Ricardo A. Sáenz

Due May 22nd Problem 1 Let $latex a\in\mathbb C$ and $latex f,g\in\mathscr O_a$. Then $latex f = g$ if and only if $latex f^{(k)}(a) = g^{(k)}(a)$ for all $latex k\in\mathbb N$. Problem 2 Let $latex \gamma_0, \gamma_1$ paths from $latex a$ to $latex b$ in $latex V$. Then $latex \gamma_0$ and $latex \gamma_1$ are path-homotopic in $latex V$ if and only if $latex \gamma_0-\gamma_1$ is homotopic to a constant in $latex V$, where $latex \gamma_0-\gamma_1:[0,1]\to\mathbb C$ is defined by $latex (\gamma_0-\gamma_1)(t) = \begin{cases}\gamma_0(2t) & 0\le t\le 1/2\\ \gamma_1(2-2t) & 1/2\le t\le 1. \end{cases}$ Problem 3 Let $latex V\subset W\subset\mathbb C$ open connected, $latex u:W\to\mathbb R$ harmonic and $latex f\in H(V)$ such that $latex \Re f = u$ in $latex V$. Then $latex (f,V)$ admits unrestricted continuation in $latex W$.

Due May 15th Problem 1 Let $latex f:[0,1]\to(0,1)$ continuous and $latex V=\{x+iy:x\in(0,1), f(x) < y < 1\}$. Then every point $latex x + if(x), x\in(0,1)$ in $latex \partial V$ is simple. Problem 2 Define $latex f\in H(\mathbb D)$ by $latex f(z) = e^{i/(z-1)^2}$ and $latex \gamma:[0,1]\to\mathbb C$ by $latex \gamma(t) = 1 + (1-t)e^{3\pi i/4}$. $latex \gamma([0,1))\subset\mathbb D$ (draw a picture) $latex \displaystyle \lim_{t\to1} f(\gamma(t))$ exists $latex \displaystyle \lim_{r\to 1} f(r)$ does not exist Find $latex \gamma_1$ such that $latex |f(\gamma_1(t))|\to\infty$ as $latex t\to1$

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.

Problem 1 Let $latex f$ be entire with $latex f(x)$ real-valued for $latex x\in\mathbb R$. Define $latex g:\mathbb R\to\mathbb R$ by $latex g(y) = \Re f(iy)$. Then $latex g$ is even. Problem 2 Let $latex f$ be entire with $latex f(x)$ real-valued for $latex x\in\mathbb R$. Define $latex h:\mathbb R\to\mathbb R$ by $latex h(y) = \Im f(iy)$. Then $latex h$ is an odd function. Problem 3 Verify the previous problems with the functions $latex \sin z, \cos z, 1 + z^2 + z^3$. Problem 4 Let $latex f$ be entire with $latex f(x)$ real-valued for $latex x\in\mathbb R$ and $latex f(iy)$ purely imaginary for $latex y\in\mathbb R$. Then $latex f$ is odd. Problem 5 What can you say if, instead, $latex f(iy)$ is also real valued for $latex y\in\mathbb R$?

Due April 24th Problem 1 Let $latex U,V\subset\mathbb C$ open, $latex f:U\to V$ holomorphic and $latex u:V\to\mathbb C$ harmonic. The $latex u\circ f$ is harmonic in $latex U$. Problem 2 Let $latex f\in C(\partial \mathbb D)$ and $latex \psi\in\text{Aut}(\mathbb D)$. Then $latex \mathscr P(f\circ\psi) = \mathscr Pf\circ\psi$. (A sketch of the proof of this result is given in the text --Theorem 10.2.0--; give the details.)  Problem 3 Explain why the following "proof" that there is no continuous function on $latex \bar{\mathbb D}$, holomorphic in $latex \mathbb D$ and equal to $latex f(e^{it}) = e^{-it}$ on the boundary, is wrong: Let $latex u\in C(\bar{\mathbb D})$ be holomorphic in $latex \mathbb D$ with $latex u|_{\partial\mathbb D} = f$. Then $latex v(z) = u(z) - 1/z$ is holomorphic in $latex \mathbb D\setminus\{0\}$, vanishing on the boundary of $latex \mathbb D$. Since the set $latex \partial\mathbb D$ has an accumulation point and $latex \mathbb D\setminu...

