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The image appears to contain the proof of a theorem regarding the analytic extension of a function \( f(z) \). I will summarize and explain the content step by step based on the current visible information:

### Theorem
If a part of an analytic function \( f(z) \) is defined in a domain \( \Omega^+ \), then \( f(z) \) has an analytic extension \( f(\overline{z}) \) that satisfies \( f(z) = f(\overline{z}) \).

### Proof Outline:
1. **Define a function \( V(z) \):**
   \[
   V(z) = \begin{cases} 
   V(z) \text{ in } \Omega^+ \\
   0 \text{ on } \sigma \\
   -V(\overline{z}) \text{ in the mirror image of } \Omega^+
   \end{cases}
   \]

2. **Claim:**
   \( V \) is harmonic on \( \sigma \) (the boundary).
   
   - Fix a point \( x_0 \in \sigma \).
   - Consider a disk with center \( x_0 \) contained in \( \Omega \).
   - Let \( P_v \) denote the Poisson integral with respect to this disk, formed with the boundary values of \( V \).

3. **Harmonicity of \( V - P_v \):**
   - The difference \( V - P_v \) is harmonic in the upper half of the disk.
   - It vanishes on the half-circle by **Schwarz's theorem**.
   - It vanishes on the diameter, because \( V \) tends to zero as expected.

4. **Symmetry and conclusion:**
   - \( P_v \) vanishes by symmetry.
   - By the **maximum and minimum principle**, we conclude that... (the proof likely continues but is cut off in this image).

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Would you like more details on harmonic functions, Poisson integrals, or Schwarz's theorem? 

Here are some relevant questions to expand on this proof:
1. How do you define harmonic functions, and why is it important in this context?
2. What is the Poisson integral formula, and how does it help in this proof?
3. What role does symmetry play in proving the vanishing of the function \( P_v \)?
4. Could you explain the maximum and minimum principles for harmonic functions?
5. How does Schwarz's reflection principle apply to this theorem?

**Tip:** Always check if boundary conditions are applied correctly when using Poisson integrals in harmonic analysis, as they often determine the uniqueness of the solution.

Part of an analytic function f(z) in a domain, then f(z) has an analytic extension which satisfies f(z) = f(\overline{z}).

Explore the proof of an analytic function's extension in a domain using harmonic function theory. The proof involves the Poisson integral, Schwarz's theorem, and the maximum and minimum principle. Suitable for advanced undergraduates and graduate-level studies in complex analysis.

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