In complex plane, paths {closed contour} can loop around origin. There is line integral {contour integration} around the path. Looping once increases integral by 2 * pi * i. Looping counterclockwise once increases integral by -2 * pi * i.
Integration paths over complex functions do not matter {Cauchy integral theorem}. For complex function f(z), integral of f(z) = F(z) = (1 / (2 * pi)) * (integral from p = -pi to p = +pi of (x * f(x) / (x - z)) * dp), where z is complex number, and x = |z| * e^(i*p) {Cauchy integral formula}. Cauchy integral formula makes series {majorant series} and has residue {integral residue}.
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Date Modified: 2022.0225