Currently, I have solved the temperature equation and I obtained the values $T(x, y)$ in the xy-space. I would like to map the xy-space onto the uv-space conformally and injective (that means that a point $(x_p, y_p)$ will map one-to-one on $(u_p, v_p)$). What will happen with the temperature value $\tilde{T}(u_{p}, v_{p})$? Will it be equal to $T(x_p, y_p)$, $ T(x_p, y_p)$ multiplied by a factor or something else?
The conformal mapping that I want to use is the Schwarz-Christoffel transformation that maps an unit disk onto a polygon.