Question

56. Consider the multiple-valued fu nction F()z1/3 that assigns to z the set of three cube roots of z. Explicitly define three distinct branches fi. fa, and fs of F, all of which have the nonnegative real axis as a branch cut.

Answer 1

Consider the multivalued function F(z) = z^1/3 consider z = pi(cos theta + i sin theta), 0 < theta < pi/2 therefore z^1/3 = pi^1/3 (cos theta + i sin theta)^1/3 By De'movier's theorem z^1/3 = pi^1/3 (cos theta + i sin theta)^1/3 z^1/3 = pi^1/3 [cos 2k pi + theta/3 + i sin 2 k pi + theta/3]^1/3 K = 0, 1, 2... consider k = 0 z^1/3 = pi^1/3 [cos theta/3 + i sin theta/3] consider k = 1 z^1/3 = pi^1/3 [cos 2 pi + theta/3 + i sin 2pi + theta/3] consider k = 2 z^1/3 = pi^1/3 [cos 4pi + theta/3 + i sin 2pi + theta/3]