= Squaring map from a quadrant to the upper half-plane
{title2=$(x+iy)^2=x^2-y^2+2ixy$}
The map $z\mapsto z^2$ is a bijective <conformal map> from the open first quadrant to the <complex upper half-plane>. In polar coordinates it doubles arguments from $(0,\pi/2)$ to $(0,\pi)$, and its derivative $2z$ never vanishes inside the quadrant. Its inverse is the branch of the <square root> with argument in $(0,\pi/2)$. By <conformal invariance of harmonicity>, a <harmonic function> $f$ on the half-plane pulls back to $g(x,y)=f(x^2-y^2,2xy)$. The two boundary rays map to the positive and negative real axes, respectively.
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