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+\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