Ir al contenido principal

## Due May 8th

### Problem 1

If $P(z)$ is a polynomial, then there exists $z_0$ such that $|z_0|=1$ and $|P(z_0) - \dfrac{1}{z_0}| \ge 1$.

### Problem 2

There exists a sequence $P_n(z)$ of polynomials such that $P_n(0) =1$ for all $n$ and $P_n(z)\to 0$ for all $z\in\mathbb C$, $z\not=0$.

### Problem 3

Assume Runge's Theorem and prove the following theorem: Let $V\subset\mathbb C$ open and $\Gamma$ a cycle in $V$. If  $\displaystyle \int_\Gamma f(z) dz = 0$ for all $f\in H(V)$ of the form $f(z) = 1/(z-a)$, then $\displaystyle \int_\Gamma f(z) dz = 0$ for all $f\in H(V)$.

### Problem 4

Let $V\subset\mathbb C$ be bounded, connected and open. There exists $f\in H(V)$ which cannot be extended to a function holomorphic in a strictly larger open set.

### Comentarios

1. Should it be z_0 in the inequality in problem 1?

2. Any hint on how to take the Kn?

1. The book says that we can consider an increasing sequence of compact subsets Kn, whose union is C\{0}

3. Algún hint extra para el 2?

1. https://i.blogs.es/bcef7b/070513_pacman/450_1000.jpg

2. otra cosa, en el 3 creo que tenemos que asumir que gamma tiene índice 0 alrededor de cualquier punto en el complemento de V

3. No, la hipótesis del problema se encarga de no necesitar eso.