Past exam of the short options course of the University of Oxford / 2017 / 5 / c / Solution

Use the squaring map from a quadrant to the upper half-plane:By conformal invariance of harmonicity, the answer isOn the horizontal axis it gives $f(x^2,0)=e^{-x^2}$, and on the vertical axis it gives $f(-y^2,0)=e^{-y^2}$. Also $|(x+iy{)}^2|=x^2+y^2$, so spatial infinity maps to spatial infinity and the required decay follows. The two boundary values agree at the origin.