Let C denote the positively oriented boundary of the half disk 0 ≤ r ≤ 1, 0 ≤ θ ≤ π, and let f (z) be a continuous function defined on that half disk by writing f (0) = 0 and using the branch f (z) = √ reiθ/2 r > 0,−π 2 < θ < 3π 2 of the multiple-valued function z1/2. Show that C f (z) dz = 0 by evaluating separately the integrals of f (z) over the semicircle and the two radii which make up C. Why does the Cauchy–Goursat theorem not apply here?