I am have a problem resolving an equation with complexe numbers on their algebrique form.
Here is it :
For all complexe z = x + iy (x and y real and different from -i) we have :
Z = (z - 1) / (iz + 1)
Find Re(Z) and Im(Z) by x and y.
I already have the answer : Re(Z) = (x+y-1)/((1-y)^2 + x) and Im(Z) = (-x^2-y^2+x+y)/((1-y)^2+x^2, but I don't know how to get to it.
I've tried to resolv it with but it doesn't seems to work :
Z = (z-1)/(iz+1)
= (x+iy-1)/(ix-y-1) , because i^2 = -1
And Re(Z)=0 <=> Z=
So : Z- = 0
(x+iy-1)/(ix-y-1) - (x-iy-1)/(ix+y-1) = 0
((x+iy-1)(ix+y-1)-(x-iy-1)(ix-y-1))/((ix-y-1)(ix+y-1)) = 0
but : (ix^2-ix^2+iy^2-iy^2-ix+ix-iy-iy+xy-xy+xy-xy-x+x-y+y+1-1) = 0
I still don't know what is Re(Z) I must have done a mistake and I don't know where or how to find the answer.
Help me understand this problem! Please.