论文标题
花生谐波扩展，用于扁平环坐标中拉普拉斯方程的基本解决方案
Peanut harmonic expansion for a fundamental solution of Laplace's equation in flat-ring coordinates
论文作者
论文摘要
我们得出了Laplace方程在三维欧几里得空间中的扁圆环坐标中的基本解的扩展。这种膨胀是一系列双系列功能的产品，它们在坐标表面的内部和外部是谐波的，这些表面是花生形的，与扁平环的表面正交。这些内部和外部花生谐波功能以拉梅 - 瓦格林功能表示。使用基本解决方案的扩展，我们根据lamé-wangerin函数中的无限序列为方位角傅立叶分量的添加定理，以奇数半数的legendre函数作为无限序列。对于三种Lamé-Wangerin函数的产物，我们还取得了第二种典型功能的整体身份。在限制情况下，我们获得了球形坐标中基本解决方案的扩展。
We derive an expansion for the fundamental solution of Laplace's equation in flat-ring cyclide coordinates in three-dimensional Euclidean space. This expansion is a double series of products of functions that are harmonic in the interior and exterior of coordinate surfaces which are peanut shaped and orthogonal to surfaces which are flat-rings. These internal and external peanut harmonic functions are expressed in terms of Lamé-Wangerin functions. Using the expansion for the fundamental solution, we derive an addition theorem for the azimuthal Fourier component in terms of the odd-half-integer degree Legendre function of the second kind as an infinite series in Lamé-Wangerin functions. We also derive integral identities over the Legendre function of the second kind for a product of three Lamé-Wangerin functions. In a limiting case we obtain the expansion of the fundamental solution in spherical coordinates.
