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Mostrando las entradas de febrero, 2020

### Homework 4: Complex Analysis

Due February 28th Problem 1 For any positive integer $n$, the polynomial $z^n(z-2)-1$ has $n$ roots in the disk $\mathbb D = D(0,1)$. Problem 2 Suppose $f:\overline{\mathbb D}\to\overline{\mathbb D}$ is continuous and holomorphic in $\mathbb D$. Then $f$ has a fixed point in $\overline{\mathbb D}$. Problem 3 Prove Hurwitz's Theorem: If $V$ is an open set, $f_n\in H(V)$ is a sequence such that $f_n\to f$ uniformly on compact subsets of $V$, $\overline{D(z,r)}\subset V$, and $f$ has no zeroes on $\partial D(z,r)$, then there exists $N$ such that, for all $n\ge N$, $f_n$ and $f$ have the same number of zeroes in $D(z,r)$. Problem 4 The series $\displaystyle \sum_{n=0}^\infty \frac{1}{z-n}$ diverges for all $z\in\mathbb C\setminus\mathbb Z$. The series $\displaystyle \sum_{n=0}^\infty \Big( \frac{1}{z-n} + \frac{1}{z+n} \Big)$ conv

### Homework 3: Complex Analysis

Due February 21st Problem 1 Verify explicitly that $\text{Ind}(\partial D(z_0,r),z_0) = 1$ for all $z_0\in\mathbb C$ and $r>0$. For $z_0\in\mathbb C$ and $r>0$, $\text{Ind}(\partial D(z_0,r),z) = \begin{cases} 1 & |z-z_0| < r\\ 0 & |z-z_0| > r. \end{cases}$ Problem 2 If $f,g$ are holomorphic near $z_0$ and $f$ has a simple zero at $z_0$, find an expression for the residue of $g/f$ at $z_0$. If $f$ has a simple pole at $z_0$ and $g$ is holomorphic near $z_0$, then $\text{Res}(fg,z_0) = g(z_0)\text{Res}(f,z_0)$. If $f$ is holomorphic near $z_0$ and $g(z) = f(z)/(z-z_0)^n$, then $\text{Res}(g,z_0) = \dfrac{f^{(n-1)}(z_0)}{(n-1)!}$. Problem 3 Use the residue theorem to show that $\displaystyle \int_{-\infty}^\infty \frac{dx}{1+x^2} = \pi$. ( Hint:  For $R>0$, consider the upper semicircle $\gamma$