scholarly journals Deriving Cauchy's Integral Formula Using Division Method

2019 ◽  
Vol 1 ◽  
pp. 259-264
Author(s):  
M Egahi ◽  
I O Ogwuche ◽  
J Ode
Keyword(s):  
Analytic Functions ◽  
Complex Analysis ◽  
Integral Formula ◽  
Integral Theorem ◽  
Cauchy’S Integral Formula ◽  
Cauchy’S Integral Theorem

Cauchy's integral theorem and formula which holds for analytic functions is proved in most standard complex analysis texts. The nth derivative form is also proved. Here we derive the nth derivative form of Cauchy's integral formula using division method and showed its link with Taylor's theorem and demonstrate the result with some polynomials.

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.

scholarly journals A further generalization of the Catalan numbers and its explicit formula and integral representation

2021 ◽  
Author(s):  
Wen-Hui Li ◽  
Feng Qi ◽  
Omran Kouba ◽  
Issam Kaddoura
Keyword(s):  
Number Theory ◽  
Integral Representation ◽  
Explicit Formula ◽  
Generating Function ◽  
Complex Analysis ◽  
Integral Formula ◽  
Catalan Numbers ◽  
Combinatorial Number Theory ◽  
Integral Formulas ◽  
Cauchy’S Integral Formula

In the paper, motivated by the generating function of the Catalan numbers in combinatorial number theory and with the aid of Cauchy’s integral formula in complex analysis, the authors generalize the Catalan numbers and its generating function, establish an explicit formula and an integral representation for the generalization of the Catalan numbers and corresponding generating function, and derive several integral formulas and combinatorial identities.

scholarly journals A further generalization of the Catalan numbers and its explicit formula and integral representation

2020 ◽  
Author(s):  
Wen-Hui Li ◽  
Feng Qi ◽  
Omran Kouba ◽  
Issam Kaddoura
Keyword(s):  
Number Theory ◽  
Integral Representation ◽  
Explicit Formula ◽  
Generating Function ◽  
Complex Analysis ◽  
Integral Formula ◽  
Catalan Numbers ◽  
Combinatorial Number Theory ◽  
Integral Formulas ◽  
Cauchy’S Integral Formula

In the paper, motivated by the generating function of the Catalan numbers in combinatorial number theory and with the aid of Cauchy's integral formula in complex analysis, the authors generalize the Catalan numbers and its generating function, establish an explicit formula and an integral representation for the generalization of the Catalan numbers and corresponding generating function, and derive several integral formulas and combinatorial identities.

scholarly journals Meromorphic or Holomorphic Completion of a Reinhardt Domain

1970 ◽  
Vol 38 ◽  
pp. 1-12 ◽  
Author(s):  
Eiichi Sakai
Keyword(s):  
Meromorphic Function ◽  
Power Series ◽  
Meromorphic Functions ◽  
Integral Formula ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Reinhardt Domain ◽  
Continuity Theorem ◽  
Cauchy’S Integral Formula ◽  
Long Time

In the theory of functions of several complex variables, the problem about the continuation of meromorphic functions has not been much investigated for a long time in spite of its importance except the deeper result of the continuity theorem due to E. E. Levi [4] and H. Kneser [3], The difficulty of its investigation is based on the following reasons: we can not use the tools of not only Cauchy’s integral formula but also the power series and there are indetermination points for the meromorphic function of many variables different from one variable. Therefore we shall also follow the Levi and Kneser’s method and seek for the aspect of meromorphic completion of a Reinhardt domain in Cn.

scholarly journals An Efficient Method to Find Solutions for Transcendental Equations with Several Roots

2015 ◽  
Vol 2015 ◽  
pp. 1-4 ◽  
Author(s):  
Rogelio Luck ◽  
Gregory J. Zdaniuk ◽  
Heejin Cho
Keyword(s):  
Null Space ◽  
Heat Conduction Problem ◽  
Hankel Matrix ◽  
Polynomial Equation ◽  
Transient Heat Conduction ◽  
Transcendental Equation ◽  
Integral Theorem ◽  
Transient Heat Conduction Problem ◽  
Cauchy’S Integral Theorem ◽  
Roots Of A Polynomial

This paper presents a method for obtaining a solution for all the roots of a transcendental equation within a bounded region by finding a polynomial equation with the same roots as the transcendental equation. The proposed method is developed using Cauchy’s integral theorem for complex variables and transforms the problem of finding the roots of a transcendental equation into an equivalent problem of finding roots of a polynomial equation with exactly the same roots. The interesting result is that the coefficients of the polynomial form a vector which lies in the null space of a Hankel matrix made up of the Fourier series coefficients of the inverse of the original transcendental equation. Then the explicit solution can be readily obtained using the complex fast Fourier transform. To conclude, the authors present an example by solving for the first three eigenvalues of the 1D transient heat conduction problem.

SYMOIN STOILOV (1887-1961): DETAILS OF SCIENTIfiC CAREER

2021 ◽  
Vol 9 (1) ◽  
pp. 152-163
Author(s):  
O. Martynyuk ◽  
I. Zhytaryuk
Keyword(s):  
Interwar Period ◽  
Analytic Functions ◽  
Complex Analysis ◽  
Topological Analysis ◽  
Second World War ◽  
Educational Process ◽  
Mathematical Science ◽  
World War ◽  
Second World ◽  
Selection Of

The present article covers topics of life, scientific, pedagogical and social activities of the famous Romanian mathematician Simoin Stoilov (1887-1961), professor of Chernivtsi and Bucharest universities. Stoilov was working at Chernivtsi University during 1923-1939 (at this interwar period Chernivtsi region was a part of royal Romania. The article is aimed on the occasion of honoring professors’ memory and his managerial abilities in the selection of scientific and pedagogical staff to ensure the educational process and research in Chernivtsi University in the interwar period. In addition, it is noted that Simoin Stoilov has made a significant contribution to the development of mathematical science, in particular he is the founder of the Romanian school of complex analysis and the theory of topological analysis of analytic functions; the main directions of his research are: partial differential equation; set theory; general theory of real functions and topology; topological theory of analytic functions; issues of philosophy and foundation of mathematics, scientific research methods, Lenin’s theory of cognition. The article focuses on the active socio-political and state activities of Simoin Stoilov in terms of restoring scientific and cultural ties after the Second World War.

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