A Laurent Expansion and Residue Theorems of k-Regular Functions in Clifford Analysis

In this context, we mainly study the behavior in the neighborhood of finite singular points for k-regular functions in R1^n with values in R0、n. We get a Laurent expansion of them in an open set, prove its uniqueness, give the definitions of k-poles, isolated and essential singular points and removable singularity, discuss some properties, and further obtain the residue theorems.

Application of the Residue Theorem to Trigonometric Sum Identities

By evaluating a contour integral with the Cauchy residue theorem, we prove a general summation formula on trigonometric sum, which contains several interesting trigonometric identities as special cases.

Residue Theorem based soft sliding mode control for wind power generation systems

This paper proposes a residue theorem based soft sliding mode control strategy for a permanent magnet synchronous generator(PMSG)based wind power generation system(WPGS),to achieve the maximum energy conversion and improved in the system dynamic performance.The main idea is to set a soft dynamic boundary for the controlled variables around a reference point.Thus the controlled variables would lie on a point inside the boundary.The convergence of the operating point is ensured by following the Forward Euler method.The proposed control has been verified via simulation and experiments,compared with conventional sliding mode control(SMC)and proportional integral(PI)control.

The Number of Representations of an Integer as a Sum of Eight Triangular Numbers

An infinite product is expanded to Laurent series by residue theorem.Applying this expansion, the formula for the number of representations of an integer as a sum of eight triangular numbers is easily obtained.

A high performance frequency offset estimator for OFDM

This paper proposes a simple method to enlarge the estimation range of conventional carrier frequency offset (CFO) estimation methods based on correlations among the identical parts of the preamble. A novel preamble is designed, which is composed of one regular OFDM training block with even numbers of identical parts and one irregular OFDM training block with odd numbers of identical parts. The initial estimates obtained over the two training blocks are next exploited to jointly estimate the CFO. By elaborately selecting the numbers of identical parts for the two training blocks, the proposed CFO estimator can estimate frequency offset over tens of the subcarrier spacing. Simulation results showed that the proposed CFO estimator satisfies the estimate range requirement for the practical OFDM systems, while achieving a very good estimate performance.

A Contour Integral Method for Linear Differential Equations in Complex Plane

This paper presents the contour integral method for solving the linear constant coefficient ordinary differential equations in complex plane,and obtains the uniform expressions of the general solutions.Firstly,by using Residue Theorem,the general form of the contour integral representation for the homogeneous complex differential equation is obtained,which can be degenerated to classical results in real line.As for inhomogeneous complex differential equations with constant coefficients,we construct the integral expression of the particular solution for any continuous forcing term,and give rigorous proof via Residue Theorem.Thus the general solutions of inhomogeneous complex differential equations are also given.The main purpose of this paper is to give a foundation for a complete theory of linear complex differential equations with constant coefficients by a contour integral method.The results can not only solve the inhomogeneous complex differential equation well,but also explain the forms that are difficult to be understood in the classical solutions.