This paper studies that the Vector Valued M analytic function with range in a Banach space which is defined by the regular solutions of the elliptic system f x+Mf y=0,where M is an m×m constant matrix without real eigenvalues and f is a m×q matrix.

The vector-valued generalized M-analytic function with rauge in a Banach space is defined by the solution of the elliptic system Lf =fx+Mfy+EPfy = 0, where M is an m×m constant matrix with no real eigenvalues, f is an m×q matrix, E is a constant nilpotent m×m matrix satisfying Er = 0(r≥2), P is a variable m×m matrix in the space Ha.

On the deducing and the teaching of Cauchy-Rieman equations of analytic function;

Some properties of p-valents analytic functions with negative coefficients;

A sufficient and necessary condition of the analytic function with the higher-order derivative;

By mirror image method and regularity of analytical function,the expression of the electric potential and intensity of a line charge within a thin cylindrical conductor are obtained,and the equations of the equipotential lines and the electric field lines are obtained.

Based on relationship between analytical functions and Bezier curves,the numerical method of conversion between them was presented.

Also, an expression formula for this analytical function is obtained.

A subclass of p-valent analytic functions defined by Ruscheweyh derivatives;

On a subclass of analytic functions with negative coefficients;

Properties of analytic functions defined by Noor integral operator;

An analytical trial function method (ATF) was developed based on the quadrilateral area coordinate system (QAC-II) for plane elements (ACATF).

Based on the analytical trial functions,two 5-node membrane elements named ATFM5-I and ATFM5-II were proposed.

Based on the analytical trial functions,a 4-node 8 degrees-of-freedom generalized conforming plane element with internal parameters is developed in this paper.