Complex Analysis

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.

This user-friendly textbook introduces complex analysis at the beginning graduate or advanced undergraduate level. Unlike other textbooks, it follows Weierstrass' approach, stressing the importance of power series expansions instead of starting with the Cauchy integral formula, an approach that illuminates many important concepts. This view allows readers to quickly obtain and understand many fundamental results of complex analysis, such as the maximum principle, Liouville's theorem, and Schwarz's lemma. The book covers all the essential material on complex analysis, and includes several elegant proofs that were recently discovered. It includes the zipper algorithm for computing conformal maps, as well as a constructive proof of the Riemann mapping theorem, and culminates in a complete proof of the uniformization theorem. Aimed at students with some undergraduate background in real analysis, though not Lebesgue integration, this classroom-tested textbook will teach the skills and intuition necessary to understand this important area of mathematics.

The constructive development of elementary complex analysis, through Cauchy’s integral formula, is a simple matter; the material seems to have a natural constructive cast. This is carried out in Sections 1, 2, 3, and 4. Next it is shown that under certain conditions it is possible to find the zeros of an analytic function. Although results of this type involve much more careful estimates than their classical counterparts, nothing basically new is required. The next section discusses singularities, and includes proofs of two constructively distinct versions of the Picard theorem on the range of a differentiable function in the neighborhood of a singularity; the proofs are based on Schottky’s theorem, the classical proof of which readily adapts to a constructive one. The last section provides a constructive version of the Riemann mapping theorem, following the classical Koebe approach as presented by Ostrowski and Warschawski. Some care must be taken in finding the right definitions, since the Riemann mapping theorem is not constructively valid without additional restrictions on the domain.KeywordsDifferentiable FunctionClosed PathCircular PathLaurent ExpansionRiemann Mapping TheoremThese 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.

For matrix A, vector b and function f, the computation of vector f(A)b arises in many scientific computing applications. We consider the problem of obtaining quantum state |f> corresponding to vector f(A)b. There is a quantum algorithm to compute state |f> using eigenvalue estimation that uses phase estimation and Hamiltonian simulation e^{\im A t}. However, the algorithm based on eigenvalue estimation needs \poly(1/\epsilon) runtime, where \epsilon is the desired accuracy of the output state. Moreover, if matrix A is not Hermitian, \e^{\im A t} is not unitary and we cannot run eigenvalue estimation. In this paper, we propose a quantum algorithm that uses Cauchy's integral formula and the trapezoidal rule as an approach that avoids eigenvalue estimation. We show that the runtime of the algorithm is \poly(\log(1/\epsilon)) and the algorithm outputs state |f> even if A is not Hermitian.

The Riemann mapping theorem has been said by some to be the greatest theorem of the nineteenth century. The entire concept of the theorem is profoundly original, and its proof introduced many new ideas. Certainly normal families and the use of extremal problems in complex analysis are just two of the important techniques that have grown out of studies of the Riemann mapping theorem.

The framework of generalized analytic functions arising from the related potentials (so-called k-harmonically analytic functions) has been developed in application to three-dimensional (3D) axially symmetric Stokes flow problems. Cauchy's integral formula for the class of k-harmonically analytic functions has been obtained, and series representations for k-harmonically analytic functions for the regions exterior to sphere and prolate and oblate spheroids have been derived. As the central result in the developed framework, a solution form representing the velocity field and pressure for 3D axially symmetric Stokes flows has been constructed in terms of two 0-harmonically analytic functions. It has also been shown that it uniquely determines an external velocity field vanishing at infinity. With the obtained solution form, the problem of 3D Stokes flows due to the axially symmetric translation of a solid body of revolution has been reduced to a boundary-value problem for two 0-harmonically analytic function...

A boundary element method based on the Cauchy's integral formulae, called the complex variable boundary element method (CVBEM), is proposed for the numerical solution of boundary value problems governing plane thermoelastic deformations of anisotropic elastic bodies. The method is applicable for a wide class of problems which do not involve inertia or coupling effects and can be easily and efficiently implemented on the computer. It is applied to solve specific test problems.

The Cauchy integral formula is a fundamental and critical formula in the complex function, and it has a variety of applications in many aspects such as physics, engineering, and mathematics. This article focuses on the applications in the functions and two main theorems in the Cauchy integral formula. It explores the formula and provides corresponding proofs of the formula. To this end, the aim of the article is to use the Cauchy integral formula to cope with some difficult and complex integrals and functions. As a result, this article denotes to use the formula to simplify the complex and difficult problems and provides a practical and fast method. Hence, the article lists some examples to demonstrate the functions’ use of the Cauchy integral formula. At last, the article may serve as a reference for relevant research and survey in the field of the complex functions and provide insights into the Cauchy integral formula’s uses.

INTRODUCTION. This note concerns Riemann mappings of rectangles onto the unit disk in the complex plane. Let's fix some notation for the discussion. For the unit circle and unit disk we'll use, respectively, S1 and D. For the rectangle parallel to the coordinate axes, centered at the origin 0 E C with height 1 and base length a > 0, so that the corners are at (?a/2, ? 1/2), we'll use R(a). In particular, it will be convenient to have R(b) c R(a) if b < a. Riemann's mapping theorem says that every simply connected plane domain Ql not equal to the whole plane can be mapped conformally one-to-one onto D. Thus this certainly holds for the interior of rectangles. These maps are thoroughly well understood from several points of view, primarily using the Schwarz-Christoffel formula (elliptic functions in a not too subtle disguise). The purpose of this note is to show that rectangles with different modulus (different values of a) can be seen as conformally different and completely characterized by sets of four distinct points on S with very little machinery. In fact, our only tools will be the Schwarz reflection principle, the Schwarz lemma (in other words, Schwarz seemed to have a pretty good sense about this sort of thing), some elementary conformal mappings, and the concept of normal families of analytic functions. Perhaps then the real purpose of this note is to serve as a reminder of the power of the profound, beautiful, and essentially elementary maximum modulus principle. The only reference necessary for this discussion is [A]. The theorem of Lowner may also be found in [C]. The proof given there is of a somewhat different flavor, and Lowner's proof [L] was roughly the same as the one given here.

