Ricardo A. Sáenz

Due March 1 Problem 1 If the sequences $latex a_n$ and $latex b_n$ are bounded, then  $latex \displaystyle u(r,\theta) = \sum_{n=0}^\infty r^n (a_n \cos(n\theta) + b_n \sin(n\theta))$ is harmonic in $latex \mathbb D$. Problem 2 For $latex m,n\in\mathbb Z$, $latex \displaystyle \int_0^{2\pi} e^{im\theta}e^{-in\theta} d\theta = \begin{cases} 2\pi & m=n\\0 & m\not=n.\end{cases}$ Problem 3 The series $latex \displaystyle \sum_{n=1}^\infty \frac{(-1)^n}{n} \sin n\theta$ converges for each $latex \theta$. Problem 4 If $latex f$ is Riemann integrable and periodic with period $latex T$, then $latex \displaystyle \int_a^{a+T} f(x) dx = \int_0^T f(x) dx$ for any $latex a\in\mathbb R$. Problem 5 Let $latex g$ be the periodic function given in $latex [-\pi,\pi)$ by $latex g(\theta) = |\theta|$. For each $latex n\in\mathbb Z$, $latex \displaystyle \hat g(n) = \begin{cases}\dfrac{\pi}{2} & n=0\\0 & \text{even } n\not=0\\- \dfra...

Notes on the Poisson kernel and the Poisson integral:  Chapter 3 .

Due February 22 Problem 1: Symmetry Lemma If $latex x\in\mathbb B$ and $latex \xi\in\mathbb S$, then $latex \displaystyle \Big| |x|\xi - \frac{x}{|x|} \Big| = |x - \xi|.$ Problem 2: Hopf Lemma If $latex u$ is a nonconstant harmonic function in $latex \mathbb B$, is continuous on $latex \bar{\mathbb B}$, and attains its maximum at $latex \zeta\in\mathbb S$, then there exists $latex c > 0$ such that $latex u(\zeta) - u(r\zeta) > c (1 - r)$ for any $latex 0 < r < 1$. Problem 3 If $latex u$ is harmonic in $latex \Omega$ and $latex \bar B_r(x_0)\subset\Omega$, then the values of $latex u$ in $latex B_r(x_0)$ are determined by its values on $latex S_r(x_0)$. Problem 4 Let $latex u_n$ be a sequence of harmonic functions in $latex \Omega$ such that $latex u_n\rightrightarrows u$ on any compact $latex K\subset\Omega$. Then $latex u$ is harmonic in $latex \Omega$. Problem 5 Let $latex u$ be harmonic ...

Notes on basic properties of harmonic functions:  Chapter02.pdf

Due February 15 Problem 1 Suppose $latex u$ is harmonic in a neighborhood of $latex \bar\Omega$, where $latex \Omega$ is a $latex C^1$ domain. Then $latex \displaystyle \int_{\partial\Omega} \partial_\nu u \, d\sigma = 0$. Problem 2 If $latex \Omega\subset\mathbb R^d$ is a bounded domain and $latex u$ is harmonic in $latex \Omega$ and continuous on $latex \bar\Omega$, then $latex u$ takes its maximum and its minimum on $latex \partial\Omega$. Problem 3 Let $latex \Omega\subset\mathbb R^d$ be a bounded domain, $latex u$ and $latex v$ harmonic in $latex \Omega$ and continuous on $latex \bar\Omega$. If $latex u=v$ on $latex \partial\Omega$, then $latex u=v$ in $latex \Omega$. Problem 4 If $latex f$ is an entire function and its real part is nonnegative, then $latex f$ is constant. Problem 5 If $latex u$ is a radial harmonic function in $latex \mathbb B$, then it is constant.

You can download the notes from Week 1 here:  Motivation and preliminaires

Las tareas de la clase de Cálculo 2 serán publicadas en el portal de Classroom. Para ingresar es necesario utilizar la cuenta universitaria.

Due February 8 Problem 1 Let $latex (r,\theta)$ be the polar coordinates of the plane. Then  $latex \displaystyle \Delta u = \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial\theta^2}.$ Problem 2 Let $latex u$ be a harmonic function in $latex \mathbb R^2$. Then there exists a conjugate harmonic function $latex v$ to $latex u$. ( Hint: Consider a line integral of the 1-form $latex - \dfrac{\partial u}{\partial y} dx + \dfrac{\partial u}{\partial x} dy$.) If $latex v_1$ and $latex v_2$ are conjugate to $latex u$ in the plane, then $latex v_1 - v_2$ is constant. Problem 3 If 0 is conjugate to $latex u$ in the plane, then $latex u$ is constant. If $latex f$ is holomorphic in $latex \mathbb C$ and real valued, then $latex f$ is constant. Problem 4 Let $latex \Gamma(s)$ be the gamma function. Integrate by parts to verify the identity $latex \Ga...