AI 中文总结
研究拉梅超椭圆周长的精确解析表示，通过高斯超几何函数项组成的两个级数分支表示，正分支收敛，负分支阿贝尔可和，公式在轴置换下不变，该族在不同\(s\)值下插值，\(s = 1\)时对应周长最短的菱形。
AI 中文摘要
我们推导了度数\(s>0\)的拉梅超椭圆周长的精确解析表示。结果用由高斯超几何函数项组成的两个级数定义的分支表示：\(0<s<1\)时为负分支，\(s>1\)时为正分支。正分支的收敛条件由莱布尼茨判别法得出；负分支虽通常发散，但阿贝尔可和。该公式在轴置换下不变，符合半轴互换的对称性。随着\(s\)变化，该族在拉梅十字和矩形之间插值，\(s = 1\)时对应菱形，是族中周长最短的过渡曲线。
英文摘要
We derive exact analytic representations for the perimeter of a Lamé superellipse of degree $s>0$. The result is expressed in terms of two branches defined by series whose terms are Gauss hypergeometric functions: a negative branch for $0<s<1$ and a positive branch for $s>1$. For the positive branch, the convergence condition follows from the Leibniz test; the negative branch, although divergent in the ordinary sense, is shown to be Abel-summable. Consistently with the symmetry under interchange of the semi-axes, the formula is invariant under axis permutation. As $s$ varies, the family interpolates between the Lamé cross and the rectangle, while the case $s=1$ corresponds to the rhombus, which acts as the transition curve with the shortest perimeter within the family.
Comments29 pages, 3 figures
