I have the Mobius transformation T(z)=z/(z-(1+i)), how do I find T(R), T(iR), T(K_1) and T(K_2) for K_1={z; |z-1|=1}, K_2={z;|z-i|=1}?
Thanks.
with z= x+ iy, any z in R (real numbers) can be written as x+ i0 so T(z)= x/(x-(1+i))= x/((x-1)-i). "Rationalize the denominator" by multiply numerator and denominator by (x- 1)+ i: T(z)= (x(x-1)+ ix)/((x-1)^2+ 1) which has real part u= x(x-1)/((x-1)^2+ 1)= (x^2- x)/(x^2- 2x+ 2) and imaginary part v= x/((x-1)^2+ 1)= x/(x^2- 2x+ 2). Can you find a relation between u and v? Does the fact this is a Mobius transformation help you?