MATH 800 Midterm: MATH 800 KU Math800 Spring 2012 Midterm
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF KANSAS
MIDTERM MATH 800 - Spring 2012
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(1) (60 points)
Suppose Ω is an open set and P0⊂D(P0, r)⊂Ω. Let f: Ω \ {P0} → Cis
holomorphic, so that for soem constant Cand for all z∈D(P0, r)\ {P0},
|f(z)| ≤ C
p|z−P0|.
Show that P0is removable singularity for f.
Document Summary
Suppose is an open set and p0 d(p0, r) . Let f : \ {p0} c is holomorphic, so that for soem constant c and for all z d(p0, r) \ {p0}, Show that p0 is removable singularity for f . p|z p0| Is there an analytic function on d, so that f : d d with f (0) = 1 f (0) = 3. Let c be an open and connected set, so that d = {z : |z| 1} . Let f : c be a non-constant holomorphic function. Show that if |f (z)| = 1 whenever |z| = 1, then f (d) d. Hint: the claim follows by establishing that for every w0 d, there is z0 d, so that f (z0) = w0. This should be done in a few steps: (a) show (by argument principle) that the equation f (z) = w0 has a solution, if f (z) = 0 has a solution.
