Math 300: Assignment #5: Due: Friday, Oct. 23. in class. 1. Section 3.3: 6. 2. Section 3.3: 16. 3. Recall that arg (z) is the branch of the argument that lies in (, + 2]. Let L be the branch of the logarithm dened using arg (z). a. Find L0 (i), L/2 (i). b. Where is L/2 (z 2 + 1) analytic? 4. Find the principal values of a. (i)i1 . b. i2/3 . 5. Let the square root be computed using the branch L/2 of the logarithm. a. Where is (z 2 i)1/2 analytic? b. Where is z(1 i/z 2 )1/2 analytic? (Hint: show that if z is in the nonanalytic locus then z 2 lies on the half-circle with center i/2 and radius 1/2. Sketch the set of such z.) 6. Where is the principal branch of tan1 (z) analytic? 7. Consider the principal branch of sin1 (z). (That means, both the logarithm and the square root are computed using the principal branches.) a. Show that if z is purely imaginary, z = ib, then so is sin1 (z)? b. If z is real, is sin1 (z) also real