compute an integral with residue I have to find the value of
$$\int_{-\infty}^{\infty}e^{-x^2}\cos({\lambda x})\,dx$$
using residue theorem. What is a suitable contour? Any help would be appreciate! Thanks...

Hint: By symmetry, we can let $\gamma$ be the path running along the real axis and get that our integral is just $$\int_\gamma e^{-z^2}e^{i\lambda z} dz.$$ Now what happens when you combine these terms and complete the square? Your answer should become a significantly simpler problem.
But be careful with the resulting path. There's still a good bit to do with this approach.
Edit, more steps: You obtain from this the integral $$\int_{\gamma'} e^{-z^2} dz,$$ where $\gamma'$ runs along the real axis, shifted by $-\frac{\lambda i}{2}$. Now, you can show that if you integrate along the real axis, your answer is $$\int_{-\infty}^\infty e^{-x^2} dx = \sqrt{\pi}.$$ This can be done by contour integration. I claim that the integral we are interested in can be obtained from this using contour integration. Relate the path we want to this path and do some estimates on the ends.
