Cauchy Integral Formula Research Articles

Residue Theorem and Logarithmic Residue Theorem

This passage is written for researching the application of residue theorem and Logarithmic Residue theorem. The residue theorem is a powerful tool for the analysis of complex functions. It also can be used to calculate the integral of the real functions. Residue theorem is the extension of the Cauchy’s integral theorem and Cauchy integral formula. Passage researched two theorems by defining the residue in two ways. Each of them is defined by Laurent series and defined by integral. Then the residue theorem is introduced with its definition and applications. The passage points out two instances for the applications of the residue theorem. After the residue theorem, the Logarithmic Residue theorem is listed with its definition and applications. There are also three instances for its applications. The residue theorem and Logarithmic Residue theorem mainly contribute for the research of the complex functions’ integral calculations. The residue theorem provides powerful mathematical tools in certain special types of real integration problems.

A Cauchy integral formula for (p, q)-monogenic functions with α-weight

A Cauchy integral formula for (p, q)-monogenic functions with α-weight

Computing the Mittag-Leffler function of a matrix argument

It is well-known that the two-parameter Mittag-Leffler (ML) function plays a key role in Fractional Calculus. In this paper, we address the problem of computing this function, when its argument is a square matrix. Effective methods for solving this problem involve the computation of higher order derivatives or require the use of mixed precision arithmetic. In this paper, we provide an alternative method that is derivative-free and works entirely using IEEE standard double precision arithmetic. If certain conditions are satisfied, our method uses a Taylor series representation for the ML function; if not, it switches to a Schur-Parlett technique that will be combined with the Cauchy integral formula. A detailed discussion on the choice of a convenient contour is included. Theoretical and numerical issues regarding the performance of the proposed algorithm are discussed. A set of numerical experiments shows that our novel approach is competitive with the state-of-the-art method for IEEE double precision arithmetic, in terms of accuracy and CPU time. For matrices whose Schur decomposition has large blocks with clustered eigenvalues, our method far outperforms the other. Since our method does not require the efficient computation of higher order derivatives, it has the additional advantage of being easily extended to other matrix functions (e.g., special functions).

A rational‐Chebyshev projection method for nonlinear eigenvalue problems

AbstractThis article describes a projection method based on a combination of rational and polynomial approximations for efficiently solving large nonlinear eigenvalue problems. In a first stage the nonlinear matrix function under consideration is approximated by a matrix polynomial in . The error resulting from this polynomial approximation is in turn approximated by rational functions with the help of the Cauchy integral formula. The two approximations are combined and a linearization is performed. A key ingredient of the proposed approach is a projection method that uses subspaces spanned by vectors of the same dimension as that of the original problem instead of that of the linearized problem. A procedure is also presented to automatically select shifts and to partition the region of interest into a few subregions. This allows to subdivide the problem into smaller subproblems that are solved independently. The accuracy of the proposed method is theoretically analyzed and its performance is illustrated with a few test problems that have been discussed in the literature.

Comparison of K-spectral set bounds on norms of functions of a matrix or operator

We use results in [M. Crouzeix and A. Greenbaum, Spectral sets: numerical range and beyond, SIAM Jour. Matrix Anal. Appl., 40 (2019), pp. 1087-1101] to derive upper bounds on the norm of a function f of a matrix or other bounded linear operator A based on the infinity-norm of f on various regions in the complex plane. We compare these results to those that can be derived from a straightforward application of the Cauchy integral formula by replacing the norm of the integral by the integral of the resolvent norm. While, in some cases, the new upper bounds on ‖f(A)‖ are much tighter than those from the Cauchy integral formula, we show that in many cases of interest, the two bounds are of the same order of magnitude, with that from the Cauchy integral formula actually being slightly smaller. We give a partial explanation of this in terms of the numerical range of the resolvent at points near an ill-conditioned eigenvalue. At such points we show that the resolvent is close to a rank one matrix whose numerical range is a disk about a point near the origin, and we argue that in this case the two bounds are almost the same.

Analytic continuation: A tool for aeromagnetic data interpretation

Extending potential-field data to vertical complex-plane slices provides opportunities, unique to analytic functions, for subsurface imaging. We begin with an existing numerical method for differentiation by integration using Cauchy's integral formula. An example from data over northern Ontario, Canada, illustrates better conditioning compared to derivatives computed using finite differences. Next, these quantities are used for analytic continuation using rational complex-series expansion (Padé approximation). This method provides stable downward continuation to distances that are significantly greater than are achievable by Taylor series and spectral methods. A synthetic test confirms that the rational approximant faithfully reproduces the magnetic response of a thin sheet. A sign reversal of the total field, coincident with the origin of the sheet, features prominently on this example. We also provide synthetic results for sheets with finite depth extent, point sources, thin slabs, and contacts. These also exhibit polarity flips near the source location. Further, these provide a template for interpreting the source geometry using the shape of the approximating function. Additional synthetic tests illustrate an automatic method for locating closely spaced random assemblages of isolated poles. We apply our methods to a deposit-scale heliborne survey over a recently discovered volcanogenic massive sulphide ore deposit in a Canadian greenstone belt. The analytically continued profile over the deposit suggests the presence of a subvertical thin sheet of unknown depth extent. This interpretation coincides with a sulphide ore body that has been previously delimited by several drill holes. Some known prospects near this deposit are also imaged on maps and profiles of the Padé approximant.

