Problem Set 1

Park City Mathematics Institute
Undergraduate Summer School 2018

Introduction to Harmonic Analysis

  1. Let R be a rotation in the plane.
    1. Consider the change of variables (\xi, \eta) = R(x,y). Then .
    2. If u is harmonic, then u\circ R is also harmonic.
  2. Let (r,\theta) be the polar coordinates of the plane. Then
  3. Let u be a harmonic function on \R^2. Then there exists a harmonic function v that is conjugate to u, so f = u + iv is holomorphic. (Hint: Consider a line integral of the 1-form \displaystyle - \frac{\partial u}{\partial y} dx + \frac{\partial u}{\partial x} dy.)
    1. If v_1 and v_2 are conjugate to u in the plane, then v_1 - v_2 is a constant.
    2. If 0 is conjugate to u in the plane, then u is a constant.
    3. If f is holomorphic in \C and real valued, then f is a constant.
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