Integral between two curve points is integral {line integral} {curvilinear integral} from a to b of p(x, f(x)) * dx, where f(x) is curve function, and p is surface function.
For closed curve, line integral over line equals double integral over closed region {Green's theorem} {Green theorem}. Line integral over regular simply connected closed curve equals zero. Line integrals over any two regular curves between two region points are equal. For closed surface, double integral over surface equals triple integral over closed volume.
Double integral, over surface, of cross product of del operator and vector function, equals line integral, over boundary curve, of vector function {Stokes theorem}.
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