\noindent{\bf Q1)}

$$w = {i-z\over i+z} = -{z-i\over z+i} = e^{i\pi}{z-i\over z-\overline i}
.$$

This, therefore, puts the upper half-plane into a circle of radius 1.

\noindent{\bf Q2)}

$$w = {z-1\over z+1} = {x+iy-1\over x+iy+1} = {(x-1+iy)(x+1-iy)\over
(x+1)^2+y^2} = {x^2 - 1 + 2iy + y^2\over (x+1)^2+y^2}.$$

$$u = {x^2 - 1 + y^2\over (x+1)^2 + y^2},\quad v = {2y\over(x+1)^2+y^2}
.$$

$$y > 0 \to v > 0,$$
because, with $x = -1,$ $v = 2/y,$ which maps positive y to positive v.

\noindent{\bf Q3)}

They are different because the transformation in Q2 isn't the same as
Q1. It could be rewritten as a rotation of $pi/2$ then a transform
similar to Q1, but this does not correspond.

\bye
