So I ran into residue theorem recently and found it to be pretty amazing, and have been trying to get some of the more fundamental aspects of Laurent series and contour integrals down to make sure I understand it properly, but there's still one big aspect that keeps confusing me majorly. According to the theorem, a contour integral on the complex plane around a set of singularities (I'm paraphrasing) like (x, y) = (0, 1) or (0, -1) in z = 1/(1 + (x + i y)^2) is 2 i pi [Sigma on i] Res(z, point i), but that implies it's constant no matter the size or shape of the chosesn contour (given the set of singularities it encircles does not change), and that any closed contour integral which does not circle a singularity must be zero? The first part is mindblowing if it's true, but it doesn't contradict in too obvious a fashion with my logic, the last point does. If you take a contour integral about a circle, each point in which has a positive z value (ie. where the surface is above the x, y plane for all of the circle) then there is no negative part to cancel anything and the integral should have a non-zero, positive value... Where am I going wrong in my thinking or understanding of the theorem? Is it to do with conflating closed contour integrals over the complex plane too closely with generalised line/path integrals? I've always understood them to essentially be synonyms, so my physical intuition of the former derives from that of the latter.