How does the Schwarz-Christoffel transformation map polygons in the z-plane to the w-plane?

The Schwarz-Christoffel transformation maps polygons in the z-plane to the upper half of the w-plane, with the polygon's boundary mapped onto the real axis of the w-plane. This transformation is defined by a specific function that depends on the vertices and internal angles of the polygon.

The transformation is designed to take any polygon in the z-plane and map it onto the entire upper half of the w-plane, ensuring that the edges of the polygon correspond to segments along the real axis in the w-plane. The function used in the transformation incorporates complex constants that are determined by the polygon's physical properties, such as its vertices and angles.