The Cauchy Integral Formula

1990 ◽  
Vol 21 (4) ◽  
pp. 327
Author(s):  
David P. Kraines ◽  
Vivian Y. Kraines ◽  
David A. Smith
Keyword(s):  
Integral Formula ◽  
Cauchy Integral Formula ◽  
Cauchy Integral

scholarly journals An integral representation, complete monotonicity, and inequalities of the Catalan numbers

Filomat ◽  
2018 ◽  
Vol 32 (2) ◽  
pp. 575-587 ◽  
Author(s):  
Feng Qi ◽  
Xiao-Ting Shi ◽  
Fang-Fang Liu
Keyword(s):  
Integral Representation ◽  
Generating Function ◽  
Integral Formula ◽  
Cauchy Integral Formula ◽  
Catalan Numbers ◽  
Complete Monotonicity ◽  
Cauchy Integral ◽  
Complex Functions

In the paper, by virtue of the Cauchy integral formula in the theory of complex functions, the authors establish an integral representation for the generating function of the Catalan numbers in combinatorics. From this, the authors derive an alternative integral representation, complete monotonicity, determinantal and product inequalities for the Catalan numbers.

scholarly journals Quaternionic G-Monogenic Mappings in Em

2018 ◽  
Vol 12 ◽  
pp. 1-34
Author(s):  
Vitalii S. Shpakivskyi ◽  
Tetyana S. Kuzmenko
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Complex Variable ◽  
Analytic Functions ◽  
Integral Formula ◽  
Cauchy Integral Formula ◽  
Cauchy Integral ◽  
Partial Differential ◽  
Morera Theorem ◽  
Classical Integral

We consider a class of so-called quaternionic G-monogenic mappings associatedwith m-dimensional (m 2 f2; 3; 4g) partial differential equations and propose a description of allmappings from this class by using four analytic functions of complex variable. For G-monogenicmappings we generalize some analogues of classical integral theorems of the holomorphic functiontheory of the complex variable (the surface and the curvilinear Cauchy integral theorems,the Cauchy integral formula, the Morera theorem), and Taylor’s and Laurent’s expansions.Moreover, we investigated the relation between G-monogenic and H-monogenic (differentiablein the sense of Hausdorff) quaternionic mappings.

A Cauchy Integral Formula for Infrapolymonogenic Functions in Clifford Analysis

2020 ◽  
Vol 30 (2) ◽  
Author(s):  
Ricardo Abreu Blaya ◽  
Juan Bory Reyes ◽  
Arsenio Moreno García ◽  
Tania Moreno García
Keyword(s):  
Clifford Analysis ◽  
Integral Formula ◽  
Cauchy Integral Formula ◽  
Cauchy Integral

scholarly journals A simple spectral algorithm for recovering planted partitions

2017 ◽  
Vol 5 (1) ◽  
pp. 139-157 ◽  
Author(s):  
Sam Cole ◽  
Shmuel Friedland ◽  
Lev Reyzin
Keyword(s):  
Adjacency Matrix ◽  
Projection Operator ◽  
High Probability ◽  
Integral Formula ◽  
Cauchy Integral Formula ◽  
Cauchy Integral ◽  
Partition Model ◽  
Spectral Algorithms ◽  
Spectral Algorithm ◽  
Proof Of Correctness

Abstract In this paper, we consider the planted partition model, in which n = ks vertices of a random graph are partitioned into k “clusters,” each of size s. Edges between vertices in the same cluster and different clusters are included with constant probability p and q, respectively (where 0 ≤ q < p ≤ 1). We give an efficient algorithm that, with high probability, recovers the clusters as long as the cluster sizes are are least (√n). Informally, our algorithm constructs the projection operator onto the dominant k-dimensional eigenspace of the graph’s adjacency matrix and uses it to recover one cluster at a time. To our knowledge, our algorithm is the first purely spectral algorithm which runs in polynomial time and works even when s = Θ (√n), though there have been several non-spectral algorithms which accomplish this. Our algorithm is also among the simplest of these spectral algorithms, and its proof of correctness illustrates the usefulness of the Cauchy integral formula in this domain.

Cauchy Integral Formula on the Integration Path C on Singularity in Mechanics of Materials

2012 ◽  
Vol 502 ◽  
pp. 124-127
Author(s):  
Xue Feng Cao
Keyword(s):  
Path Integral ◽  
Singular Integrals ◽  
Function Analysis ◽  
Integral Formula ◽  
Cauchy Integral Formula ◽  
Integration Path ◽  
Cauchy Integral ◽  
Mechanics Of Materials ◽  
New Formula ◽  
Integrand Function

This paper is a combination of conditions and the knowledge of singular integrals, the integrand function analysis of the deformation, the singularity in the integral for a class on the path integral come up with a complex new formula for the solution,the formula can be used in these areas,such as mechanics of materials.

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