Confusing Conformal Maps: Exploring Complex Analysis

[Grufey]

Hello!, I was studing the conformal maps in complex analysis, I don't understand this definition:

Definition: A map [tex]f:A\rightarrow\mathbb{C}[/tex] is called conformal at [tex]z_0[/tex] if there exist a [tex]\theta\in[0,2\pi][/tex] and [tex]r>0[/tex] such that for any curve [tex]\gamma(t)[/tex] which is differentiable at [tex]t=0[/tex], for which [tex]\gamma(t) \in A[/tex] and [tex]\gamma(0)=z_0[/tex], and which satisfisfies [tex]\gamma\prime(0)\neq0[/tex] the curve [tex]\sigma(t)=f(\gamma(t))[/tex] is differentiable at [tex]t=0[/tex] and, setting [tex]u=\sigma\prime(0)[/tex] and [tex]v=\gamma\prime(0)[/tex], we have [tex]\left|u\right|=r\left|v\right|[/tex] and [tex]\arg u =\arg v + \theta (\mod 2\theta)[/tex]

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

 

[jostpuur]

[tex]\left|u\right|=r\left|v\right|[/tex]

This says that the tangent vector of the curve is scaled by some number that does not depend on the direction of the tangent vector.

[tex]\arg u =\arg v + \theta (\mod 2\theta)[/tex]

This says that f rotates the tangent vector by some constant angle.

Suppose you had two curves [itex]\gamma_1[/itex] and [itex]\gamma_2[/itex], so that [itex]\gamma_1(0)=\gamma_2(0)[/itex], their tangent vectors [itex]u_1:=\gamma'_1(0)[/itex] and [itex]u_2:=\gamma'_2(0)[/itex], and tangent vectors of the image paths [itex]v_1:=(f\circ\gamma_1)'(0)[/itex] and [itex]v_2:=(f\circ\gamma_2)'(0)[/itex].

The result

[tex]
\arg u_2 - \arg u_1 = \arg v_2 - \arg v_1\quad\mod\;2\pi
[/tex]

comes quite easily, and that is what the angle preserving means.

 

[tacman]

for your question and for sharing your confusion about the definition of conformal maps in complex analysis. This topic can be quite challenging and it is not uncommon to have difficulty understanding the various definitions and concepts involved.

To simplify the definition, a conformal map is a function that preserves angles between curves. In other words, if two curves intersect at a certain angle in the domain, the transformed curves will also intersect at the same angle in the range.

The definition you provided is a bit more technical and formal, but it essentially means that for a map to be conformal at a point z_0, there must exist a specific angle and a radius of a circle centered at z_0, such that when the map is applied to a curve passing through z_0, the angle between the tangent vectors of the original curve and the transformed curve is the same as the angle between the tangent vectors of the original curve and the circle.

To better understand this definition, it might be helpful to look at some examples of conformal maps and see how they preserve angles. You can also try working through some practice problems to solidify your understanding. Additionally, seeking help from a professor or a tutor can also be beneficial in clarifying any confusion you may have.

Keep practicing and don't get discouraged, complex analysis can be challenging but with persistence and effort, you will eventually grasp the concept of conformal maps. Good luck!