greetings , we have the following integral :
$$I_{1}(x)=\lim_{T\rightarrow \infty}\frac{1}{2\pi i}\int_{\gamma-iT}^{\gamma+iT}\sin(n\pi s)\frac{x^{s}}{s}ds$$
n is an integer . and $\gamma >1$
if $x>1$ we can close the contour to the left . namely, consider the contour :
$$C_{a}=C_{1}\cup C_{2}\cup C_{3}\cup C_{4}$$
where :
$$C_{1}=\left [ \gamma-iT,\gamma+iT \right ]$$
$$C_{2}=\left [ \gamma+iT,-U+iT \right ]$$
$$C_{3}=\left [ -U+iT ,-U-iT \right ]$$
$$C_{4}=\left [ -U-iT ,\gamma-iT \right ]$$ and $U>>\gamma$.
then by couchy's theorem :
$$I(x)=I_{1}+I_{2}+I_{3}+I_{4}=0$$
if$x<1$, we can close the contour to the right via the following contour :
$$C_{b}=C_{1}\cup C_{2}\cup C_{3}\cup C_{4}$$
$$C_{1}=\left [ \gamma-iT,\gamma+iT \right ]$$
$$C_{2}=\left [ \gamma+iT,U+iT \right ]$$
$$C_{3}=\left [ U+iT ,U-iT \right ]$$
$$C_{4}=\left [ U-iT ,\gamma-iT \right ]$$
then also by couchy's theorem :
$$I(x)=0$$
the plan is to give an estimate of the integrals along the segments of the rectangular contour, and calculate $I_{1}$ in both cases via the result obtained by cauchy's theorem . however, i don't have the first clue on how to do that, hence the quest !
