# Thread: Mapping of complex numbers

1. ## Mapping of complex numbers

Hi, I'm not sure of the answer to this question (or even if belongs in this forum, apologies):

Let ϕ: C → C be the rotation of angle θ around the point $z_0$ ∈ C.
Express ϕ(z ) in terms of z , $z_0$ and e^iθ.

I know that z = (r, θ) in polar coordinates means z=re^iθ , but I don't know how to express ϕ(z). My best guess is ϕ(z) = $z_0$e^iθ but I don't really understand how to visualize this rotation or where z is.

Thanks so much for any help.

2. Originally Posted by kimberu
Hi, I'm not sure of the answer to this question (or even if belongs in this forum, apologies):

Let ϕ: C → C be the rotation of angle θ around the point $z_0$ ∈ C.
Express ϕ(z ) in terms of z , $z_0$ and e^iθ.

I know that z = (r, θ) in polar coordinates means z=re^iθ , but I don't know how to express ϕ(z). My best guess is ϕ(z) = $z_0$e^iθ but I don't really understand how to visualize this rotation or where z is.
Do it in three steps: first, translate the plane so that $z_0$ comes to lie at the origin, 0. Second, rotate the plane around the origin by $\theta$, and finally shift the origin back to $z_0$.
So the first step is $z\mapsto z-z_0$, combined with the second, you have $z\mapsto (z-z_0)e^{i\theta}$, and combined with the third, you get $z\mapsto \phi(z) := (z-z_0)e^{i\theta}+z_0$.