The Poisson integral vanishes by symmetry in the context provided for the following reasons:

In the scenario described in your notes, you are working with a function $V(z)$ that is harmonic on a domain (likely a half-disk or a related domain) and the Poisson integral $P_V$ is used to reconstruct $V$ from its boundary values. Specifically, the function $V$ seems to vanish on a symmetric boundary (like the real axis or a diameter of a disk), and this symmetry plays a role in the vanishing of the Poisson integral.

The symmetry argument can be understood as follows:

1. Symmetric Boundary Conditions: The function $V(z)$ is defined to have specific properties across a mirror boundary (such as the real axis). Since $V(z)$ tends to zero as you approach the symmetric boundary, this symmetry implies that the values of $V$ on one side of the boundary mirror those on the other side, either exactly or as a reflection.

2. Schwarz Reflection Principle: This principle allows for extending a harmonic or analytic function across a boundary if the function satisfies certain symmetry conditions. In this case, since $V$ vanishes on the boundary and has a symmetric setup, it can be concluded that the Poisson integral constructed with these boundary conditions will also respect this symmetry.

3. Poisson Integral Properties: The Poisson integral formula reconstructs harmonic functions inside a disk from their boundary values. If the boundary values (such as $V$) are symmetric or vanish on part of the boundary, the resulting function inside the domain will also reflect this behavior. Therefore, the Poisson integral $P_V$ will vanish on the symmetric boundary due to the boundary conditions that force it to zero on that section.

4. Harmonicity and Symmetry: The harmonic function $V - P_V$ is said to vanish on the diameter because of the symmetric boundary values and the vanishing of $V$ on that line. This enforces the Poisson integral to vanish by symmetry as well.

In summary, the Poisson integral vanishes by symmetry because the boundary conditions on $V(z)$ are symmetric, particularly vanishing on a specific boundary (the diameter in this case), and the Poisson integral respects this symmetry.

Would you like a deeper explanation of any part of this?

Here are five relative questions to consider:

1. How does the Poisson kernel help in solving the Dirichlet problem for harmonic functions?
2. What is the role of the maximum and minimum principles in the proof shown in your notes?
3. How does the Schwarz reflection principle extend the domain of an analytic or harmonic function?
4. What are the properties of harmonic functions that allow the use of the Poisson integral?
5. How do boundary conditions affect the behavior of harmonic functions inside a domain?

Tip: Symmetry in harmonic functions often simplifies the problem significantly by reducing the domain or boundary conditions, so always look for symmetric properties in these types of problems.