关于椭圆积分

kivy

第二类椭圆积分加法公式是怎么推导的？

mathe

这个需要用到椭圆函数的性质，要看懂不容易

葡萄糖

\[ E(u+v,\color{gray}{k})=E(u,\color{gray}{k})+E(v,\color{gray}{k})-{\color{gray}{k}}^2\operatorname{sn}(u,\color{gray}{k})\operatorname{sn}(v,\color{gray}{k})\operatorname{sn}(u+v,\color{gray}{k}) \]
\[ \Pi(\color{gray}{n},u+v,\color{gray}{k})=\Pi(\color{gray}{n},u,\color{gray}{k})+\Pi(\color{gray}{n},v,\color{gray}{k})-\frac{1}{2}\ln\frac{1+{\color{gray}{k}}^2\operatorname{sn}(\color{gray}{n},\color{gray}{k})\operatorname{sn}(u,\color{gray}{k})\operatorname{sn}(u+v+a,\color{gray}{k})}{1-{\color{gray}{k}}^2\operatorname{sn}(\color{gray}{n},\color{gray}{k})\operatorname{sn}(u,\color{gray}{k})\operatorname{sn}(u+v-a,\color{gray}{k})} \]

葡萄糖

可以构造差函数来求导证明第二类椭圆积分加法公式