"Definition: A map ƒ: A ⊂ ℂ→ ℂ is called conformal at z0, if there exists an angle θ ∈[0,2Pi) and an r > 0 such that for any curve γ(t) that is differentiable at t=0, for which γ(t)∈ A and γ(0)= z0, and that satisfies γ ' ≠0, the curve σ(t) = ƒ(γ(t)) is differentiable at t=0 and, setting u = σ'(0) and v = γ'(0), we have |u| = r |v| and arg(u)= arg(v)+ θ (mod 2π). A map is called conformal when it is conformal at every point." I am having a lot of trouble understanding this definition. First, what exactly is θ? Is it related v=γ'(0) or u=σ '(0)? Relative to what is θ being measured, or what two rays is θ being defined by? Also, how do I determine arg(u) and arg(v)? Why must γ(0) be z0? Why must we have σ(t) and γ(t) to begin with? I'm having a lot of trouble understanding this definition and what it exactly means. Could someone please explain and elaborate the statements made in this definition. I've read it plenty of times, but it just doesn't make any sense - I've been wrestling with this for sometime now. I looked at other sources, but they don't seem to be as detailed as this definition - which seems to offer more insight into the concept once understood. I kind of understand that the general point being made is that ƒ is a rotation through an angle θ and magnitude |ƒ ' (z0)|, however, I don't see how the definition shows this.I also don't understand where equation arg(u)= arg(v)+ θ comes from. Is ƒ a sort of matrix transformation with the properties that rotates and changes magnitudes? This definition was taking out of Marsden and Hoffman's Basic Complex Analysis, 3rd edition, definition 1.5.6 - in case you happen to have the book.