椭圆中带有狄利克雷边界条件的本原拉普拉斯特征值
The fundamental Laplacian eigenvalue of the ellipse with Dirichlet boundary conditions
AI总结：
研究椭圆中狄利克雷拉普拉斯特征值随偏心率变化的特性，通过扩展现有解析展开式并结合高精度计算与LLL算法获得高精度系数。
摘要：
在本项目中，我考察了椭圆内拉普拉斯算子的最低狄利克雷特征值作为偏心率的函数。两种现有的解析展开式被扩展：在接近圆（偏心率接近零）时，向马尔库特级数添加九项；在接近无限条带（偏心率接近单位）时，向渐近展开式添加四项。过去，其他方法如边界变分技术已被用于解决此问题，但我采用了一种不同的方法——不仅提供了现有结果的独立验证，还扩展了这些结果。我的起点是为选定的偏心率值进行高精度计算，然后将这些数据拟合到适当的参数多项式中，得到高精度系数，再通过受先前结果引导的LLL整数关系算法进行处理。
英文摘要：
In this project, I examine the lowest Dirichlet eigenvalue of the Laplacian within the ellipse as a function of eccentricity. Two existing analytic expansions of the eigenvalue are extended: Close to the circle (eccentricity near zero) nine terms are added to the Maclaurin series; and near the infinite strip (eccentricity near unity) four terms are added to the asymptotic expansion. In the past, other methods, such as boundary variation techniques, have been used to work on this problem, but I use a different approach -- which not only offers independent confirmation of existing results, but extends them. My starting point is a high precision computation of the eigenvalue for selected values of eccentricity. These data are then fit to polynomials in appropriate parameters yielding high-precision coefficients that are fed into an LLL integer-relation algorithm with forms guided by prior results.
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