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
= (x-iy-1)/(i(x-iy)+1) no, because the denominator contains another
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.
Now you could use the complex conjugate of the denominator to make it real, so multiply by