M300 Section 201 Assignement 4 due Thursday, February 25 at the beginning of class In this homework, log may be either the multi-valued logarithm or one of its branches: we specify which one for each question. The last question marked by a star is optional and will not be marked. 1. Let P (z) = a0 + a1 z + ad z d be a nonconstant polynomial of degree with real coecients ai . (i) Show that if C is a root of P (z) of multiplicity n N, then is also a root of P (z), with same multiplicity. When do we have = ? (ii) Show that P (z) has a factorization of the form P (z) = Q1 (z) Q (z), where the Qi are polynomials with real coecients of degree one or two. 2. Show that the zeros of cos z in C are z = + 2k, k Z. 2 3. By rst computing the k-th derivative of eiz , show the following formulas: cos(k) z = cos(z + k ), 2 sin(k) z = sin(z + k ). 2 4. Show that the following formulas hold for all z C. (i) cos2 z + sin2 z = 1. 1 . cos2 z 5. Here log is the principal branch of the logarithm. Compute the following quantities: (ii) tan z = 1 + tan2 z = 11 (i) log(2ei 6 ). (ii) log(1 + i). 6. Here log is the principal branch of the logarithm. Show that log(z 2 + 1) is well-dened and analytic on = { z C {0} : Arg z ( , ) }. You may argue by drawing. 4 4 7. Here log is the multi-valued logarithm and logprinc is its principal branch. Let f (z) denote a branch of log z on C (, 0]. (i) Show that there exists a continuous function k : C, taking only integer values, such that f (z) = logprinc z + 2ik(z) for all z . (ii) Let z0 . Show that k is constant on a suciently small neighborhood of z0 . 1 2 8. Here log is the multi-valued logarithm. Find a branch of log near 0 and takes the value 3i at 0. 1 which is analytic z1 9. Here log is the principal branch of the logarithm. (i) Determine the image of (0, +) by the non-increasing function g(x) = (ii) Determine the domain of analycity of f (z) = log 10. () 1 . 1 1 + 1. x z2 We dene if z = 0, 0 f (z) = 1 e z4 if z = 0. 1 1 You may use the fact that limt0 |t| e t4 = 0 for real t. We view f as a function from R2 to C when we write f or f . Show that: x y (i) f is analytic on C {0}. (ii) The partial derivatives of f at 0 exist and are equal to 0. (iii) The partial derivatives of f exist everywhere and satisfy the Cauchy-Riemann equations at every point. (iv) Writing z = x + iy, we have 1 4 f (x, y) = 5 e z4 , x z i f 4i 1 (x, y) = 5 e z4 . y z (v) For t R, |f (te 8 )/t| = 1/|t|. Deduce that f does not have a complex derivative at 0. Considering (iii), does this contradict a theorem seen in class? (vi) Reasoning as in (v), show that f is not continuous at zero. x