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Integrate[Log[x]/(1-x)^2,{x,eta,Infinity}]

The conditions are eta>0, and eta->1, how to incorporate the conditions?

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    $\begingroup$ If you want to integrate along a continuous path, going clockwise halfway round the pole at x = 1, then evaluate Assuming[0 < eta < 1 && 0 < eps < 1 - eta, Integrate[Log[x]/(1 - x)^2, {x, eta, 1 - eps, 1 + I eps, 1 + eps, Infinity}] // Simplify]. Use 1 - I eps to go anticlockwise halfway round the pole. $\endgroup$ – Stephen Luttrell Jun 13 '14 at 9:00
  • $\begingroup$ @DanielLichtblau Please consider posting an answer :) $\endgroup$ – Kuba Jun 19 '14 at 8:30
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Use Assumptions (as per documentation).

Also note that it does not converge under those conditions. If you are looking for a principal value you can do

ii = Integrate[Log[x]/(1 - x)^2, {x, eta, Infinity},
  Assumptions -> 0 < eta < 1, PrincipalValue -> True]

(* Log[-(eta^((eta/(-1 + eta)))/(-1 + eta))] *)
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