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Mostrando las entradas de febrero, 2019

Harmonic Analysis: Week 1 notes

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

Tareas de la clase de Cálculo 2

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

Problem set 1, Harmonic Analysis

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