contour integral logarithm
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When calculating integrals like $int_{0}^{infty} R(x)log(x) dx$ with R(x)=P(x)/Q(x) a rational function i use the keyhole contour as in the example 4 of this link https://en.wikipedia.org/wiki/Contour_integration, with the argument of the logarithm between o and 2 $pi$ . Now everything is fine if Q doesn't have non negative zeros, but if it has i haven't found anything on the internet. I think i should change the argument and choose the one between - $pi$ and $pi$ , but in that case i think i end up with $int_{0}^{infty} R(-x)(logx+ i{pi})^2 dx$ - $int_{0}^{infty} R(-x)(logx- i{pi})^2 dx$ . Now that R(-x) is my mistake, it is actually R(x) and everything is fine or we can use this type of contour only for special cases like if R is even or odd? And if Q has both positive and nega...