Math 520a - Midterm take home exam Do 5 out of the 6 problems ...
Math 520a - Midterm take home exam Do 5 out of the 6 problems ...
Math 520a - Midterm take home exam Do 5 out of the 6 problems ...
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5. Let log(z) be <strong>the</strong> branch <strong>of</strong> <strong>the</strong> log with branch cut along <strong>the</strong> negativereal axis. It has a power series ab<strong>out</strong> z 0 = −1 + i:log(z) =∞∑a n (z − z 0 ) nn=0A confused pr<strong>of</strong>essor claims that since <strong>the</strong> distance from z 0 to <strong>the</strong> branchcut is 1, <strong>the</strong> radius <strong>of</strong> convergence <strong>of</strong> <strong>the</strong> power series is 1. Explain what iswrong with this reasoning and find (with justification) <strong>the</strong> correct radius <strong>of</strong>convergence.6. Suppose that f is analytic in a disc <strong>of</strong> radius ρ ab<strong>out</strong> 0 and has a zero <strong>of</strong>order n at 0. Suppose that f has no o<strong>the</strong>r zero in this disc. Define a closedcontour by γ(t) = f(re it ) for 0 ≤ t ≤ 2π, with r < ρ. Compute <strong>the</strong> windingnumber <strong>of</strong> γ ab<strong>out</strong> 0 in terms <strong>of</strong> n and r and justify your computation.2
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