MATH 113 Midterm: MATH 113 Harvard 113 Fall 01113hw4
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Due thursday, october 18: let f : c c be a continuous function, and let c be a smooth curve in. C parametrized by z(t) : [a, b] c. let p = (a0 = a, a1, a2, . , an = b) be a partition of [a, b], let zj = z(aj) for j = 0, . , n, and let rp (f ) = Explain why rc f rp (f ) if maxj|aj aj 1| is small. Formulate and prove a more precise version of this statement: we know from lecture that if f (z) is analytic on the closed disc {|z| r} then r|z|=r f (z)dz = 0. Let be a smooth, simple closed curve, oriented counterclockwise, which surrounds the region d in. Denote by d the closure of d, i. e. , d = d . Let p, q : d r2 be continuously di erentiable (c 1) real-valued functions.
