Complex Analysis and Analytic Functions

1997 ◽  
pp. 36-51
Author(s):  
Benjamin Fine ◽  
Gerhard Rosenberger
Keyword(s):  
Analytic Functions ◽  
Complex Analysis

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.

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 Supersymmetric Polynomials on the Space of Absolutely Convergent Series

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1111 ◽  
Author(s):  
Farah Jawad ◽  
Andriy Zagorodnyuk
Keyword(s):  
Banach Space ◽  
Analytic Functions ◽  
Complex Analysis ◽  
Convergent Series ◽  
Infinite Dimensional

We consider an algebra H b s u p of analytic functions on the Banach space of two-sided absolutely summing sequences which is generated by so-called supersymmetric polynomials. Our purpose is to investigate H b s u p and its spectrum with using methods of infinite dimensional complex analysis and the theory of Fréchet algebras. Some algebraic bases of H b s u p are described. Also, we show that the spectrum of the algebra of supersymmetric analytic functions of bounded type contains a metric ring M . We prove that M is a complete metric (nonlinear) space and investigate homomorphisms and additive operators on this ring. Some possible applications are discussed.

Ideal structure of rings of analytic functions with non-Archimedean metrics

2021 ◽  
Author(s):  
Nicholas Bruno
Keyword(s):  
Analytic Functions ◽  
Complex Analysis ◽  
Entire Functions ◽  
Prime Ideals ◽  
Ideal Structure ◽  
Finitely Generated ◽  
Finite Radius ◽  
Maximal Ideals ◽  
Divisibility Properties ◽  
The Ideal

The work of Helmer [Divisibility properties of integral functions, Duke Math. J. 6(2) (1940) 345–356] applied algebraic methods to the field of complex analysis when he proved the ring of entire functions on the complex plane is a Bezout domain (i.e. all finitely generated ideals are principal). This inspired the work of Henriksen [On the ideal structure of the ring of entire functions, Pacific J. Math. 2(2) (1952) 179–184. On the prime ideals of the ring of entire functions, Pacific J. Math. 3(4) (1953) 711–720] who proved a correspondence between the maximal ideals within the ring of entire functions and ultrafilters on sets of zeroes as well as a correspondence between the prime ideals and growth rates on the multiplicities of zeroes. We prove analogous results on rings of analytic functions in the non-Archimedean context: all finitely generated ideals in the ring of analytic functions on an annulus of a characteristic zero non-Archimedean field are two-generated but not guaranteed to be principal. We also prove the maximal and prime ideal structure in the non-Archimedean context is similar to that of the ordinary complex numbers; however, the methodology has to be significantly altered to account for the failure of Weierstrass factorization on balls of finite radius in fields which are not spherically complete, which was proven by Lazard [Les zeros d’une function analytique d’une variable sur un corps value complet, Publ. Math. l’IHES 14(1) (1942) 47–75].

scholarly journals Solving Riemann-Hilbert problems with meromorphic functions

2019 ◽  
Vol 11 (1) ◽  
pp. 117-130
Author(s):  
Dan Kucerovsky ◽  
Aydin Sarraf
Keyword(s):  
Meromorphic Function ◽  
Complex Plane ◽  
Blaschke Product ◽  
Analytic Functions ◽  
Complex Analysis ◽  
Meromorphic Functions ◽  
Factorization Theorem ◽  
Simply Connected ◽  
The Given

Abstract In this paper, we introduce the use of a powerful tool from theoretical complex analysis, the Blaschke product, for the solution of Riemann-Hilbert problems. Classically, Riemann-Hilbert problems are considered for analytic functions. We give a factorization theorem for meromorphic functions over simply connected nonempty proper open subsets of the complex plane and use this theorem to solve Riemann-Hilbert problems where the given data consists of a meromorphic function.

scholarly journals A discrete analogue of the Paley-Wiener theorem for bounded analytic functions in a half plane

1977 ◽  
Vol 23 (3) ◽  
pp. 376-378
Author(s):  
Doron Zeilberger
Keyword(s):  
Analytic Function ◽  
Half Plane ◽  
Analytic Functions ◽  
Complex Analysis ◽  
Discrete Analogue ◽  
Bounded Analytic Functions ◽  
Bounded Analytic Function ◽  
Wiener Theorem ◽  
Upper Half Plane

In this note we prove a discrete analogue to the following Paley–Weiner theorem: Let f be the restriction to the line of a bounded analytic function in the upper half plane; then the spectrum of f is contained in ([0, ∈). The discrete analogue of complex analysis is the theory of discrete analytic functions invented by Lelong-Ferrand (1944) and developed by Duffin (1956) and others.

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