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## help with a complex integral [closed]

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 !

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 You may want to ask contour integral questions at math.stackexchange.com – S. Carnahan♦ May 25 2012 at 3:08