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Homework for Math 103
Assignment Four
Due 6 December 1999
Work problems 2, 3, 4, 5, 13, 16, 18, 20 on pages 227-230 of your text (end of chapter ten).

You will want to consider the hints below. In addition, I found the following useful on problem 16. Recall that if $ a,b\in\mathbf{C}$ then $ [a,b]^*=\lbrace\,z\in\mathbf{C}:z=tb+(1-t)a\hbox{ for }
t\in[0,1]\,\rbrace $.
\begin{ques}
Suppose that $\Omega$\ is a domain, that $f\in H(\Omega)$, and that...
...ve, it is possible to show
that the inequality in Rudin's hint holds.
\end{ques}


HINT FOR #2: Either use the Baire Category Theorem or the observation that if $ f$ is not a polynomial, then given $ n\in\mathbf{Z}^{+}$, $ a\in\mathbf{C}$, and $ r>0$, there is a $ z\in D_{r}(a)$ such that $ f^{(n)}(z)\not=0$. HINT FOR #4: Estimate $ \vert f^{k+1}(z)\vert$. HINT FOR #5: The hypotheses of the problem don't allow us to conclude even that the limit function $ f=\lim_{n}f_{n}$ is continuous. Instead, you'll have to prove that $ \lbrace\,f_{n}\,\rbrace $ is uniformly Cauchy. You may want to use (and prove) that if $ g_{n}(z)\le M$ for all $ z\in\gamma^{*}$ and $ g_{n}$ converges pointwise to a 0, then

$\displaystyle \int_{\gamma}g(z)\,dz\to0.$    

HINT FOR #13: Use the residue theorem and #8. HINT FOR #18: This is a straightforward modification of the argument used in class for the case $ p=0$. HINT FOR #20: Notice that $ f_{n}'\to f'$ uniformly on compact subsets of $ \Omega$. Show this implies $ f_{n}'/f'\to f'/f$ uniformly on any $ \gamma^{*}$ provided $ f\not=0$ on $ \gamma^{*}$.




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Math 103 Fall 1999
1999-11-23