove and obtained some very good results. This note gives a necessary and sufficient condition for the existence of the composition of some formal power series. By means of the theorems established in this note, the existence of the composition of a nonunit formal power series is a special case.

XXII.—On Methods of Summability Based on Power Series

1957 ◽  
Vol 64 (4) ◽  
pp. 342-349
Author(s):  
D. Borwein
Keyword(s):  
Power Series ◽  
Summability Method ◽  
Radius Of Convergence ◽  
Main Concern ◽  
Image Position

SynopsisGiven a power series with real non-negative coefficients and having radius of convergence p, a summability method P is defined as follows:The main concern of this note is to establish conditions sufficient for one such method to include another.

Accurate power series for eigenvalues of spheroidal angle functions and their convergence radii

2011 ◽  
Vol 89 (11) ◽  
pp. 1083-1099
Author(s):  
Tam Do-Nhat
Keyword(s):  
Power Series ◽  
Least Squares ◽  
Complex Plane ◽  
Branch Point ◽  
Upper Bound ◽  
Least Squares Method ◽  
Radius Of Convergence ◽  
New Method ◽  
Recursion Relations

In this paper, the radius of convergence of the spheroidal power series associated with the eigenvalue is calculated without using the branch point approach. Studying the properties of the power series using the recursion relations among its coefficients in the new method offers some insights into the spheroidal power series and its associated eigenfunction. This study also used the least squares method to accurately compute the convergence radii to five or six significant digits. Within the circle of convergence in the complex plane of the parameter c = kF, where k is the wavenumber and F is the semifocal length of the spheroidal system, the extremely fast convergent spheroidal power series are computed with full precision. In addition, a formula for the magnitude of the upper bound of the error is obtained.

RESTRICTED UNIVERSALITY OF POWER SERIES

2001 ◽  
Vol 33 (5) ◽  
pp. 543-552 ◽  
Author(s):  
JEAN-PIERRE KAHANE ◽  
ANTONIOS D. MELAS
Keyword(s):  
Power Series ◽  
Finite Union ◽  
Radius Of Convergence ◽  
Nonempty Interior ◽  
Partial Sums ◽  
Universal Approximation ◽  
Compact Set ◽  
Approximation Properties

We prove the existence of a power series having radius of convergence 0, whose partial sums have universal approximation properties on any compact set with connected complement that is contained in a finite union of circles centred at 0 and having rational radii, but do not have such properties on any compact set with nonempty interior. This relates to a theorem of A. I. Seleznev.

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