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## Park City Mathematics InstituteUndergraduate 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 $\displaystyle \frac{\partial^2 u}{\partial\xi^2} + \frac{\partial^2 u}{\partial\eta^2} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}$.

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

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

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.