The fundamental theorems of complex analysis

This paper is the Cauchy integral theorem and integral of complex function carried out a comparative analysis, summarized in the Cauchy integral theorem and Cauchy integral formula, higher-order derivative formula, residue theorem and the relationship between the derivations to be proved, the formula can be used in these areas, such as mechanics of materials.

We develop a linear theory of discrete complex analysis on general quad-graphs, continuing and extending previous work of Duffin, Mercat, Kenyon, Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our approach based on the medial graph yields more instructive proofs of discrete analogs of several classical theorems and even new results. We provide discrete counterparts of fundamental concepts in complex analysis such as holomorphic functions, derivatives, the Laplacian, and exterior calculus. Also, we discuss discrete versions of important basic theorems such as Green's identities and Cauchy's integral formulae. For the first time, we discretize Green's first identity and Cauchy's integral formula for the derivative of a holomorphic function. In this paper, we focus on planar quad-graphs, but we would like to mention that many notions and theorems can be adapted to discrete Riemann surfaces in a straightforward way. In the case of planar parallelogram-graphs with bounded interior angles and bounded ratio of side lengths, we construct a discrete Green's function and discrete Cauchy's kernels with asymptotics comparable to the smooth case. Further restricting to the integer lattice of a two-dimensional skew coordinate system yields appropriate discrete Cauchy's integral formulae for higher order derivatives.

Recently J. Miao proved that if is a holomorphic function with Hadamard gaps on the open unit disc D then f ∈ Xp if and only if f ∈ Bp if and only if if and if only if where Xp, Bp and denote respectively the class of holomorphic functions on D which satisfy |f′(z)|p(1 − |z|2)p − 1dxdy is a finite measure, a Carleson measure and a little Carleson measure on D. In this paper we give a higher-dimensional version of Miao's result.

This article is concerned with the asymptotic behaviour of the Mahler measure (defined for the occasion) of functions of the form \(z^n + f(z)\), where f(z) is a holomorphic function on an open subset of \({\mathbb {C}}\) such that \(f(0) \ne 0\).

Cauchy Integral Theorem and Cauchy Integral Formula

Equivalence, linearization, and decomposition of holomorphic operator functions

We introduce a generalized Cauchy integral formula for special classes of functions, holomorphic in several complex variables. These functions also satisfy a generalized Cachy-Riemann equation. The integral formula is then applied to demonstrate that these classes of holomorphic functions are Fréchet modules over complex Clifford algebras, and to provide maximal domains of convergence for Taylor expansions, with homogeneous terms grouped together, of real analytic solutions to Maxwell's equations in vacuo.

By using the differential operator Dm f(z), z∈ U (Definition 1), we introduce a class of holomorphic functions, denoted by , and we obtain some inclusion relations.

Various BMO-type characteristics are given for some class of holomorphic functions with a mixed norm. Some criteria are established for Blaschke products to belong to analytic Besov spaces.

In this paper we introduce and examine the differential subordination related to the geometric mean of the form [p(z)]1-γp(z)+zp′(z)φp(z),zp′(z)γ≺h(z),z∈D,where D:={z∈C:|z|<1} and h is a convex univalent function with 0∈∂h(D). In particular, the above differential subordination generalizes the well-known Briot-Bouquet differential subordination. At the same time new type of ordinary differential equation has been proposed for study. The main results allow us to construct non-obvious subclasses in the class of holomorphic functions on D.

On N-Widths of Holomorphic-Functions of Several Variables

Let be a point sequence in the unit disc and the Nevanlinna characteristic of the sequence , . In terms of the Nevanlinna characteristic one finds estimates for the slowest possible growth of the characteristic as in the class of holomorphic functions in vanishing on . Let be a meromorphic function in . In terms of the Nevanlinna characteristic function of one finds estimates for the slowest possible growth of the characteristics and in the class of pairs of holomorphic functions and such that .

In this chapter we use the Cauchy theory to study functions that are holomorphic on an annulus, and analytic functions with isolated singularities. We describe a classification for isolated singularities. Functions that are holomorphic on an annulus have Laurent series expansions, an analogue of power series expansions for holomorphic functions on discs. Holomorphic functions with a finite number of isolated singularities in a domain can be integrated using the residue theorem, an analogue of the Cauchy integral formula. We discuss local properties of these functions. The study of zeros and poles of meromorphic functions leads to a theorem of Rouché that connects the number of zeros and poles to an integral. The theorem is not only aesthetically pleasing in its own right but also allows us to give alternate proofs of many important results. In the penultimate section of this chapter we illustrate the use of complex function theory in the evaluation of real definite integrals. The appendix discusses Cauchy principal values, a way to integrate functions with certain singularities.KeywordsCauchy TheoryHolomorphic FunctionReal Definite IntegralsLaurent Series ExpansionNonconstant Meromorphic FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Over 100 years ago Harald Bohr identified a deep problem about the convergence of Dirichlet series, and introduced an ingenious idea relating Dirichlet series and holomorphic functions in high dimensions. Elaborating on this work, almost twnety years later Bohnenblust and Hille solved the problem posed by Bohr. In recent years there has been a substantial revival of interest in the research area opened up by these early contributions. This involves the intertwining of the classical work with modern functional analysis, harmonic analysis, infinite dimensional holomorphy and probability theory as well as analytic number theory. New challenging research problems have crystallized and been solved in recent decades. The goal of this book is to describe in detail some of the key elements of this new research area to a wide audience. The approach is based on three pillars: Dirichlet series, infinite dimensional holomorphy and harmonic analysis.

A function with complex values and at every point of the specific domain contains a derivative is commonly known as analytic functions which is also referred as holomorphic functions. We begin by interpreting \(A\) as the class for all holomorphic functions \(L(\xi)\) with a Taylor series expansion written in the form: \[L(\xi) = \xi + \sum_{i=2}^{\infty} x_i \xi^i\] where \(x_i \in \mathbb{C}\) and \(\xi \in D\). \(D\) is the open unit disk where \(D = \{\xi : \xi \in \mathbb{C}, |\xi| &lt; 1 \}\). Furthermore, we suggest the subclass of \(A\) that is univalent in \(D\) represented as \(S\). It is commonly known that the main subclasses of class \(S\) are the class of starlike functions and the class of convex functions. To develop and analyze the Fekete-Szegö problems, the theory of geometric function contributes significantly to this. Moreover, the frequent use of \(q\)-calculus as a general direction of research among mathematicians has caught our attention. In this research, we attained the initial coefficients, \(x_2\) and \(x_3\), and the upper bound for the functional \(|x_3 - \nu x_2^2|\) of functions \(L\) in the two new subclasses that are introduced by involving the Sălăgeăn \(q\)-differential operator, \(M_q^\eta L(\xi)\) and the definition of subordination.