Find the limit $$\lim_{z \to \infty}\int_{\mathbb C}|w|e^{-|z-w|^2}dA(w) $$
where A is area measure such that dA=rdrd$\theta$
Please help me, I did four page computation by changing to polar cordinates and using modified Bessel formula. My computation shows limit does not exists but I am not about limit. This integration is identical to integration I asked some time ago. This is part of my research so please help me.
The limit is $+\infty$. You can see this either by making the change of variables $w\mapsto z-w$ in the integral (as PhoemueX commented), or by the simple inequality $$ \int_{\Bbb C} |w|e^{-|z-w|^2} dA(w) \ge \int_{\{w\colon |z-w|\le 1\}} |w|e^{-|z-w|^2} dA(w) \ge e^{-1}(|z|-1) \int_{\{w\colon |z-w|\le 1\}} dA(w) = \frac{\pi(|z|-1)}e. $$