greetings , we have the following integral :(adsbygoogle = window.adsbygoogle || []).push({});

[tex]I(x)=\lim_{T\rightarrow \infty}\frac{1}{2\pi i}\int_{\gamma-iT}^{\gamma+iT}\sin(n\pi s)\frac{x^{s}}{s}ds[/tex]

n is an integer . and [itex]\gamma >1 [/itex]

if [itex] x>1[/itex] we can close the contour to the left . namely, consider the contour :

[tex] C_{a}=C_{1}\cup C_{2}\cup C_{3}\cup C_{4}[/tex]

where :

[tex]C_{1}=\left [ \gamma-iT,\gamma+iT \right ] [/tex]

[tex]C_{2}=\left [ \gamma+iT,-U+iT \right ] [/tex]

[tex]C_{3}=\left [ -U+iT ,-U-iT \right ] [/tex]

[tex]C_{4}=\left [ -U-iT ,\gamma-iT \right ] [/tex]

and [itex]U>>\gamma [/itex]

then by couchy's theorem :

[tex] I(x)=I_{1}+I_{2}+I_{3}+I_{4}=0[/tex]

if [itex] x<1[/itex], we can close the contour to the right via the following contour :

[tex] C_{b}=C_{1}\cup C_{2}\cup C_{3}\cup C_{4}[/tex]

[tex]C_{1}=\left [ \gamma-iT,\gamma+iT \right ] [/tex]

[tex]C_{2}=\left [ \gamma+iT,U+iT \right ] [/tex]

[tex]C_{3}=\left [ U+iT ,U-iT \right ] [/tex]

[tex]C_{4}=\left [ U-iT ,\gamma-iT \right ] [/tex]

then also by couchy's theorem :

[tex] I(x)=0[/tex]

the plan is to give an estimate of the integrals along the segments of the rectangular contour, and calculate [itex] I_{1}[/itex] 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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# Help with a complex integral

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