Problem 1 Let $latex V\subset \mathbb C$ be open and connected and $latex \mathcal F\subset H(V)$. If $latex \mathcal F$ is a normal family, then $latex \mathcal F' = \{f': f\in\mathcal F\}$ is also a normal family. Problem 2 The converse to the result in Problem 1 is false. Find a "small" hypothesis to make the converse true. Problem 3 Let $latex V\subset \mathbb C$ be open and connected and $latex \mathcal F\subset H(V)$ a normal family. Let $latex f_n\in \mathcal F$ a sequence such that $latex f_n(z)\to f(z)$ for each $latex z\in S\subset V$, where $latex S$ has a limit point in $latex V$ and $latex f\in H(V)$. Then $latex f_n\to f$ in $latex H(V)$. Problem 4 Let $latex V\subset \mathbb C$ be open, $latex M>0$, and  $latex \displaystyle\mathcal F = \Big\{ f\in H(V): \iint_V |f(z)|^2 dxdy \le M \Big\}$. Then $latex \mathcal F$ is a normal family. Problem 5 Let $latex V\subset \mathbb C$ be open and connected and $latex \{f_n\}\subset H(...

Problem 1 Consider the function on $latex [0,\infty)$ given by $latex \psi(t) = \dfrac{t}{1+t}$. $latex \psi$ is concave and increasing $latex \psi$ is continuous at $latex 0$ For any $latex \varepsilon > 0$ there exists $latex \delta > 0$ such that $latex \psi(t) < \delta$ implies $latex 0\le t < \varepsilon$ Problem 2 Let $latex d_j$ be a sequence of quasimetrics on $latex X$ that separates points, and $latex d$ the metric constructed from the $latex d_j$ as in class. Let, for each positive integer $latex N$, $latex x\in X$ and $latex \varepsilon>0$, $latex B_N(x,\varepsilon) = \{ y\in X: d_j(x,y) < \varepsilon, j = 1, 2, \ldots, N\}.$ Then $latex B_N(x,\varepsilon)$ is open with respect to the metric $latex d$. Problem 3 $latex S\in X$ is totally bounded in $latex (X,d)$ if and only if for any $latex \varepsilon > 0$ and any positive integer $latex N$ there exist $latex x_1, x_2, \ldots, x_m$ such that $latex \displaystyle S \subset \bigc...

Problem 1 The map $latex \Phi(z) = i\dfrac{1+z}{1-z}$ is a biholomorphic mapping from the unit disk $latex \mathbb D$ onto the upper half-plane $latex \mathbb H$. $latex \Phi$ is called the Cayley transform . Problem 2 Let $latex A=\begin{pmatrix} a & b\\c & d \end{pmatrix}\in SL(2,\mathbb R)$ and $latex \phi_A$ the Möbius transformation $latex \phi_A(z) = \dfrac{az+b}{cz+d}$. Then $latex \phi_A\in\text{Aut}(\mathbb H)$. Problem 3 For $latex a\in\mathbb H$, give an explicit $latex A\in SL(2,\mathbb R)$ so that $latex \phi_A$ takes $latex a$ to $latex i$. Problem 4 If $latex \psi\in\text{Aut}(\mathbb H)$, then there exists $latex A\in SL(2,\mathbb R)$ so that  $latex \psi = \phi_A$. ( Hint:  Consider the matrix $latex A = \begin{pmatrix} \cos\theta & -\sin\theta\\\sin\theta & \cos\theta \end{pmatrix}$ and verify that $latex \Phi^{-1}\circ\phi_A\circ\Phi$ is a rotation in $latex \mathbb D$. Use the previous problem and the discus...

