1)  Elliptic/ Quasi-Elliptic function

2)  Quasi-elliptic function

1.
A general design method of quasi-elliptic function bandpass filters(BPFs) is introduced,and an example of an S-band asymmetric quasi-elliptic function filter coaxial BPF with one transmission zero is given.

2.
Recent years many results have been obtained on the design of HTS filter,and our work focused on the design of HTS filter based on quasi-elliptic function.

3.
The quasi-elliptic function filter with one transmission zero has the better selectivity than the general Chebyshev filter.

3)  elliptic function

1.
By using the elliptic function and conformal transformation theory,a close form solution to this problem is obtained.

2.
Phase plane properties of an electron in Wiggler field are analysed by using Jacobian elliptic function and elliptic integral based on the pendulum equation for FEL.

3.
A class of new doubly periodic wave solutions for(2+1)-dimensional breaking soliton equation are obtained by introducing appropriate Jacobi elliptic function and Weierstrass elliptic function in the general solution(contains two arbitrary functions)got by means of multilinear variable separation approach for(2+1)-dimensional breaking soliton equation.

4)  Jacobi elliptic functions
Jacobi椭圆函数
1.
Some new exact solutions of the Jacobi elliptic functions of NLS equation;

2.
Using Jacobi elliptic functions expansion method and a modified Hyperbolic-Tan function method,homogenous balancing method,construct the exact solution of nonlinear evolution equations;then using Mathematica software,the solitary wave solutions of the kind of nonlinear evoluation equations are obtained successfully.

3.
By using Mathematica and the F-expansion method recently proposed on the base of analogic method,homogeneous balance method and Jacobi method,the double periodic wave solutions expressed by Jacobi elliptic functions for the(n+1)-dimensional Sinh-Gordon equation .

5)  Jacobi elliptic function
Jacobi椭圆函数
1.
Exact solutions of jacobi elliptic function for boussinesq equation;
Boussinesq方程的Jacobi椭圆函数精确解
2.
Jacobi elliptic function envelope solutions of nonlinear Schringer equation;

3.
A solution of a nonlinear simple pendulum using Jacobi elliptic function;

6)  Jacobian elliptic function
Jacobi椭圆函数
1.
A coupled KdV system was solved by using the generalized Jacobian elliptic function expansion method.

2.
Its exact trave l ing wave solutions, which included rational form solutions, solitary wave soluti ons, triangle function periodic solutions, polynomial type Jacobian elliptic fun ction periodic solutions and fractional type Jacobian elliptic function periodic solutions, were given.

 椭圆函数elliptic function   在有限复平面上亚纯的双周期函数。所谓双周期函数是指具有两个基本周期的单复变函数，即存在ω1，ω2两个非0复数，，而对任意整数n，m，有f（z＋nω1＋mω2）＝f（z）  ，于是｛nω1＋mω2｜n，m为整数｝构成f（z）的全部周期，在复平面上任取一点a，以a，a＋ω1，a＋ω1＋ω2，a＋ω2为顶点的平行四边行的内部  ，再加上两个相邻的边及其交点，这样构成的一个半开的区域称为f（z）的一个基本周期平行四边形，将它平行移动nω1＋mω2，当n，m取遍所有整数时，即得一覆盖整个复平面的周期平行四边形网，f（z）在每一个周期平行四边形中的性质都和它在基本周期平行四边形中的一样。在基本周期平行四边形中，f（z）有以下性质：非常数椭圆函数一定有极点，且极点留数之和必为零，因而不可能只有一个一阶极点，有n个极点的椭圆函数称为n阶椭圆函数  ，它在基本周期平行四边形内取任一值n次，即对任意复数A，f（z）－A在基本周期平行四边形内有且仅有n个零点  ，且f（z）的零点之和与极点之和的差必等于一个周期。   在以上性质的规范下  ，有两大类重要的椭圆函数：①魏尔斯特拉斯-δ函数。它表作，其中ω＝2nω1＋2mω2，Σ'表n，m取遍全部整数之和  ，但要除去ω＝0的情形。这是一个二阶椭圆函数，在周期平行四边形中，仅有一个ω是二阶极点，ω＝δ（z）满足微分方程（ω′）2＝4ω3－g2ω－g3，其中g2=60Σ'g3=140Σ'，由此可见ω＝δ（z）是的反函数，右边的积分称为椭圆积分。可以证明，所有的椭圆函数都可以用δ（z）函数来表示  ，而每一个椭圆函数都一定满足一个常系数一阶的代数微分方程。②雅可比椭圆函数。它定义为椭圆积分的反函数  ，记作ω＝J（z），J（z）的基本周期平行四边形是一个矩形  ，其基本周期是4K与2iK′，此处,,其二阶极点为iK′，而k是一个实常数。