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Homework 12: Complex Analysis

Due May 8th

Problem 1

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

Problem 2

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

Problem 3

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

Problem 4

Let $latex V\subset\mathbb C$ be bounded, connected and open. There exists $latex f\in H(V)$ 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?

    ResponderBorrar
  2. Any hint on how to take the Kn?

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

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

      Borrar
    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

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

      Borrar

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