Due March 13th Problem  1 Let $latex z\mapsto P(z)$ be the stereographical projecton. Then $latex P(z)$ and $latex P(w)$ are antipodal points if and only if $latex z\bar w = -1$. Problem 2 Let $latex f(z)$ be an entire function such that $latex \lim_{z\to\infty} f(z) = \infty$. Prove, without any use of the Riemann sphere, that $latex f(z)$ is a polynomial. Problem 3 Use the previous problem (and not the theorems seen in class) to prove that $latex \text{Aut}(\mathbb C)$ is equal to the set of nonconstant affine maps. Problem 4 Use the previous problem (and not the theorems seen in class) to prove that $latex \text{Aut}(\mathbb C_\infty)$ is equal to the set of all linear-fractional transformations. Problem 5 For a non-singular complex $latex 2\times 2$ matrix $latex A$, let $latex \phi_A$ be the linear-fractional transformation seen in class. Verify directly that $latex \phi_A\circ\phi_B = \phi_{AB}$. For$latex A\in GL_2(\mathbb C)$, let $latex P_A$ be ...

Due March 6th Problem 1 Suppose $latex f\in H(\mathbb C)$, $latex f(z+1) = - f(z)$ for all $latex z\in\mathbb C$, $latex f(0) = 0$ and $latex |f(z)| \le e^{\pi|\Im z|}$ for all $latex z\in\mathbb C$. Then $latex f(z) = c\sin\pi z$ for some constant $latex c$. ( Hint:  Use Problem 4 of Homework 2.) Problem 2 Suppose $latex f:\mathbb C\to\mathbb C$ is continuous, $latex f(z + 2) = f(z)$ and $latex |f(z)| \le e^{\pi|z|}$ for all $latex z\in\mathbb C$. Then there exists a constant $latex c$ such that $latex |f(z)| \le c e^{\pi|\Im z|}$ for all $latex z\in\mathbb C$. Problem 3 Find $latex \displaystyle \sum_{n=1}^\infty \frac{1}{n^2}$ using the series for $latex \cot\pi z$. Problem 4 Prove that Theorem 7.5 in the text follows from Theorem 7.6. Problem 5 Suppose $latex f\in H(V)$, $latex W\subset \mathbb C$ is open, and $latex h:W\to V$ satisfies $latex f(h(z)) = z$ for all $latex z\in W$. Give an example where it is not true that $latex h\in H(W)$. If $lat...

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...

Esta es una lista de las principales ideas vistas en el curso, y que sirve como referencia para prepararse para cualquiera de los exámenes finales (ordinario, extraordinario, regularización). Números reales Propiedad arquimidiana Equivalencias entre los enunciados de completitud Principio de intervalos encajados Axioma del supremo Teorema de Bolzano-Weierstrass Criterio de convergencia de Cauchy Convergencia de series absolutamente convergentes Series Convergencia y criterio de Cauchy Convergencia absoluta Convergencia condicional Truco de Abel Criterio de Dirichlet Reordenamientos Teorema de Riemann Series de funciones Convergencia uniforme Criterio de Cauchy Continuidad Diferenciabilidad Integración Convergencia dominada Criterio M de Weierstrass Series de potencias Radio de convergencia $latex limsup$  Teorema de Abel Series de Fourier Núcleo de Dirichlet Lema de Riemann Teorema de Dirichlet Teoremas del cálculo V...

Como se establece en el programa del curso, 50% de la calificación ordinaria consiste en el desarrollo de un proyecto final. El proyecto debe ser entregado el 9 de diciembre, antes del examen ordinario escrito. A continuación, la lista de proyectos finales a desarrollar. Continuidad de funciones aditivas Fernando Cedeño Teorema de Tauber Carolina Estévez Incontabilidad de los reales Uri Gallegos Funciones trigonométricas Cristina Núñez

El segundo examen parcial es este viernes, y aquí tienen una guía del material cubierto que puede ayudarles a prepararse. Completitud de los reales Equivalencias entre los enunciados de completitud Principio de intervalos encajados Axioma del supremo Teorema de Bolzano-Weierstrass Criterio de convergencia de Cauchy Convergencia de series absolutamente convergentes Series Convergencia y criterio de Cauchy Convergencia absoluta Criterios de convergencia por comparación Criterios del cociente y la raíz Criterio de condensación de Cauchy; criterio p Criterio de la integral Convergencia condicional Series alternantes Truco de Abel Criterio de Dirichlet Reordenamientos Teorema de Riemann Series hipergeométricas Criterio de Gauss Series de funciones Convergencia uniforme Criterio de Cauchy Continuidad Diferenciabilidad Integración Convergencia dominada Criterio M  de Weierstrass Series de potencias Radio de convergencia $latex ...

