Sequences and Series of Analytic Functions, the Residue Theorem

2009 ◽  
pp. 1-88
Author(s):  
Eberhard Freitag ◽  
Rolf Busam
Keyword(s):  
Analytic Functions ◽  
Residue Theorem

scholarly journals Valiron’s Interpolation Formula and a Derivative Sampling Formula in the Mellin Setting Acquired via Polar-Analytic Functions

2020 ◽  
Vol 20 (3-4) ◽  
pp. 629-652
Author(s):  
Carlo Bardaro ◽  
Paul L. Butzer ◽  
Ilaria Mantellini ◽  
Gerhard Schmeisser
Keyword(s):  
Analytic Functions ◽  
Main Tool ◽  
Interpolation Formula ◽  
Residue Theorem ◽  
Differentiation Formula ◽  
Sampling Formula ◽  
Bernstein Spaces ◽  
Derivative Sampling ◽  
Classical Interpolation ◽  
Identity Theorem

AbstractIn this paper, we first recall some recent results on polar-analytic functions. Then we establish Mellin analogues of a classical interpolation of Valiron and of a derivative sampling formula. As consequences a new differentiation formula and an identity theorem in Mellin–Bernstein spaces are obtained. The main tool in the proofs is a residue theorem for polar-analytic functions.

scholarly journals Integration of polar-analytic functions and applications to Boas’ differentiation formula and Bernstein’s inequality in Mellin setting

2020 ◽  
Vol 13 (4) ◽  
pp. 503-514 ◽  
Author(s):  
Carlo Bardaro ◽  
Paul L. Butzer ◽  
Ilaria Mantellini ◽  
Gerhard Schmeisser
Keyword(s):  
Analytic Functions ◽  
Integral Formula ◽  
Type Inequality ◽  
General Version ◽  
Residue Theorem ◽  
Bernstein Type ◽  
Bernstein’S Inequality ◽  
Differentiation Formula ◽  
Bernstein Type Inequality ◽  
Cauchy’S Integral Formula

Abstract We establish a general version of Cauchy’s integral formula and a residue theorem for polar-analytic functions, employing the new notion of logarithmic poles. As an application, a Boas-type differentiation formula in Mellin setting and a Bernstein-type inequality for polar Mellin derivatives are deduced.

A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable

2020 ◽  
Vol 17 (2) ◽  
pp. 256-277
Author(s):  
Ol'ga Veselovska ◽  
Veronika Dostoina
Keyword(s):  
Complex Plane ◽  
Complex Variable ◽  
Analytic Functions ◽  
Combinatorial Identities ◽  
Independent Interest ◽  
Closed Curves ◽  
In Series ◽  
Derivatives Of

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

A NEW SUBCLASS OF ANALYTIC FUNCTIONS DEFINED BY OPOOLA DIFFERENTIAL OPERATOR

2020 ◽  
Vol 9 (8) ◽  
pp. 5343-5348 ◽  
Author(s):  
T. G. Shaba ◽  
A. A. Ibrahim ◽  
M. F. Oyedotun
Keyword(s):  
Differential Operator ◽  
Analytic Functions
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