Yes, your reasoning is fundamentally correct. A Möbius transformation, given by $f(z) = \frac{az + b}{cz + d}$, is a conformal map (analytic and preserves angles) that maps circles and lines to circles and lines. In your scenario:

1. Mapping $z_0$ to the origin: A Möbius transformation can always be adjusted to map $z_0$ to $0$, ensuring that the region is positioned suitably in the complex plane.

2. Mapping $z$ on the boundary to $z' = e^{i\theta}$: The boundary of the region is mapped to the unit circle, where points are represented as $e^{i\theta}$, which is consistent with Möbius transformations.

3. Mapping infinity to infinity: Möbius transformations can also handle infinity due to their projective nature. Mapping infinity to itself is an important property for preserving the structure of certain domains.

4. Analyticity: Since Möbius transformations are rational functions with non-zero denominators, they are analytic everywhere in the complex plane except possibly at isolated singularities (e.g., where the denominator is zero). In this case, your described mapping avoids singularities in the domain of interest, ensuring analyticity everywhere.

5. Conclusion: The mapping is analytic everywhere within the region because Möbius transformations are globally analytic except at poles, which are outside the mapped region in your case.

Would you like further elaboration or graphical representations of Möbius transformations?

Five Related Questions:

1. How do Möbius transformations preserve angles in complex mappings?
2. What are the key properties of Möbius transformations regarding circles and lines?
3. How does the conformal nature of Möbius transformations relate to analytic continuation?
4. Can you construct the specific Möbius transformation for your case?
5. What happens to the mapping when $z_0$ is moved to other points in the region?

Tip:

Always verify if the region of interest avoids singularities (e.g., poles) of the Möbius transformation to ensure analyticity.