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About 10,000 papers for “Holomorphic function”

Some Conclusions on Holomorphic Function and Analytic Function

Lijiang Zeng · DEStech Transactions on Social Science Education and Human Science · 2017

In many applied sciences, such as physics, engineering, etc. the holomorphic function plays an extremely important role, and at the same time, the analytic function plays an extremely important role. In this article, we tries to clarify, under certain conditions, they are the same kind of function. We will see that a function is analytic if and only if it is holomorphic. More importantly, in this process, the mathematical tools be used by us have a lot of application in many applied sciences.

Cited by 0Open access

Iterated function systems of holomorphic maps

Marco Abate, Ian Short · Advances in Mathematics · 2026

We unify and advance a host of works on iterated function systems of holomorphic self-maps of hyperbolic Riemann surfaces. Our foremost result is a generalisation to left iterated function systems of an unpublished and little known theorem of Heins on iteration in the unit disc. Applications abound -- to work of Benini et al. on transcendental dynamics, to the theory of hyperbolic steps of holomorphic maps, and to left semiconjugacy in the unit disc. We extend other work of Benini et al. and Ferreira on relatively compact left iterated function systems, and we prove a hyperbolic distance inequality for holomorphic maps that generalises a theorem of Bracci, Kraus, and Roth. Additionally, we strengthen results of the first author and Christodoulou on left iterated function systems, removing the need for Bloch domains, and we answer an open question from their work. Finally, we establish a version of the Heins theorem for right iterated functions systems, and we generalise theorems of Beardon and Kuznetsov on right iterated function systems in relatively compact semigroups of holomorphic maps.

Cited by 0Open access

Invariant Differential Operators and Holomorphic Function Spaces

Zhimin Yan · Journal of Lie theory · 2000

v r # r T 8 8 P r # r # r { r 8 r # r 8 h r h 8 Y # r r 8 # 8 U w r r 8 8 8 r { # 8 # r 8 r ' 8 # 4 r 8 # & u r T & r " r 8 r # 4 r 8 { 8 I # v 8 r r P r r 8 8 # 0 2 r # r r 8 T r w 8 r r 8 8 w r 8 8 w # 8 8 # r T 8 r r 8 r ' 8 # S r # 8 # u 8 r T r # 8 8 Y 8 8 { # r r { 8 r 8 # 8 # T r {h W Wd f h m p R

Cited by 15Open access

Charaterization of Holomorphic function using power series

Xiaohan Lin · Highlights in Science Engineering and Technology · 2023

Complex analysis is a subfield of mathematical analysis that focuses on the operations of complex numbers. The theory of functions of a complex variable is another name for it. Holomorphic functions, is the key object studied in the filed of complex analysis. These functions can be differentiated, have negative values, and are defined on the complex plane. Some of the ideas, concepts, and techniques frequently utilized in research include the residue theorem, Cauchy integral formula, Laurent series expansion, etc.In particular, algebraic geometry, fluid dynamics, quantum mechanics, and other related sciences have made substantial use of complex analysis in mathematics, physics, and engineering over the years. Girolamo Cardano and Raphael Bombelli, two Italian mathematicians, initially recognized complex numbers while attempting to solve a particular algebra in the 16th century, while Cauchy and Riemann further improved it in the 19th century.This essay explores the arithmetic features of complex numbers firstly. Secondly, a thorough interpretation of the characteristics of holomorphic functions is provided. Finally, using power series methods, the holomorphic function on the disc is characterized.

Cited by 0Open access

The dilatation function of a holomorphic isotopy

Samuel L. Krushkal · Functiones et Approximatio Commentarii Mathematici · 2009

Every nonvanishing univalent function $f(z)$ in the disk $\Delta^* = \widehat{\mathbb{C}} \setminus \overline{\Delta}, \Delta = \{|z| < 1\}$, for example, with hydrodynamical normalization, generates a complex isotopy $f_t (z) = t f(t^{-1} z): \Delta^* \times \Delta \to \widehat{\mathbb{C}}$, which is a special case of holomorphic motions and plays an important role in many topics. Let $q_f$ denote the minimal dilatation among quasiconformal extensions of $f$ to $\widehat{\mathbb{C}}$. In 1995, R. Kühnau raised the questions whether the dilatation function $q_f(r) = q_{f_r}$ is real analytic and whether the function $f$ can be reconstructed if $q_f(r)$ is given. The analyticity of $q_f$ was known only for ellipses and for the Cassini ovals. Our main theorem provides a wide class of maps with analytic dilatations and implies also a negative answer to the second question.

