如何证明一个二元调和函数的任意阶偏导数也是调和函数?

一个二元调和函数 $u(x,y)$ 满足以下条件：$$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$$

现在我们来证明一个二元调和函数的任意阶偏导数也是调和函数。我们首先考虑二阶偏导数：$$\frac{\partial^2}{\partial x^2}\Big(\frac{\partial u}{\partial x}\Big)+\frac{\partial^2}{\partial y^2}\Big(\frac{\partial u}{\partial x}\Big)=\frac{\partial}{\partial x}\Big(\frac{\partial^2 u}{\partial x^2}\Big)+\frac{\partial}{\partial y}\Big(\frac{\partial^2 u}{\partial x\partial y}\Big)$$

由于 $u(x,y)$ 是调和函数，所以 $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$，因此：$$\frac{\partial^2}{\partial x^2}\Big(\frac{\partial u}{\partial x}\Big)+\frac{\partial^2}{\partial y^2}\Big(\frac{\partial u}{\partial x}\Big)=\frac{\partial}{\partial x}\Big(\frac{\partial^2 u}{\partial x^2}\Big)+\frac{\partial}{\partial y}\Big(\frac{\partial^2 u}{\partial x\partial y}\Big)=\frac{\partial}{\partial x}\Big(-\frac{\partial^2 u}{\partial y^2}\Big)+\frac{\partial}{\partial y}\Big(\frac{\partial^2 u}{\partial x\partial y}\Big)=0$$

因此，$\frac{\partial^2}{\partial x^2}\Big(\frac{\partial u}{\partial x}\Big)+\frac{\partial^2}{\partial y^2}\Big(\frac{\partial u}{\partial x}\Big)=0$，也就是说，$\frac{\partial^2 u}{\partial x^2}$ 是一个调和函数。

同理，我们可以证明任意阶偏导数都是调和函数。因此，一个二元调和函数的任意阶偏导数也是调和函数。