mr. vodka said:Oh, does a conformal map have to be bijective?
(in that case: the inverse would be bounded ==(Liouville)==> constant, contradiction)
(Now I'm also assuming differentiability! So does a conformal map have to be bijective AND/OR analytical?)
A conformal map is a function that preserves angles between intersecting curves at each point. In other words, it preserves the shape of curves, but not necessarily their size or orientation.
Proving the non-existence of a conformal map is important because it helps us understand the limitations of complex analysis. By identifying cases where a conformal map does not exist, we can gain a deeper understanding of the properties and behaviors of complex functions.
The non-existence of a conformal map can be proven using various techniques, including the Cauchy-Riemann equations, the Schwarz lemma, and the maximum modulus principle. These methods involve analyzing the behavior of complex functions and their derivatives to determine if they satisfy the conditions for conformality.
Yes, a function can be conformal in some regions but not others. In order for a function to be conformal, it must satisfy the Cauchy-Riemann equations at every point in the region of interest. If there are points where these equations are not satisfied, the function will not be conformal in those regions.
Conformal maps have many practical applications in fields such as engineering, physics, and computer graphics. They are used to model and analyze complex systems, such as fluid flow, electromagnetic fields, and 3D objects. They are also used in map-making and navigation, as they preserve the shape of land masses and coastlines.