Grufey
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Hello!, I was studing the conformal maps in complex analysis, I don't understand this definition:
Definition: A map f:A\rightarrow\mathbb{C} is called conformal at z_0 if there exist a \theta\in[0,2\pi] and r>0 such that for any curve \gamma(t) which is differentiable at t=0, for which \gamma(t) \in A and \gamma(0)=z_0, and which satisfisfies \gamma\prime(0)\neq0 the curve \sigma(t)=f(\gamma(t)) is differentiable at t=0 and, setting u=\sigma\prime(0) and v=\gamma\prime(0), we have \left|u\right|=r\left|v\right| and \arg u =\arg v + \theta (\mod 2\theta)
I only know about the conformal maps, that the angle between the curves after the transform is equal to the before of the transformation. But I cannot find the relation, with the definition. I think that I don't undertand the definition.
Thanks
Definition: A map f:A\rightarrow\mathbb{C} is called conformal at z_0 if there exist a \theta\in[0,2\pi] and r>0 such that for any curve \gamma(t) which is differentiable at t=0, for which \gamma(t) \in A and \gamma(0)=z_0, and which satisfisfies \gamma\prime(0)\neq0 the curve \sigma(t)=f(\gamma(t)) is differentiable at t=0 and, setting u=\sigma\prime(0) and v=\gamma\prime(0), we have \left|u\right|=r\left|v\right| and \arg u =\arg v + \theta (\mod 2\theta)
I only know about the conformal maps, that the angle between the curves after the transform is equal to the before of the transformation. But I cannot find the relation, with the definition. I think that I don't undertand the definition.
Thanks