Conformal Mapping: Is Non-Analytic Point Conformal?

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SUMMARY

The discussion centers on the conditions for conformal mapping in relation to analytic functions. It establishes that a mapping w=f(z) is conformal at a point z0 if the derivative df/dz at z0 is non-zero. The conversation highlights that if df/dz does not exist at z0, the point cannot be considered conformal, as this indicates the function is non-analytic. Therefore, the theorem regarding conformal mappings strictly applies to analytic functions only.

PREREQUISITES
  • Understanding of analytic functions and their properties
  • Knowledge of complex differentiation
  • Familiarity with the concept of conformal mappings
  • Basic grasp of complex variables
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  • Study the properties of analytic functions in complex analysis
  • Learn about the implications of non-analytic points in complex mappings
  • Explore the relationship between differentiability and conformality
  • Investigate examples of conformal mappings in various contexts
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Students of complex analysis, mathematicians focusing on conformal mappings, and anyone interested in the properties of analytic functions.

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A theorem I took down in class says:

Consider the analytic function f(z). The mapping w=f(z) is conformal at the point z0 if and only if df/dz at z0 is non-zero.

However, if df/dz does not exist at that point z0, is that point still a conformal mapping? That would make the function non-analytic and this wouldn't apply right?
 
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Right. An analytic function is differentiable everywhere by definition.
 

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