MATH C34 FALL 2017 PROBLEM SET 2 GIULIO TIOZZO (1) Compute lim supn an for the following sequences. (Note: it may be infinite.) (a) an = (1)n + (1)3n (b) (1)n n2 + n an = (c) an = ein (d) \u001a an = n2 if n even if n odd. 1 n (2) Determine the radius of convergence of the following series: (a) X zn (1)n 2 n +1 n=1 (b) X n=1 zn +n 2n (c) X 3n n5 z n n=1 (d) X log(n) z n n=1 (Recall that log(n) n for n sufficiently large). (3) (a) Prove that if |z| < 1 then \u0012 \u00132 X 1 nz n1 = 1z n=1 (b) Write down a power series which equals the function f (z) = for all z with |z| < 1. (4) Consider the function f : C C given by f (z) = z 3 . 1 \u0010 1 1z \u00113 2 GIULIO TIOZZO (a) Under the identification of C with R2 , write this function as a function F : R2 R2 . That is, find a function F : R2 R2 such that F (x, y) = (Re f (x + iy), Im f (x + iy)) for all x, y R. (b) Compute the Jacobian matrix of F , and check that the Cauchy-Riemann equations hold. (5) Let f : C C be a holomorphic function, and suppose that f (C) R, i.e. the function only takes real values. (a) Using the Cauchy-Riemann equations, prove that f 0 (z) = 0 for all z C. (b) Prove that f is constant. Due Thursday, September 28, at 3 PM