The Cauchy integral formula for (p, q)-monogenic functions with α-weight in superspace

First, some properties for -monogenic functions with α-weight in superspace are obtained. Then the Cauchy–Pompeiu formula in superspace is given. Finally, the Cauchy integral formula and the Cauchy integral theorem for -monogenic functions with α-weight in superspace are proved.

General procedure for free-surface recovery from bottom pressure measurements: application to rotational overhanging waves

A novel boundary integral approach for the recovery of overhanging (or not) rotational water waves (with constant vorticity) from pressure measurements at the bottom is presented. The method is based on the Cauchy integral formula and on an Eulerian–Lagrangian formalism to accommodate overturning free surfaces. This approach eliminates the need to introduce a priori a special basis of functions, thus providing a general means of fitting the pressure data and, consequently, recovering the free surface. The effectiveness and accuracy of the method are demonstrated through numerical examples.

Mild Solutions of Fractional Integrodifferential Diffusion Equations with Nonlocal Initial Conditions via the Resolvent Family

Fractional integrodifferential diffusion equations play a significant role in describing anomalous diffusion phenomena. In this paper, we study the existence and uniqueness of mild solutions to these equations. Firstly, we construct an appropriate resolvent family, through which the related equicontinuity, strong continuity, and compactness properties are studied using the convolution theorem of Laplace transform, the probability density function, the Cauchy integral formula, and the Fubini theorem. Then, we construct a reasonable mild solution for the considered equations. Finally, we obtain some sufficient conditions for the existence and uniqueness of mild solutions to the considered equations by some fixed point theorems.

Integral theorems in finite-dimensional commutative algebra

For monogenic (continuous and differentiable in the sense of G\^ateaux) functions given in special real subspaces of an arbitrary finite-dimensional commutative associative algebra over the complex field and taking values in this algebra, we establish basic properties analogous to properties of holomorphic functions of a complex variable. Methods for proving results are based on a representation of monogenic functions via holomorphic functions of complex variables that allows to establish analogues of Cauchy-Riemann conditions and the continuity of G\^ateaux derivatives of all orders for monogenic functions. In such a way, analogues of a number of classical theorems of complex analysis (the Cauchy integral theorem for a curvilinear integral, the Cauchy integral formula, the Morera theorem, the Taylor theorem) are proved and different equivalent definitions for the mentioned monogenic functions are established. An analogue of the Cauchy theorem for an integral over non piecewise smooth surfaces is proved.

A Local Cauchy Integral Formula for Slice-Regular Functions

AbstractWe prove a local Cauchy-type integral formula for slice-regular functions. The formula is obtained as a corollary of a general integral representation formula where the integration is performed on the boundary of an open subset of the quaternionic space, with no requirement of axial symmetry. As a step towards the proof, we provide a decomposition of a slice-regular function as a combination of two axially monogenic functions.

Complex Analysis and Residue Theorem

The study of the properties of analytical functions is described as complex analysis. The residue theorem is an important conclusion in complex analysis. This paper introduces the origin of imaginary numbers from Cardano Formula defines the conversion of complex number formats from the Euler Theorem, and proves the intermediate theorem Cauchy Integral Formula before reaching our final conclusion and the goal of the paper, Residue Theorem. The name of the theorem comes from the concept of residue, which is defined using a functions Laurent series. We could then derive the Residue Theorem from the Cauchy Integral Theorem, also called the Cauchy-Goursat Theorem. We will be able to formalize our prior, ad hoc method of computing integrals over contours encompassing singularities. Additionally, it is a theorem that may be applied to zero-pole qualities and curved integral properties. The Residue Theorem is the basis of many essential mathematical facts revolving around line integrals, particularly in solving ODEs and PDEs and describing physics models.