It has long been customary in the theory of linear transformations-integral equations, linear equations in an infinite number of unknowns, etc.-to make use of certain notions from the theory of functions of a complex variable. In particular, the well known Neumann expansion of the resolvent is a formal power series whose coefficients, instead of being numbers, are iterates of a certain linear transformation.' These resemblances to the classical CauchyWeierstrass theory can be described accurately in terms of a theory of analytic functions whose values lie in a complex Banach space. The possibility of such a theory, derived in the usual manner from Cauchy's theorem and integral formula, was recognized by Wiener.2 Still further generalizations, in which the independent variable may also range over a complex Banach space, have been considered by a number of people.3 It therefore seems desirable to outline a general abstract theory of linear operations depending analytically on a parameter, which we shall here suppose to be a complex variable. DEFINITION 1.1 A function f(X) defined in an open set A of the complex plane, with values in a complex Banach space E, is called analytic in A if the limit

In my talk I will attempt to draw some connections between complex analysis and computational geometry, particularly between conformal mappings, hyperbolic geometry, the medial axis and optimal meshing. The Riemann mapping theorem says that there is a conformal (angle preserving) map of the unit disk, DISK, to the interior Ω of any simple n-gon. How much work is needed to compute this map? [MR2671015] We can compute the conformal map f: DISK -> Ω to within error e in time O(n ⋅ log 1/e log log 1/e).The proof uses an iteration that converges quadratically to f, but it needs a good initial guess that is close to the correct answer. The medial axis allows us to quickly construct a map Ω -> DISK that is close to Riemann's map with precise estimates. This gives our starting point, and the time estimates in the theorem depend on time needed to compute the medial axis (linear by work of Chin, Snoeyink and Wang).Conversely, the proof of Theorem 1 uses ideas from analysis that may be of interest in CG. For example, the dome of a planar domain Ω is the surface in R+3 given by the upper envelope of all hemispheres whose base on R2 is a medial axis disk of Ω. Domes arise in the study of 3-manifolds, and a fundamental result of Dennis Sullivan about convex sets in hyperbolic 3-space lies at the heart of Theorem 1. The proof of Theorem 1 also introduces the thick-thin decomposition of a polygon, inspired by the thick-thin decomposition of a hyperbolic manifold. The thin parts correspond to long, narrow pieces of the polygon (but not in the obvious way; the precise definition uses a conformal invariant called extremal length). This decomposition plays in important role in proving: [MR2671014] Any simple n-gon has a O(n) quadrilateral mesh with all angles ≤ 120O and all new angles ≥ 60O.The sharp upper bound is due to Bern and Eppstein; the lower bound is the novel part here. The result is also true for PSLGs with at most n edges and vertices, although simple examples show the O(n) must be replaced by O(n2): [Bishop-Quad] Every PSLG has an O(n2) quadrilateral mesh with all angles ≤ 120O and all new angles ≥ 60O. Only O(n/e) angles satisfy |θ-90O| > e, for any e >0.

The history of complex function can be dated back to about 18 century when Euler came up with two equations derived from integrals of complex function. Over the age, mathematicians have been trying to explore and discuss more in this mysterious field, and this new branch gradually became prevalent in the 19 century mathematics like when calculus ruled the 18 century mathematics. In particular, the concept of residue was invented and became an important part in complex function, being applied in some special types of real integral. Also it is a generalization of Cauchy integral theorem and Cauchy integral formula. Nowadays, the area of complex function is filled with crystallization of the wisdom of countless scholars and researchers, and the precious mathematical treasure can also meet the needs of other academic areas like physics and biology. Summarizing the various approaches and examples of application in fields of mathematics and others can be a valuable topic. In this paper, the history of application and some examples for references of complex functions, and introduce the main concepts and formulas about them will be briefly discussed, including the area of complex numbers, analytic function, and residue. There are some relationships between some of the applications to be dig out and considerable real-world problems, and they are supposed to the generalization of those subjects can provide certain enlightenment to people interested in or during the study in fields of complex function.

A novel integral equation method is developed in this paper for the analysis of two‐dimensional general anisotropic elastic bodies with cracks. In contrast to the conventional boundary integral methods based on reciprocal work theorem, the present method is derived from Stroh's formalism for anisotropic elasticity in conjunction with Cauchy's integral formula. The proposed boundary integral equations contain boundary displacement gradients and tractions on the non‐crack boundary and the dislocations on the crack lines. In cases where only the crack faces are subjected to tractions, the integrals on the non‐crack boundary are non‐singular. The boundary integral equations can be solved using Gaussian‐type integration formulas directly without dividing the boundary into discrete elements. Numerical examples of stress intensity factors are given to illustrate the effectiveness and accuracy of the present method.

Application of Wiener–Hopf technique to linear nonhomogeneous integral equations for a new representation of Chandrasekhar's H-function in radiative transfer, its existence and uniqueness

Remarks on Cauchy's Integral Formula in Matric Spaces