This is Complex Variable. In detailed explanation please. I am a returning student, and My Calc class was in 2007. Thank you.

A 'F.T.C.'for contour integrals. Let F: A - C cont. with A, a domain. The following are equal. 1) There exists a F holomorphic (Complex, differentiable) s. t. F: A - C and F' = F i) [ F are path independent. i.e. if ya and yo are contours from zo to z, then [ F=S F iii) if y is a closed curve, then [ F = 0 PF: 1) - ii) Let F be s. t. F' = F then, (F=[ f(2) dz = [of(r(t) )' (t)at = S a f(r(t) )at = F(Y(1)|2=F(v(b)) -F(y(a)) F(r(b)) - F(y(a)) does not depend on path from y, to Ya PF: ti) - iii) [ F is path independent = [ F = 0 Let y be a closed curve contained in a 4. so, y = V1+12 note V1: 20 - Z1: -V2: Zo - 21: since contour integral are assumed to be path independent. PF: iii) - ii) Assume [, F = 0 for y C A, fix zo, Z, E A and let y, 1, be any path contours from Zo, to z1. Note: yo -yz is a close curve so, F = 0 = [ F - S F and finally, [ F = J F PROVE THAT: (ii - Dii) = i fix zo E A. Define F(z) = [_ f(s)as| F(z) is well defined by i), ds is w. r. t. arclength. remains to be shown F' (z) = f(z). h-co hint: F(t) = [ f(x)dx F'(t) = lim (t+h)-8(0) Use the fundamental theorem idea. (tth)-t