In a microwave discharge, the profile of electron number density inside a cylindrical plasma with elliptic cross section is analytically calculated. Using separation of variables way for the electron continuity equation (as function of position and time) and so solving it, the electron number density profile is calculated. It is shown that the profile of electron number density in a cylindrical plasma with elliptical cross section is in the form of Mathieu functions. The characteristic diffusion length for the elliptical cylinder plasma is obtained, and it is seen that it is different with the characteristic diffusion length for the circular cylinder plasma. In the steady state, the normalized density profiles of lowest even and odd modes are plotted. Furthermore, the components of normalized electron number density gradient are plotted. In the limit that plasma cylinder with elliptical cross section being converted to plasma cylinder with circular cross section, the obtained results are confirmed.

在微波放电中，解析地计算出具有椭圆形横截面的圆柱形等离子体内部的电子数密度分布。使用电子连续性方程的变量分离方法（作为位置和时间的函数）并求解，可以计算出电子数密度分布。结果表明，在具有椭圆形横截面的圆柱形等离子体中，电子数密度的分布呈Mathieu函数形式。获得了椭圆形等离子体的特征扩散长度，并且发现其与圆柱形等离子体的特征扩散长度不同。在稳态下，绘制了最低偶数和奇数模式的归一化密度分布图。此外，绘制了归一化电子数密度梯度的成分。