Function that transforms (half) plane to annulus

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SUMMARY

The discussion centers on the existence of a complex function that conformally transforms a polygonal region of the plane into an annulus. The solution provided is the exponential function, exp(z), with the critical condition that the domain must be limited to a size of 2π to prevent overlap. This transformation is significant in complex analysis and has practical implications in various mathematical applications.

PREREQUISITES
  • Complex analysis fundamentals
  • Understanding of conformal mappings
  • Knowledge of the exponential function in complex variables
  • Familiarity with polygonal domains in the complex plane
NEXT STEPS
  • Study the properties of conformal mappings in complex analysis
  • Explore the implications of the exponential function in complex transformations
  • Research the concept of annuli in complex geometry
  • Investigate other complex functions that can perform similar transformations
USEFUL FOR

Mathematicians, students of complex analysis, and anyone interested in the applications of conformal mappings in geometry and analysis.

meldraft
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Hey all,

I am looking for a complex function that conformally transforms a part of the plane (don't really care which part, as long as it's a polygon (even with infinite sides)) to an annulus (that is, a disk with a hole in the middle). My books on complex analysis conveniently skip it, but I'm pretty sure it must exist. Does anyone know of such a function??
 
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Nevermind, found it :-p

In case another person is interested, it's exp(z), but the size of the domain must be 2*pi, otherwise the domain overlaps with itself
 

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