Cauchy’s Integral Theorem. Proof of Theorem 3.1. Cauchy’s Integral Formula

2017 ◽  
pp. 77-85
Author(s):  
Alexander Isaev
Keyword(s):  
Integral Formula ◽  
Integral Theorem ◽  
Cauchy’S Integral Formula ◽  
Cauchy’S Integral Theorem ◽  
Theorem Proof

scholarly journals Deriving Cauchy's Integral Formula Using Division Method

2019 ◽  
Vol 1 ◽  
pp. 259-264
Author(s):  
M Egahi ◽  
I O Ogwuche ◽  
J Ode
Keyword(s):  
Analytic Functions ◽  
Complex Analysis ◽  
Integral Formula ◽  
Integral Theorem ◽  
Cauchy’S Integral Formula ◽  
Cauchy’S Integral Theorem

Cauchy's integral theorem and formula which holds for analytic functions is proved in most standard complex analysis texts. The nth derivative form is also proved. Here we derive the nth derivative form of Cauchy's integral formula using division method and showed its link with Taylor's theorem and demonstrate the result with some polynomials.

scholarly journals Meromorphic or Holomorphic Completion of a Reinhardt Domain

1970 ◽  
Vol 38 ◽  
pp. 1-12 ◽  
Author(s):  
Eiichi Sakai
Keyword(s):  
Meromorphic Function ◽  
Power Series ◽  
Meromorphic Functions ◽  
Integral Formula ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Reinhardt Domain ◽  
Continuity Theorem ◽  
Cauchy’S Integral Formula ◽  
Long Time

In the theory of functions of several complex variables, the problem about the continuation of meromorphic functions has not been much investigated for a long time in spite of its importance except the deeper result of the continuity theorem due to E. E. Levi [4] and H. Kneser [3], The difficulty of its investigation is based on the following reasons: we can not use the tools of not only Cauchy’s integral formula but also the power series and there are indetermination points for the meromorphic function of many variables different from one variable. Therefore we shall also follow the Levi and Kneser’s method and seek for the aspect of meromorphic completion of a Reinhardt domain in Cn.

scholarly journals An Efficient Method to Find Solutions for Transcendental Equations with Several Roots

2015 ◽  
Vol 2015 ◽  
pp. 1-4 ◽  
Author(s):  
Rogelio Luck ◽  
Gregory J. Zdaniuk ◽  
Heejin Cho
Keyword(s):  
Null Space ◽  
Heat Conduction Problem ◽  
Hankel Matrix ◽  
Polynomial Equation ◽  
Transient Heat Conduction ◽  
Transcendental Equation ◽  
Integral Theorem ◽  
Transient Heat Conduction Problem ◽  
Cauchy’S Integral Theorem ◽  
Roots Of A Polynomial

This paper presents a method for obtaining a solution for all the roots of a transcendental equation within a bounded region by finding a polynomial equation with the same roots as the transcendental equation. The proposed method is developed using Cauchy’s integral theorem for complex variables and transforms the problem of finding the roots of a transcendental equation into an equivalent problem of finding roots of a polynomial equation with exactly the same roots. The interesting result is that the coefficients of the polynomial form a vector which lies in the null space of a Hankel matrix made up of the Fourier series coefficients of the inverse of the original transcendental equation. Then the explicit solution can be readily obtained using the complex fast Fourier transform. To conclude, the authors present an example by solving for the first three eigenvalues of the 1D transient heat conduction problem.

scholarly journals On the Surface Fields of a Small Circular Loop Antenna Placed on Plane Stratified Earth

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Mauro Parise
Keyword(s):  
Analytical Method ◽  
Integral Formula ◽  
Integral Representations ◽  
Loop Antenna ◽  
Circular Loop ◽  
Time Harmonic ◽  
Hankel Functions ◽  
Cauchy’S Integral Formula ◽  
Em Field ◽  
Surface Fields

An analytical method is presented which makes it possible to derive exact explicit expressions for the time-harmonic surface fields excited by a small circular loop antenna placed on the top surface of plane layered earth. The developed procedure leads to casting the complete integral representations for the EM field components into forms suitable for application of Cauchy’s integral formula. As a result, the surface fields are expressed as sums of Hankel functions. Numerical simulations are performed to show the validity and accuracy of the proposed solution.

Cauchy’s Integral Formula

2011 ◽  
pp. 111-115
Author(s):  
Ravi P. Agarwal ◽  
Kanishka Perera ◽  
Sandra Pinelas
Keyword(s):  
Integral Formula ◽  
Cauchy’S Integral Formula
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