When computing the characteristic function of Cauchy distribution, we applied the Cauchy Integration theorem:
$ \int_{C_{R}}\frac{e^{i\alpha z}}{z^{2}+1}dz=\int_{-R}^{R}\frac{e^{i\alpha z}}{z^{2}+1}dz+\int_{\Gamma_{R}}\frac{e^{i\alpha z}}{z^{2}+1}dz=I_{R}+J_{R} $
We assume $\alpha>0$ and we use the curve from $(-R,0)$ to $(R,0)$ and back to $(-R,0)$ counter clockwise from the positive half plane (imaginary part >0) .
$C_{R}$ is the counter-clockwise contour, $\Gamma_{R}$ is the counterclockwise half circle on the upper half plane. The final result is: $ \pi e^{-\alpha} $
I am curious where did we use the condition $\alpha>0$ in this derivation?
I suspect it to be the choice of the integration contour. But how? Can we choose $\alpha<0$ while integrate over the same positive contour and get the same result but it is unstable for $\alpha<0$
Thanks in advance.