i'm confused about this integral

mmzaj

i'm trying to prove - or disprove ! - the following :
[tex] -\ln x\frac{\left \{ x^{1/n} \right \}}{2n^{3}}=\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{s}{\left((ns)^{2}-1\right)^{2}} x^{s}ds[/tex]
where [itex]\left \{ x^{1/n} \right \} [/itex] is the fractional part of [itex] x^{1/n}[/itex]
for [itex] x\in \mathbb{R}:x>1 [/itex], [itex] n\in \mathbb{Z}^{+}[/itex]
i'm confused about where to close the contour: to the right , or to the left of the imaginary axis. because the integrand has poles at [itex]n^{-1} [/itex] and [itex] -n^{-1}[/itex]. and by the reside theorem, i get two different results!!

jackmell

Maybe you're getting different results because the results are different. Are you sure the contribution along the half-circle arc is zero whether you go around the left half plane or the right half plane?

Just compute it numerically to see if there's a difference, then if there is, try and show it analytically.

mmzaj

maybe i was closing the contour the wrong way!! i didn't use half circles, i closed it using straight segments parallel to the real/imaginary lines. thanks for the remark . however, i still have doubts about the 'steppy' nature of the result - if correct !! - .

haruspex

Is it that you get the opposite sign? If so, maybe you forgot to flip the bounds of the linear integral in order to go anticlockwise around the pole.