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Problem 19

Suppose \(\left\\{f_{n}(z)\right\\}\) is a sequence of analytic functions that converge uniformly to \(f(z)\) on all compact subsets of a domain \(D\). Let \(f_{n}\left(z_{n}\right)=0\) for every \(n\), where each \(z_{n}\) belongs to \(D .\) Show that every limit point of \(\left\\{z_{n}\right\\}\) that belongs to \(D\) is a zero of \(f(z)\).

Expert verified

The limit point \(a\) of the sequence \(\{z_{n}\}\) that belongs to the domain \(D\) is a zero of the function \(f(z)\) because every subsequence \(\{z_{n_k}\}\) converging to \(a\) satisfies \(f_{n_k}(z_{n_k}) = 0\) due to the uniform convergence of \(\{f_n(z)\}\) to \(f(z)\) over the domain \(D\).

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Chapter 9

Suppose \(f(z)\) has a pole of order \(m\) at \(z=z_{0}\), and \(P(z)\) is polynomial of degree \(n\). Show that \(P(f(z))\) has a pole of order \(m n\) at \(z=z_{0}\).

Chapter 9

If \(f(z)\) is analytic and nonzero in the disk \(|z|<1\), prove that for \(0<\) \(r<1\) $$ \exp \left(\frac{1}{2 \pi} \int_{0}^{2 \pi} \log \left|f\left(r e^{i \theta}\right)\right| d \theta\right)=|f(0)| . $$

Chapter 9

If \(a>1\), prove that \(f(z)=z+e^{-z}\) takes the value \(a\) at exactly one point in the right half-plane.

Chapter 9

Show that the polynomial \(z^{4}+4 z-1\) has one root in the disk \(|z|<1 / 3\) and the remaining three roots in the annulus \(1 / 3<|z|<2\).

Chapter 9

Let \(F_{1}(z)=z^{5}+z+16, F_{2}(z)=z^{7}-5 z^{3}-12\) and $F_{3}(z)=z^{7}+6 z^{3}+12$. Determine whether all zeros of these functions lie in the annulus \(1<\) \(|z|<2\)

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