Question

Show that d dz (z 2 z ∗ ) does not exist anywhere.

Answer 1

Writing z = x + iy, we have 
z^2 * z_bar 
= (x + iy)^2 * (x - iy) 
= ((x^2 - y^2) + 2ixy) * (x - iy) 
= (x^3 + xy^2) + i(x^2 y + y^3). 

Hence, u = x^3 + xy^2 and v = x^2 y + y^3. 

Now, we check the Cauchy-Riemann equations. 
u_x = v_y ==> 3x^2 + y^2 = x^2 + 3y^2 ==> x^2 = y^2 
u_y = -v_x ==> 2xy = -2xy ==> xy = 0. 

So these equations are satisfied precisely when x = y = 0. 
==> z = x + iy = 0 + 0i = 0. 

(Moreover, f '(0) = u_x (0, 0) + i v_x (0, 0) = 0.) 
---- 
Double check (at z = 0): 
f '(0) = lim(Δz→0) (f(0 + Δz) - f(0))/Δz 
........= lim(Δz→0) [(Δz)^2 * (Δz)_bar) - 0]/Δz 
........= lim(Δz→0) (Δz) * (Δz)_bar 
........= 0.