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Problem Set 1

Park City Mathematics Institute
Undergraduate Summer School 2018

Introduction to Harmonic Analysis

  1. Let $latex R$ be a rotation in the plane.

    1. Consider the change of variables $latex (\xi, \eta) = R(x,y)$. Then $latex \displaystyle \frac{\partial^2 u}{\partial\xi^2} + \frac{\partial^2 u}{\partial\eta^2} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}$.

    2. If $latex u$ is harmonic, then $latex u\circ R$ is also harmonic.

  2. Let $latex (r,\theta)$ be the polar coordinates of the plane. Then
    $latex \displaystyle \Delta u = \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial\theta^2}$

  3. Let $latex u$ be a harmonic function on $latex \R^2$. Then there exists a harmonic function $latex v$ that is conjugate to $latex u$, so $latex f = u + iv$ is holomorphic. (Hint: Consider a line integral of the 1-form $latex \displaystyle - \frac{\partial u}{\partial y} dx + \frac{\partial u}{\partial x} dy$.)

    1. If $latex v_1$ and $latex v_2$ are conjugate to $latex u$ in the plane, then $latex v_1 - v_2$ is a constant.

    2. If 0 is conjugate to $latex u$ in the plane, then $latex u$ is a constant.

    3. If $latex f$ is holomorphic in $latex \C$ and real valued, then $latex f$ is a constant.