The Schwarz-Christoffel formula provides a way to map the upper half-plane (or unit disk) conformally onto a polygon in the complex plane. When dealing with pre-vertices, the situation where one pre-vertex is at infinity corresponds to mapping onto a closed polygon, as you described. This case has subtle differences but is conceptually tied to adjusting the interior angles of the polygon.

Key Insights:

1. Pre-vertex at infinity: Placing a pre-vertex at infinity does not directly correspond to setting an angle to zero. Instead, it modifies the integral formulation of the Schwarz-Christoffel mapping. In particular:

   • The angle at infinity is implicitly associated with the behavior of the integrand near the infinite point.
   • You essentially remove a vertex in the finite domain and consider how the boundary behavior contributes to the shape of the polygon.

2. Practical implications: When one pre-vertex is sent to infinity, the Schwarz-Christoffel formula simplifies: $f(z) = C \int (z - z_1)^{\alpha_1 - 1} (z - z_2)^{\alpha_2 - 1} \dots \, dz + D,$ where one $z_j$ is at infinity. In such a case:

   • The pre-vertex at infinity drops out of the product.
   • The other terms must compensate to ensure the correct asymptotic behavior and angles of the polygon.

3. Angle at infinity: In some interpretations, the angle at infinity can be thought of as the "external angle" complementing the total sum of finite interior angles of the polygon. However, it is not equivalent to setting an angle to zero at a finite vertex. Zero angles in finite locations would imply a degenerate vertex.

Conclusion:

Mapping a pre-vertex to infinity does not mean one angle is zero; rather, it adjusts the integral and boundary conditions to represent the correct polygon shape with a closed contour. The treatment of the angle at infinity ensures the overall polygon is consistent with the desired geometry.

Would you like further elaboration on:

1. Derivation of the Schwarz-Christoffel formula with a pre-vertex at infinity.
2. How the mapping changes when different pre-vertices are at infinity.
3. Examples of polygons requiring a pre-vertex at infinity.
4. Role of the constant $C$ in the formula when $z_j = \infty$.
5. Application of this in solving practical conformal mapping problems.

Tip: Always verify the boundary behavior and asymptotics of the Schwarz-Christoffel mapping to ensure it adheres to the geometry of your polygon, especially when dealing with infinity.