If you plot in the x-direction and the principal argument in the y-direction, then for the right half unit circle, won't the argument go from to ? So that's a range in the y direction from to . Now, the x-direction: when r is very small, then is very big and when r=1, then . Isn't that then a block from zero to infinity along the x-axis with the upper boundary at and lower one at ? And since , then it's that block except for the left side at x=0.