please explain thoroughly

0. (4 points) M0 Section 2.4 We have a statement \"If f(z) is not an entire function, then g(z) = f*(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) = /z, assuming that it is a branch that takes 1 to 7. Then it is not analytic on the branch cut, but g(z) = f?(z) = = is obviously an entire function. B: 1 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. . (3 points) M Section 2.4 If both f(z) and f(z) are analytic, what can you say about f(z)? Prove your claim. . (15 points) M 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) = ; and v(x, y) = ey are orthogonal. More precisely. at any point of intersection z; of the two curves, the tangent lines to the two curves are perpendicular, if f'(zy) # 0. (Hint: Compute the gradient vectors of u and v and use the Cauchy-Riemann equations. ) b. Let f(z) = 2. Show that at = = 0 (f'(0) = 0). in order for the two sets of curves have intersection that passing through = 0, ; = 9 = 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'(z9) # U in the previous part invalid.) . (3 points) M Section 2.5 Let u(x,y) = x y* 2z + 1. Prove that it is harmonic and find a harmonic conjugate v(z, y) so that f(z) = u(x,y) + iv(x, y) is analytic