Find the ?rst four nonzero Euler numbers. SolutionWhat is the radius of convergence of the series. What is the radius ofFind the ?rst three Bernoulli numbers. Explain why there are no odd terms in this series. ThenSuppose there are a ?nite number of points on the boundary suchShow that jf (z)j \ufffdM on D. This is a technical improvement on the preceding exerciseSolution. Seip)Suppose that lim sup jf (z)j \ufffd M as z approaches any point of theRemark. R ! 1, unless p (z) is a constant.SolutionD, which can only happens if f (z) is a constant, by the maximum principle.Suppose that f (z) extends continuously to the imaginary axis andSolutionSolution (K. Then 1Thus, p (z) is constant, aSolution (K. Note that ifAssume p (z) is non-zero everywhere. It is su? cient to show that any p (z) has one root, for by division we can. This show that a \ufffd u \ufffd b on D.Solution (A. Then itSo \ufffdu \ufffd \ufffda on D, equivalently a \ufffd u \ufffd b on D. Prove MV theorem for harmonic functions in Rn, as follows u (R\ufffd!x ) \ufffdBetter: Apply stroke?s theorem to shell f\ufffd0 b fore some z0 2 D.
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