advertisement

M519 Sample Final All problems would be weighted equally, 10 Problems would be graded. ♯ 1: Find the Laurent series for f (z), |z| > R, R > 0, where Z ∞ −1/t e f (z) = dt. t2 z Classify the singularity at z = 0 and determine the residue. ♯ 2: Let P (z) = a0 + a1 z + a2 z 2 + · · · + an z n be a polynomial that satisfies |P (z)| 6 M for |z| = 1. Show that each coefficient ak satisfies |ak | 6 M . ♯ 3: Classify the singular points of f (z) in the extended complex plane and give the residue for all isolated singularities. Give the order of the pole in the case of a pole. (a) f (z) = ez −1 . z3 (b) f (z) = z(z−π)2 . (sin z)2 ♯ 4: Find the Laurent series in the annulus 1 < |z| < 2 for f (z) = ♯ 5: Evaluate R∞ x sin(x) dx. −∞ 4+x2 1 . (z − 1)2 (z − 2) Justify the use of any theorems from class. R∞ √ ♯ 6: Evaluate 0 1+xx2 dx by using an appropriate “keyhole contour”. Justify the use of any theorems from class. ♯ 7: Evaluate R π/2 0 dθ . 1+cos2 θ ♯ 8: Show that the equation 2z 5 + 8z − 1 = 0 has (a) no roots in |z| > 2, (b) one root in |z| < 1 and that this root is real and positive, (c) exactly four roots in the annulus 1 < |z| < 2. ♯ 9: Let P (z) = an z n + · · · + a1 z + a0 with an , a0 6= 0, and assume P (z) 6= 0 on |z| = 1. Let P ∗ (z) = a0 z n + a1 z n−1 + · · · + an . Prove that if |a0 /an | > 1, then P (z) has the same number of roots in |z| < 1 as does the polynomial a0 P (z) − an P ∗ (z). ♯ 10: Consider the birational transformation w = f (z) ≡ following regions under f (z)? (a) {reiθ | 0 6 r 6 1, 0 6 θ 6 π}. (b) {reiθ | 0 6 r 6 1, π 6 θ 6 2π}. z−i . z+i What is the image of the ♯ 11: Find a bilinear transformation that maps the half–plane above the line y = x onto the interior of the circle |w − 1| = 1.