Approximation by Rational Functions

Prove that every meromorphic function on $S^{2}$ is rational.

Let $\Omega = \left\{ z:\left|z\right| < 1\text{ and }\left|2 z-1\right| > 1\right\}$, and suppose $f \in H(\Omega )$.

Is there a sequence of polynomials $P_{n}$ such that $P_{n}(0) = 1$ for $n = 1,2,3, \ldots$, but $P_{n}(z) \rightarrow 0$ for every $z \neq 0$, as $n \rightarrow \infty$?

Is there a sequence of polynomials $P_{n}$ such that

For $n = 1,2,3, \ldots$, let $\Delta_{n}$ be a closed disc in $U$, and let $L_{n}$ be an arc (a homeomorphic image of $[0,1])$ in $U-\Delta_{n}$ which intersects every radius of $U$. There are polynomials $P_{n}$ which are very small on $\Delta_{n}$ and more or less arbitrary on $L_{n}$. Show that $\left\{ \Delta_{n}\right\} ,\left\{ L_{n}\right\}$, and $\left\{ P_{n}\right\}$ can be so chosen that the series $f = \Sigma P_{n}$ defines a function $f \in H(U)$ which has no radial limit at any point of $T$. In other words, for no real $\theta$ does $\lim_{r \rightarrow 1} f\left(r e^{i \theta }\right)$ exist.

Here is another construction of such a function. Let $\left\{ n_{k}\right\}$ be a sequence of integers such that $n_{1} > 1$ and $n_{k+1} > 2 k n_{k}$. Define

Prove that the series converges if $\left|z\right| < 1$ and prove that there is a constant $c > 0$ such that $\left|h(z)\right| > c \cdot 5^{m}$ for all $z$ with $\left|z\right| = 1-\left(1 / n_{m}\right)$.

Hence $h$ has no finite radial limits. Prove also that $h$ must have infinitely many zeros in $U$. (Compare with Exercise 15, Chap. 12). In fact, prove that to every complex number $\alpha$ there correspond infinitely many $z \in U$ at which $h(z) = \alpha$.

Show that in Theorem 13.9 we need not assume that $A$ intersects each component of $S^{2}-\Omega$. It is enough to assume that the closure of $A$ intersects each component of $S^{2}-\Omega$.

Prove the Mittag-Leffler theorem for the case in which $\Omega$ is the whole plane, by a direct argument which makes no appeal to Runge's theorem.

Suppose $\Omega$ is a simply connected region, $f \in H(\Omega ), f$ has no zero in $\Omega$, and $n$ is a positive integer. Prove that there exists a $g \in H(\Omega )$ such that $g^{n} = f$.

Suppose $\Omega$ is a region, $f \in H(\Omega )$, and $f \not\equiv 0$. Prove that $f$ has a holomorphic logarithm in $\Omega$ if and only if $f$ has holomorphic $n$th roots in $\Omega$ for every positive integer $n$.

Suppose that $f_{n} \in H(\Omega )(n = 1,2,3, \ldots ), f$ is a complex function in $\Omega$, and $f(z) = \lim_{n \rightarrow \infty } f_{n}(z)$ for every $z \in \Omega$. Prove that $\Omega$ has a dense open subset $V$ on which $f$ is holomorphic. Hint: Put $\varphi = \sup \left|f_{n}\right|$. Use Baire's theorem to prove that every disc in $\Omega$ contains a disc on which $\varphi$ is bounded. Apply Exercise 5, Chap. 10. (In general, $V \neq \Omega$. Compare Exercises 3 and 4).

Suppose, however, that $f$ is any complex-valued measurable function defined in the complex plane, and prove that there is a sequence of holomorphic polynomials $P_{n}$ such that $\lim_{n \rightarrow \infty } P_{n}(z) = f(z)$ for almost every $z$ (with respect to two-dimensional Lebesgue measure).