Calculate complex integral $int_{-infty}^infty e^{-ix^2}d x=?$
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How to calculate this elementary complex integral? This is what we would encounter if we are studying the Green's function for Schroedinger's equation. $$int_{-infty}^infty e^{-ix^2}d x=?$$ However, I think there should be someone that posted similar question on Math SE, though I don't know how to search by equation. Thank you very much if you can help me out! And I would be grateful if you can give more than one approach P.S.: The equation $int _{-infty}^{infty}e^{-kt^2}d sqrt{k}t=sqrt{pi}$ surely comes to my mind, but I don't know why it holds for $kinmathbb{C}$ , because for me, the above integral is over real line, however, the question here is like integral on $y=e^{i pi/4}x$ ( So I think it's the problem with my complex integral knowledge.) I tried to rotate th