SUMMARY
Every conformal self-map of the complex plane C is expressed in the form f(z) = az + b, where a ≠ 0. The proof hinges on demonstrating that the isolated singularity of f(z) at infinity is a simple pole. To establish this, one must show that infinity is neither a removable singularity nor an essential singularity, which can be achieved through specific characterizations of these singularities.
PREREQUISITES
- Understanding of conformal mappings in complex analysis
- Familiarity with the concept of singularities in complex functions
- Knowledge of the definitions of removable and essential singularities
- Basic proficiency in complex function theory
NEXT STEPS
- Study the properties of singularities in complex analysis
- Learn about the classification of singularities, focusing on removable and essential types
- Explore conformal mappings in the context of the complex plane
- Investigate the implications of poles and their orders in complex functions
USEFUL FOR
Students of complex analysis, mathematicians focusing on function theory, and anyone interested in the properties of conformal mappings in the complex plane.