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Homework 2: Complex Analysis

Due February 14th

Problem 1

Explain why the function $latex f(z) = z^2\sin(1/z)$ for $latex z\not=0$, $latex f(0) = 0$, 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 $latex \displaystyle \int_{\partial T}\frac{1}{z}dz = 2\pi i$ for the triangle $latex T = [1, -1+i, -1-i]$.

Problem 3

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

Problem 4

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

Problem 5

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