Cited by 2Open access

A criterion to justify a holomorphic function

Feng Qi, Bai‐Ni Guo · Global Journal of Mathematical Analysis · 2017

In the paper, the authors present a criterion, free of considering differentiability and any partial derivative, to justify a holomorphic function and illustrate the criterion by several examples.

Cited by 3Open access

Interactions between function theory and holomorphic dynamics

Alexandre Erëmenko · arXiv (Cornell University) · 2020

This is a survey talk on the ICMS conference "Resonances of complex dynamics", 9-13 July 2018. Three examples which show how dynamical considerations lead to new results in complex function theory are discussed. These examples are: a) a refinement of of the Wiman-Valiron method, b) the proof of Hayman's conjecture on the value distribution of derivatives of meromorphic functions, and c) a new simplified proof of the result of Lin and Wang on the number of critical points of the Green function on a torus.

Cited by 2Open access

A holomorphic function having a discontinuous inverse

T. J. Suffridge · Proceedings of the American Mathematical Society · 1972

An example is given of a function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is holomorphic in the open unit ball of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l Superscript normal infinity"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>l</mml:mi> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{l^\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and which extends in a natural way to the closed unit ball. The extended function gives a one-toone correspondence of the closed ball with itself but the inverse function fails to be continuous on the image of the open ball under the map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

Cited by 4Open access

Analytic families of holomorphic iterated function systems

Mario Roy, Hiroki Sumi, Mariusz Urbański · Nonlinearity · 2008

This paper deals with analytic families of holomorphic iterated function systems. Using real analyticity of the pressure function (which we prove), we establish a classification theorem for analytic families of holomorphic iterated function systems which depend continuously on a parameter when the space of holomorphic iterated function systems is endowed with the "$\lambda$-topology". This classification theorem allows us to generalize some geometric results from the paper ("Lambda-Topology vs. Pointwise Topology", to appear in Ergod. Th. & Dynam. Sys.) of the authors, and gives us a better and clearer understanding of the global structure of the space of conformal IFSs.

Cited by 6Open access

NORMALITY CRITERIONS ON SHARING A HOLOMORPHIC FUNCTION

Poincare Journal of Analysis and Applications · 2024

Let F be a family of holomorphic functions in a domain D, whose zeros have multiplicity at least 4 in D. Let ( 0) be a holomorphic function in D, whose zeros have multiplicity at most 2. If for each pair of functions {f, g} F , f and g share in D, then F is normal in D.

Cited by 0Open access

Holomorphic extension of the de Gennes function

Virginie Bonnaillie‐Noël, Frédéric Hérau, Nicolas Raymond · Annales mathématiques Blaise Pascal · 2017

This note is devoted to prove that the de Gennes function has a holomorphic extension on a half strip containing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℝ</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:math> .

Cited by 2Open access

Holomorphic function spaces on the Hartogs triangle

Alessandro Monguzzi · BOA (University of Milano-Bicocca) · 2021

The definition of classical holomorphic function spaces such as the Hardy space or the Dirichlet space on the Hartogs triangle is not canonical. In this paper we introduce a natural family of holomorphic function spaces on the Hartogs triangle which includes some weighted Bergman spaces, a candidate Hardy space and a candidate Dirichlet space. For the weighted Bergman spaces and the Hardy space we study the (Formula presented.) mapping properties of Bergman and Szegő projection respectively, whereas for the Dirichlet space we prove it is isometric to the Dirichlet space on the bidisc.

Cited by 9

Holomorphic extension of the de Gennes function

Virginie Bonnaillie‐Noël, Frédéric Hérau, Nicolas Raymond · arXiv (Cornell University) · 2016

This note is devoted to prove that the de Gennes function has a holomorphic extension on a strip containing the real axis.

Cited by 0Open access