Complex Analysis based on Harmonic Function and Partial Derivatives

Question

Let u(x, y) be a harmonic function Show that u(x, y) satisfy the differential equation (∂²u)/∂z∂z* = 0

Where z is any complex number and z is it's conjugate*

My process is Since u(x, y) is harmonic, thus Uxx + Uyy = 0

Now I am just stuck at Expanding (∂u)/∂z∂z*

Please help

Answer 1

First observe $x=\frac{z+\bar{z}}{2}$ & $y=\frac{z-\bar{z}}{2i}$

Now, $\frac{\partial u}{\partial \bar{z}}$ =$\frac{\partial u}{\partial x}\times\frac{\partial x}{\partial \bar{z}}$ +$\frac{\partial u}{\partial y}\times\frac{\partial y}{\partial \bar{z}}$

Similarly use chain rule in next stage to get answer 0