What is the normal form of the Euclidean transformation Tzei

What is the normal form of the Euclidean transformation Tz=e^i z+b

Solution

z is of the form x+iy=r(cost + isint) where r = (x^2+y^2)^(1/2)

where t = arctan y/x

T(z) = (cost + isint )(r(cost + isint))+b

= r(cos2t+isin2t) +b

Hence transformation is doubling the argument by 2 and increase z by b units.