Ricardo A. Sáenz

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

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

Due February 14th Problem 1 Explain why the function $latex f(z) = z^2\sin(1/z)$ for $latex z\not=0$, $latex f(0) = 0$, is not a counterexample to the result that, if a function is differentiable in a neighborhood of a point, then it is infinitely differentiable at that point. Problem 2 Verify explicitly that $latex \displaystyle \int_{\partial T}\frac{1}{z}dz = 2\pi i$ for the triangle $latex T = [1, -1+i, -1-i]$. Problem 3 Prove the mean value property : If $latex V\subset\mathbb C$ is open, $latex f\in H(V)$ and $latex \overline{D(z_0,r)}\subset V$, then $latex \displaystyle f(z_0) = \frac{1}{2\pi} \int_0^{2\pi} f(z_0 + r e^{it}) dt$. Problem 4 Let $latex f\in H(\mathbb C)$. If $latex |f(z)| \le e^{\Re z}$ for all $latex z$, then there exists a constant $latex c$ such that $latex f(z) = c e^z$. If there is $latex n\in\mathbb Z_+$ such that $latex |f(z)| \le (1 + |z|)^n$ for all $latex z$, then $latex f$ is a polynomial. If $latex f(n) = 0$ for all $latex n\i...

Due February 7th Problem 1 If $latex f$ is differentiable at $latex z$, then $latex f$ is continuous at $latex z$. Problem 2 Let $latex T:\mathbb C \to \mathbb C$ be $latex \mathbb R$-linear. $latex T$ is $latex \mathbb C$-linear if and only if $latex T(iz) = iTz$ for all $latex z\in\mathbb C$. If $latex T$ has matrix $latex \begin{pmatrix}a & b\\c & d\end{pmatrix}$, then $latex T$ is $latex \mathbb C$-linear if and only if $latex a = d$ and $latex b = -c$. Problem 3 Define $latex f:\mathbb C\to\mathbb C$ by $latex f(x+iy) = \begin{cases} 0 & x=0\\ 0 & y=0\\ 1 & \text{otherwise.}\end{cases}$ Then $latex f$ satisfies the Cauchy-Riemann equations at the origin, but it is not differentiable at the origin. Problem 4 Define $latex f:\mathbb C\to\mathbb C$ by $latex f(z) = \begin{cases} |z|^2\sin(1/|z|) & z\not=0\\ 0 & z=0.\end{cases}$ Then $latex f$ is differentiable at the origin, but the partial derivative $latex \partial_x u...