complex analysisI want more details
HOMEWORK 2 DUE FEB 20. 1. Let u : D ! R be any function where D C is open. Show that u depends only on the distance to origin if and only if z @u @z 2 R. 2. For what values of z 2 C is the series 1Xn=0 zn 1 + z2n convergent? 3. (a) Describe the set fz 2 C : sinz = 0g. (b) Discuss the mapping properties of sinz. 4. (a) Give the principal branch of p1 ?? z. (b) Show that the real part of pz is always positive. 5. Let L(z) = z ?? z2 z ?? z3 z1 ?? z3 z1 ?? z2 be the cross ratio of z2; z3; z1 and z: Prove that if z1; z2; z3; z4 are on a circle in this order then L(z4) > 0: 6. Construct a linear fractional map T which maps the disk D(i; 1) centered at i and of radius r = 1; to the upper half plane Im (z) > 0; so that T (0) = 1; T (2i) = 0 and T (??1 + i) = 3. 7. Let f : G ! C be a one-to one analytic map of an open set G onto the unit disk U = fz : jzj