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## Due February 14th

### Problem 1

Explain why the function $f(z) = z^2\sin(1/z)$ for $z\not=0$, $f(0) = 0$, is not a counterexample to the result that, if a function is differentiable in a neighborhood of a point, then it is infinitely differentiable at that point.

### Problem 2

Verify explicitly that $\displaystyle \int_{\partial T}\frac{1}{z}dz = 2\pi i$ for the triangle $T = [1, -1+i, -1-i]$.

### Problem 3

Prove the mean value property: If $V\subset\mathbb C$ is open, $f\in H(V)$ and $\overline{D(z_0,r)}\subset V$, then
$\displaystyle f(z_0) = \frac{1}{2\pi} \int_0^{2\pi} f(z_0 + r e^{it}) dt$.

### Problem 4

Let $f\in H(\mathbb C)$.
1. If $|f(z)| \le e^{\Re z}$ for all $z$, then there exists a constant $c$ such that $f(z) = c e^z$.
2. If there is $n\in\mathbb Z_+$ such that $|f(z)| \le (1 + |z|)^n$ for all $z$, then $f$ is a polynomial.
3. If $f(n) = 0$ for all $n\in\mathbb Z$, then all singularities of $f(z)/\sin \pi z$ are removable.

### Problem 5

Let $f\in H(D'(z_0,r))$ such that, for some $\alpha, c >0$,
$|f(z)| \le c |z-z_0|^{-\alpha}$.
If $\alpha < N+1$ for some $N\in\mathbb N$, then $f$ has a removable singularity or a pole of order no larger than $N$ at $z_0$.