I'm reading the book "complex variables Demystified" from David McMahon and right now I'm in the chapter 12 where he explains how to solve Boundary values problems using comformal mappings
He shows a few examples; In the first of them he solves the Laplace equation defined in the right upper plane with Dirichlet boundaries conditions given by $f(0,y)=1$ and $f(x,0)=0$. He uses the mapping function $z^2$ to extend the region to the full upper plain and based on this he gets the new boundaries conditions $g(u,0)=1$ for negative values of x and $g(u,0)=0$ for the positive values of $x$. He explains this like that: "This is because when $x=0$, we have $u=-y^2, v=0$. Given the range of $0<y<\infty$ this fixes the boundary condition to $1$ when $−\infty<u<0$ and 0 when $0<u <\infty$."
I just don't get it.
In the next example he mapps the unit circle again into the full upper plain and he assigns the value of $g(\theta)$ as $1$ if $0<\theta<\pi$ and $0$ if $\pi<\theta<2\pi$. He doesn't explain how he change this boundaries to apply it to the Poisson's formula for the half plane
Id appreciate and answer in plain english without abstract symbols. Thanks in advance