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

Hint: you may want to look here:https://math.stackexchange.com/questions/2126598/how-does-the-derivative-with-respect-to-the-complex-conjugate-even-make-sense — Sean Roberson, May 28 '23 at 06:16
For some basic information about writing mathematics at this site see, e.g., here, here, here and here. — Another User, May 28 '23 at 06:53

score 0 · Answer 1 · answered May 28 '23 at 06:22

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

Complex Analysis based on Harmonic Function and Partial Derivatives

1 Answers1