we have w = a + bi = z/(1+ zi) = (x+yi)/(1+(x+yi)i) = (x+yi)/(1-y+xi)
Multiply top and bottom by (1-y-xi) then
a+bi = (x+yi)(1-y-xi)/((1-y)2 + x2 ) = (x + (y - x2 -y2)i)/((y-1)2 + x2 )
= x/((y-1)2 +x2 ) - i(x2 + y2 -y)/((y-1)2 +x2 )
Equate real and imaginary parts:
a = x/((y-1)2 +x2 )
b = -(x2 + y2 -y)/((y-1)2 +x2 )