For an integral of the form $$ I(t)\;=\;\int_{-\infty}^{+\infty}\frac{e^{i \omega t}}{1+i \omega}\,\mathrm{d} \omega $$ say, with $t>0$, a standard application of the residue calculus with a semi-circle contour integration along the upper half plane around the first-order pole $\omega=i$ yields $$ I(t)\;=\;2\pi i \;\lim_{\omega\to i}\Big((\omega-i)\,\frac{e^{i\omega t}}{1+i\omega}\Big)\;=\;2\pi \,e^{-t}\,. $$ Now, what can one do when the pole is not simple (or double, etc.), say, when one has to compute $$ I_\alpha(t)\;=\;\int_{-\infty}^{+\infty}\frac{e^{i \omega t}}{1+(i \omega)^\alpha}\,\mathrm{d} \omega \,,\qquad \alpha\in(0,1)\,,\qquad t>0$$ is residue calculus still applicable and how?
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