What is the solution of arg(e^z), where z=x + yi. Find not only the solution, but also the proof.

arg(e^z)
=arg(e^(x+yi))
=arg(e^x) + arg(e^yi)
=(0) + arg(cos(y) + i(sin(y))) Note:arg(e^x)= 0
=tan-1(sin(y)/cos(y)) Note:cos(y) is the horizontal component
=tan-1(tan(y)) and sin(y) is the vertical component
=y

here's a picture i drew to aid you in your understanding.

(this is the edited one to fit with the background of my blog, the original one was drawn with black lines against a white background. this is edited using photoshop.)