semicircle centered at (0,0) with radius 1 in the upper half of the plane.
"geometric method" : Arg[(z-1)/(z+1)] = pi/2
so Arg(z-1) - Arg(z+1) = pi/2
so the angle in between the line from 1 to z and the line from -1 to z is pi/2
so it's a circle, but the direction forces it to be in the upper half.
rough symbolic proof: arg(x) = pi/2 is same as saying that x = ki, where k is a positive number.
so (z-1)/(z+1) = ki
z-1 = kzi +ki
z(1 - ki) = 1 +ki
z = (1 + ki)/(1 - ki)
then rearrange
z = (1+ki)^2 / (1+k^2)
z = (1-k^2)/(1+k^2) + [2k/(1+k^2) ] i
and you can verify that y>0 and
x^2+y^2 = 1
etc.
