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From the provided image, the notes discuss extending a function \( v \) to a disk, and its conjugate harmonic function \( u_0 \), which is related to the reflection of a certain analytic function. Let's break it down to understand why the constant \( u_0 \) is evidently zero.

1. **Conjugate Harmonic Function**: The function \( v \) has a conjugate harmonic function \( u_0 \), which together with \( v \) forms a harmonic pair. That is, \( u_0 \) and \( v \) satisfy the Cauchy-Riemann equations in their domain of definition.
   
2. **Reflection across the Real Axis**: The function \( u_0 \) is symmetric with respect to the real axis in the sense that \( u_0(z) = \text{Re}(f(z)) \) where \( f(z) \) is analytic. The real part of \( f(z) \) is harmonic, and therefore, \( u_0(x,y) \) and \( v(x,y) \) satisfy the harmonic conditions. 

3. **Boundary Conditions and Cauchy-Riemann**: It is given that \( u_0(x,y) - u_0(x,-y) = 0 \). This reflects the fact that \( u_0 \) is symmetric with respect to the real axis. Moreover, we see from the equations that:
   \[
   \frac{\partial u_0}{\partial x} = 0 \quad \text{and} \quad \frac{\partial u_0}{\partial y} = 0
   \]
   on the real axis. These derivatives being zero on the real axis implies that \( u_0 \) vanishes on this axis.

4. **Conclusion that \( u_0 \) is Constant**: Since \( u_0 \) vanishes on the real axis and satisfies the Cauchy-Riemann equations, it must be constant throughout the domain. The text concludes that this constant is zero, since a harmonic function that is constant and vanishes at some point must be zero everywhere.

Thus, the constant \( u_0 \) is evidently zero because:
- \( u_0 \) satisfies the Cauchy-Riemann equations.
- \( u_0 \) vanishes on the real axis.
- By the reflection principle and harmonicity, \( u_0 \) is constant across the entire disk, and since it vanishes at a point, it must be zero everywhere.

Let me know if you'd like further clarification on any point.

Here are five related questions you can explore:

1. What are the Cauchy-Riemann equations and how do they relate to harmonic functions?
2. What is the reflection principle in complex analysis?
3. How does harmonicity constrain the behavior of solutions to boundary value problems?
4. Why must a harmonic function that vanishes at a point be zero everywhere under certain conditions?
5. What role does the real axis play in this construction for \( u_0 \)?

**Tip**: The symmetry of harmonic functions with respect to the real axis is a common technique used in solving problems in complex analysis and potential theory.

Explore the proof of why the constant u0 is zero in harmonic function analysis, using concepts like the Cauchy-Riemann equations, reflection principle, and harmonic function properties. This analysis is suitable for advanced undergraduate or graduate students studying complex analysis.

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