Angle between Harmonic functions I have a question about a practice prelim problem:
What is the angle between the curves $Re(z^3) = 1$ and $Re(z^3) = Im(z^3)$?  
Also, What is the angle between the curves $Re(z^3) = 0$ and $Re(z^3) = Im(z^3)$? 
So for the first part I believe that we can use conformal maps preserve angles so we are just finding the angle between $x=1$ and $x=y$ in the $x-y$ plane where $z^3 = w = x+iy$.  However, this only works for $f(z) =z^3 $ and $f'(z) \neq 0$.  Thus, we cannot use this in the second part.  Any ideas for the second part would be greatly appreciated.
Thanks!
 A: 
What is the angle between the curves $\operatorname{Re}(z^3)=1$ and $\operatorname{Re}(z^3)=\operatorname{Im}(z^3)$?

The problem is poorly phrased, since these sets are not curves. The first one is the disjoint union of three simple unbounded curves. The second one is the union of three lines passing through the origin. 
Your idea to use conformality is correct, though.  The lines $\operatorname{Re}(w)=1$ and $\operatorname{Re}(w)=\operatorname{Im}(w)$ cross at an angle $\pi/4$ and since $w=z^3$ is a conformal transformation outside of the origin, at every intersection of  $\operatorname{Re}(z^3)=1$ and $\operatorname{Re}(z^3)=\operatorname{Im}(z^3)$ the angle is $\pi/4$.

What is the angle between the curves $\operatorname{Re}(z^3)=0$ and $\operatorname{Re}(z^3)=\operatorname{Im}(z^3)$?

Again, I object to the statement. Each set is the union of three lines through $0$. The    common point of these sets is also $0$. There is an assortment of angles to choose from. The smallest of them is indeed $\pi/12$. There are also $\pi/3\pm \pi/12$, $2\pi/3\pm \pi/12$, and $\pi-\pi/12$ angles there (not counting those over $\pi$ radian).
