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Problem set 1, Harmonic Analysis

Due February 8

Problem 1

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

Problem 2

  1. Let $latex u$ be a harmonic function in $latex \mathbb R^2$. Then there exists a conjugate harmonic function $latex v$ to $latex u$. (Hint: Consider a line integral of the 1-form $latex - \dfrac{\partial u}{\partial y} dx + \dfrac{\partial u}{\partial x} dy$.)
  2. If $latex v_1$ and $latex v_2$ are conjugate to $latex u$ in the plane, then $latex v_1 - v_2$ is constant.

Problem 3

  1. If 0 is conjugate to $latex u$ in the plane, then $latex u$ is constant.
  2. If $latex f$ is holomorphic in $latex \mathbb C$ and real valued, then $latex f$ is constant.

Problem 4

Let $latex \Gamma(s)$ be the gamma function.
  1. Integrate by parts to verify the identity $latex \Gamma(s+1) = s \Gamma(s)$.
  2. For every $latex n\in\mathbb Z_+$, $latex \Gamma(n) = (n-1)!$.

Problem 5

  1. Use polar coordinates to verify the identity $latex \displaystyle \int_{\mathbb R^2} e^{-\pi|x|^2} dx = 1$.
  2. For every dimension $latex d$, $latex \displaystyle \int_{\mathbb R^d} e^{-\pi|x|^2} dx = 1$.
  3. Use spherical coordinates to verify $latex \displaystyle \omega_d = \frac{2\pi^{d/2}}{\Gamma(d/2)}$.