9. (4 points) Section 2.4 We have a statement "If f(z) is not an entire function, then g(z) = f2(z) cannot be an entire function.". Two students and a professor had the following conversation: A: The statement is false. I can give a counter-example. Let f(z) = Vz, assuming that it is a branch that takes -1 to i. Then it is not analytic on the branch cut, but g(z) = f2(z) = z is obviously an entire function. B: I think this is not a counter-example, so we cannot conclude the statement is false yet. Professor: Good! B is correct. The statement is false. Now I give you an assignment: (a) Show that student A's example does not work as a counter-example. (b) Give a (correct) counter-example. Now you are asked to complete this short assignment posted by the professor. 10. (3 points) Section 2.4 If both f( z) and - f(z) are analytic, what can you say about f ( z)? Prove your claim. 11. (15 points) Sections 2.3-2.4 Suppose f(z) = u(x, y) + iv(x, y) is analytic in a domain in the complex plane. a. Show that the family of level curves u(x, y) = c1 and v(x, y) = C2 are orthogonal. More precisely, at any point of intersection zo of the two curves, the tangent lines to the two curves are perpendicular, if f'(zo) # 0. (Hint: Compute the gradient vectors of u and v and use the Cauchy-Riemann equations.) b. Let f(z) = 22. Show that at z = 0 (f'(0) = 0), in order for the two sets of curves have intersection that passing through z = 0, c1 = C2 = 0 is the only possibility. Explain why the above result fails. (Hint: Don't forget the curves must pass through z = 0 in order to make the condition f'(zo) + 0 in the previous part invalid.) 12. (3 points) Section 2.5 Let u(x, y) = x2 - y2 - 2x + 1. Prove that it is harmonic and find a harmonic conjugate v(x, y) so that f(z) = u(x, y) + iv(x, y) is analytic