Fecha de entrega: 29 de noviembre Problema 1 Sean $latex f, g$ continuas en $latex [a,b]$ tales que $latex \displaystyle \int_a^b f = \int_a^b g$. Muestra que existe $latex c\in[a,b]$ tal que $latex f(c) = g(c)$. Problema 2 Sea $latex \phi:[a,b]\to[c,d]$ diferenciable, inyectiva y creciente, tal que $latex \phi(a) = c$ y $latex \phi(b) = d$. Si $latex f$ es integrable en $latex [c,d]$, entonces $latex (f\circ\phi)\phi'$ es integrable en $latex [a,b]$ y  $latex \displaystyle \int_c^d f(x) dx = \int_a^b f(\phi(t))\phi'(t) dt$. Problema 3 Sean $latex f,g$ integrables en $latex [a,b]$. Muestra que $latex \displaystyle \Big( \int_a^b fg \Big)^2 \le \int_a^b f^2 \cdot \int_a^b g^2$. Muestra que $latex \displaystyle \Big( \int_a^b (f+g)^2 \Big)^{1/2} \le \Big( \int_a^b f^2 \Big)^{1/2} + \Big( \int_a^b g^2 \Big)^{1/2}$ Problema 4 Sea $latex f$ periódica en $latex \mathbb R$, con periodo $latex T$, e integrable en $latex [0,T]$. Entonces, para cualquier $latex a...

Fecha de entrega: 22 de noviembre Problema 1 Sean $latex f, g$ diferenciables en $latex [a,b]$ tales que $latex f', g'$ son integrables. Muestra que $latex f'g, fg'$ son integrables y $latex \displaystyle \int f'g = f(b)g(b) - f(a)g(a) - \int g'f.$ Problema 2 Sea $latex f$ integrable en $latex [a,b]$ tal que $latex 1/f$ es acotada. Muestra que $latex 1/f$ es integrable en $latex [a,b]$. Problema 3 Sea $latex f$ continua en $latex [0,1]$ y, para cada $latex n$, define en $latex [0,1]$ la función $latex g_n(x) = f(x^n)$. Muestra que $latex \displaystyle \int g_n \to f(0).$ Problema 4 Muestra que $latex \displaystyle \frac{1}{3\sqrt 2} \le \int_0^1 \frac{x^2}{\sqrt{1+x^2}}dx \le \frac{1}{3}$. Problema 5 Sea $latex f$ continua en $latex [a,b]$ y $latex M$ su valor absoluto máximo. Muestra que $latex \displaystyle \Big( \int |f|^n \Big)^{1/n} \to M$.

Fecha de entrega: 15 de noviembre Problema 1 Muestra de manera directa que, si $latex f$ es continua en $latex [a,b]$, entonces $latex L(f) = U(f)$. Problema 2 Muestra que, si $latex f$ es acotada y continua en $latex [a,b]$ excepto en un punto $latex x_0\in[a,b]$, entonces $latex L(f) = U(f)$. Problema 3 Considera la función $latex f(x) = x$ en $latex [a,b]$. Muestra que, para cualquier partición $latex \mathcal P$,  $latex L(f,\mathcal P) \le \dfrac{b^2-a^2}{2} \le U(f,\mathcal P)$. Concluye que $latex \displaystyle \int f = \frac{b^2-a^2}{2}$. Problema 4 Sea $latex f$ integrable en $latex [0,1]$. Muestra que $latex \displaystyle \frac{1}{n}\sum_{k=1}^n f\Big(\frac{k}{n}\Big) \to \int f.$ Problema 5 Sea $latex f$ integrable en $latex [-a,a]$. Si $latex f$ es impar, muestra que $latex \displaystyle  \int_{-a}^a f = 0$. Si $latex f$ es par, muestra que $latex \displaystyle  \int_{-a}^a f = 2\int_0^a f$.