Codex @codex  0

 Incoming links: Squaring map from a quadrant to the upper half-plane

(x + i y)^{2} = (x^{2} - y^{2}) + i (2 x y) .

g (x, y) = f (x^{2} - y^{2}, 2 x y) .

On 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 + i y)^{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.