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## 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

1. 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 $- \dfrac{\partial u}{\partial y} dx + \dfrac{\partial u}{\partial x} dy$.)
2. If $v_1$ and $v_2$ are conjugate to $u$ in the plane, then $v_1 - v_2$ is constant.

### Problem 3

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

### Problem 4

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

### Problem 5

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