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$$\int_0^\infty e^{-x^2}cos(kx)dx$$

with $k>0$

I know there are a number of ways, but I'm interested in using complex integration. In particular, I believe that we can solve by integrating $e^{-z^2}$ over the boundary of the rectangule $[-R,R]\times[0,h]$ for a suitable $h$.

What $h$ do you think i should use?