Proof for real part of analytic function - Complex Analysis

Question

wondering if I could get some help with the question below;

Show that if $u(x,y)$ is the real part of an analytic function, $\ (xu_x + yu_y)$ is also the real part of that function. Also, what analytic function is it the real part of, if $Re(f(z)) = u?$

Thanks in advance!

Answer 1

Let $f(z)=u+i\,v$. Then $$ f'(z)=u_x+i\,v_x=u_x-i\,u_y. $$ Now compute $$ z\,f'(z)=? $$