Jordan Lemma not applying

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so I was trying to evaluate the integral of

{1/(x-ia) dx}

from -infinity to +infinity and where a>0

Of course, Jordan's lemma doesn't apply (it's what I used before, this is the case where the exponential equals to 1)

I'm really stuck...I've tried to use Cauchy's theorem but it just gives me zero and I'm quite sure it's not correct...

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1 Answer

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Hint: $ 1/(x - ai) + 1/(-x - ai) = 2ai/(x^2 + a^2) $, so what happens if you integrate $ 1/(x- ai) $ from $ x = -L $ to $ x = L $ for some finite $ L $?

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