In the complex $z$ plane, we choose the branch cut in the upper-half plane. Consider some complex function $z^{iw/a}$ where $w$ and $a$ are some real quantities. Then the author claims that since the function can be analytically continued in the lower half plane, we can write $$z^{iw/a}=\int^\infty_0 d\nu \chi(\nu)e^{-i\nu z}$$ for some $\chi(\nu)$.
I do not understand why we should have this expansion.