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