Travelling anti-clockwise, define the interior angle of the polygon to be
αjπ and the exterior angle to be
βjπ=π−αjπ. This means that at a corner when we turn left,
βj will be positive, and
βj will be negative when we turn right at a corner.
In your question, we want to map (from the z plane to the w plane), z = -1 to w = -1 and z = 1 to w = 0. So the Schwarz Christoffel mapping will be of the form
f(z)=A+C∫z(t+1)−β1(t−1)−β2dx,
where I've left the betas, A and C for you to determine. [Hint:].
Technically speaking, we need a third point (by the Riemann Mapping Theorem, we can fix the pre-images of 3 boundary points, i.e. 3 of the xj, and therefore a third angle (in order to close the polygon, imagine going all the way back round at (0, -infty) and (-1, infty)). But we can choose
x3=∞ and "ignore" this point in the formula.