integration - How do I evaluate $int_{0}^{infty} u^{z-1}(e^{iu}-1) , du$?

I am trying to evaluate the following integral that shows up in this paper

http://arxiv.org/pdf/1103.4306v1.pdf

$I=\int_{0}^{\infty} u^{z-1}(e^{iu}-1)du= \Gamma(z)e^{\frac{iz\pi}{2}}$

for $-1

I have no idea what type of contour I can choose. Can someone please help me?

Thank you in advance.

Answer

Consider the quarter of circle $\Gamma$ of radius $M$ centred at the origin and lying in the first quadrant, thus made of the radius $0$-$M$ the arc $M$-$iM$ and the radius $iM$-$0$.
Consider the function of the complex plane $\zeta=x+iy$
$$
f(\zeta) =\zeta^{z-1}\left( e^{i\zeta}-1\right);
$$
choosing the logarithm branch cut in the negative $x$ axis, $f$ is holomorphic within $\Gamma$, thus

$$
\oint_\Gamma f(\zeta) d\zeta=0
$$
hence
$$
\int_0^M x^{z-1}\left(e^{ix}-1\right)dx +
\int_0^{\pi/2} \left(M e^{i\varphi}\right)^{z-1} \left(e^{iMe^{i\varphi}}-z\right) iM e^{i\varphi}d\varphi
-i^{z}\int_0^M y^{z-1}\left( e^{-y}-1\right)dy=0.
$$
The contribution of the arc is under control, since letting $\Re (z)=\xi$, $\Im (z)=\eta$,

$$
\left|\int_0^{\pi/2} \left(M e^{i\varphi}\right)^{z-1} \left(e^{iMe^{i\varphi}}-z\right) iM e^{i\varphi}d\phi\right|
\le
M^\xi \int_0^{\pi/2}e^{-\eta\varphi}\left| e^{iM\cos\varphi - M\sin\varphi}-1\right|d\varphi\\
\le 2M^\xi \int_0^{\pi/2}e^{-\eta\varphi}d\varphi=2M^{\xi}\frac{1-e^{-\eta\pi/2}}{\eta}\xrightarrow[M\to\infty]{}0,
$$
since $\xi<0$.
Terefore
$$
\int_0^\infty x^{z-1}\left(e^{ix}-1\right)dx=i^{z}\int_0^\infty y^{z-1}\left( e^{-y}-1\right)dy=e^{i\pi z/2}\int_0^\infty y^{z-1}\left( e^{-y}-1\right)dy.

$$
Integrating by parts
$$
\int_0^\infty y^{z-1}\left( e^{-y}-1\right)dy=
\frac{y^z}{z}\left(e^{-y}-1\right)\Big|_0^\infty+\frac{1}{z}\int_0^\infty y^z e^{-y}dy=\frac{1}{z}\Gamma(z+1)=\Gamma(z),
$$
where we have used
$$
\lim_{y\to\infty}\frac{y^z}{z}\left(e^{-y}-1\right)=0
$$

$$
\lim_{y\to0}\frac{y^z}{z}\left(e^{-y}-1\right)=0,
$$
the second one following from the request $\xi>-1$. Putting everything back together,
$$
\int_0^\infty x^{z-1}\left(e^{ix}-1\right)dx=e^{i\pi z/2}\Gamma(z)
$$
if $-1<\Re (z)<0$.