I have the integral
${\operatorname{Im}} \left (\int^\infty_0 e^{ix} x^{s-1} \, \mathrm{d} x \right)$
and I wish to redefine $x \to iy$ but I am unsure of how to justify this using contour integration arguments. I would be very grateful for any help. Thanks.
An idea:
$$\int\limits_0^\infty e^{ix}x^{s-1}dx=\int\limits_0^\infty \left(x^{s-1}cos x+ix^{s-1}\sin x\right)dx\implies$$
$$\implies\operatorname{Im}\left(\int\limits_0^\infty e^{ix}x^{s-1}dx\right)=\int\limits_0^\infty x^{s-1}\sin x\,dx$$
The last integral converges for $\,s<1\,$