Comparison of the Cauchy Integral Formula and Residue Theorem on the Calculation of Definite Integrals

Integration is a useful mathematical tool which is applied in a wide range of studies and assists people to solve many problems. Despite methods of real analysis, complex integrations in complex analysis are more beneficial and more convenient than real integrations under certain circumstances. Cauchy’s Residue Theorem and Cauchy Integral Formula are two methods used to solve complex integrals. Cauchy’s Residue Theorem helps to solve complex integrals via the calculation involving singular points, whereas Cauchy Integral Formula is implemented to solve complex integrals by finding out poles enclosed by the contour. Herein, the comparison and analysis of the process of solving both real and complex integrations between Cauchy’s Residue Theorem and Cauchy Integral Formula are demonstrated through sample questions involving exponential integrations and polynomial integrations. This passage also includes definitions for Cauchy’s Residue Theorem, Cauchy Integral Formula, corollary of Cauchy Integral Formula, steps and solutions of sample contour integral questions.

Monogenic functions and harmonic vectors

We consider special topological vector spaces with a commutative multiplication for some of elements of the spaces and monogenic functions taking values in these spaces.Monogenic functions are understood as continuous and differentiable in the sense of G\^ateaux functions.We describe relations between the mentioned monogenic functions and harmonic vectors in the three-dimensional real space and establish sufficient conditions for infinite monogeneity of functions. Unlike the classical complex analysis, it is done in the case where the validity of the Cauchy integral formula for monogenic functions remains an open problem.

Complex Functions and Residue Theorem

The features and connections of the mappings between complex number fields are thoroughly and methodically summarized and inferred by the complex function.The functional relationship of the mapping between the fields of complex numbers is the subject of study. A complex function uses a complex number as both its independent and dependent variable. The remainder theorem generalizes both the Cauchy's integral theorem and the Cauchy's integral formula. Numerous complex calculations are resolved by the theory of Complex functions and Residue theorem, which has a wide range of applications. This article begins by defining ideas and formulas, demonstrating how formulas connect to concepts, and outlining the procedure and approaches used by mathematicians when working with formulas. The current aspect and the signal aspect are two of the most intimate relationships between Complex Functions and daily living.

Complex number and its discovery history

The operations of complex numbers are the main subject of the mathematical analysis area known as complex analysis. It is also known as the theory of functions of a complex variable. The primary research topic in the field of complex analysis is holomorphic functions. These functions are defined on the complex plane, have differentiable properties, and allow for negative values. The residue theorem, the Cauchy integral formula, the Laurent series expansion, etc. are a few concepts, ideas, and methods that are commonly used in research. In particular, over the years, complex analysis in mathematics, physics, and engineering has been extensively used in algebraic geometry, fluid dynamics, quantum mechanics, and other related areas. Two Italian mathematicians, Girolamo Cardano and Raphael Bombelli, discovered complex numbers in the 16th century while attempting to solve a particular algebra, and Cauchy and Riemann extended it in the 19th century. This essay begin with investigation of arithmetic propertity of comlex numbers and then fully explores history development of complex numbers.

Relationship Between Cauchy Integral Theorem and Residue Theorems

Cauchy integral theorem belongs to an extremely important part of complex functions, which is a fundamental bridge, and people can derive Cauchy integral theorem from the residue theorem. Cauchy's integral theorem is generally applied in many higher mathematics, is an important theorem concerning path integrals of fully pure functions. It claims that if there are two different paths from a point to another point and the function is fully pure between these two different paths, then it can be derived that the two path integrals of the function are equal. It is widely believed that the a generalization of Cauchy integral theorem and Cauchy integral formula is just like the residue theorem, and the flexible use of the residue theorem can easily solve some difficult problems in complex functions. Therefore, this paper will use the definition and derivation process of Cauchy residue theorem and integral theorem to demonstrate the specific connection between them and their practical application.

Charaterization of Holomorphic function using power series

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.

The Application of Complex Variables Function and Residue Theorem

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.

Analytical Solution to DGLAP Integro-Differential Equation in a Simple Toy-Model with a Fixed Gauge Coupling

We consider a simple model for QCD dynamics in which DGLAP integro-differential equation may be solved analytically. This is a gauge model which possesses dominant evolution of gauge boson (gluon) distribution and in which the gauge coupling does not run. This may be N=4 supersymmetric gauge theory with softly broken supersymmetry, other finite supersymmetric gauge theory with a lower level of supersymmetry, or topological Chern–Simons field theories. We maintain only one term in the splitting function of unintegrated gluon distribution and solve DGLAP analytically for this simplified splitting function. The solution is found using the Cauchy integral formula. The solution restricts the form of the unintegrated gluon distribution as a function of momentum transfer and of Bjorken x. Then, we consider an almost realistic splitting function of unintegrated gluon distribution as an input to DGLAP equation and solve it by the same method which we have developed to solve DGLAP equation for the toy-model. We study a result obtained for the realistic gluon distribution and find a singular Bessel-like behavior in the vicinity of the point x=0 and a smooth behavior in the vicinity of the point x=1.