Math 185 Assignment 10

1. Laurent series

   Recall from Assignment 7 that if $0 \lt r \lt R \lt \infty$ and $f$ is holomorphic on (an open region containing) $\overline{A_{r, R}(0)}$ then for $z \in A_{r, R}(0)$ we have \[f(z) = \frac1{2\pi i}\paren{\int_{\partial B_R(z_0)} \frac{f(w)}{w-z}\,dw - \int_{\partial B_r(z_0)} \frac{f(w)}{w-z}\,dw}.\]
   1. Use the generalized Cauchy Integral Formula to produce a much simpler proof of this fact.
   2. Let $r \gt 0$ and $C_r = \partial B_r(0)$ the circle centred at $0$ of radius $r$. Suppose that $f$ is holomorphic on some open set containing $C_r$. Show that \[g_r : z \longmapsto \frac1{2\pi i}\int_{C_r} \frac{f(w)}{w-z}\,dw\] is holomorphic on $\C \setminus C_r$, with a removable singularity at $\infty$ so that $g(\infty) = 0$.
   3. The right-hand side of the identity \[f(z) = \frac1{2\pi i}\paren{\int_{\partial B_R(z_0)} \frac{f(w)}{w-z}\,dw - \int_{\partial B_r(z_0)} \frac{f(w)}{w-z}\,dw}\] makes sense for any $z \in \C \setminus (C_r \cup C_R)$. Does it agree with $f$ outside of $\overline{A_{r, R}(0)}$?
   4. Use the power series representation of $g_r(1/z)$ to show that there are coefficients $(a_n)_{n\lt0}$ so that for $|z|\gt r$, \[g_r(z) = \sum_{n=-\infty}^{-1} a_nz^n,\] and the convergence is uniform on compact subsets of $\C\setminus\overline{B_r(0)}$.
   5. Conclude that if $f$ is holomorphic on an open set containing $\overline{A_{r, R}(0)}$ then there are coefficients $(a_n)_{n\in\Z}$ so that \[f(z) = \sum_{n=-\infty}^{\infty} a_nz^n\] uniformly on compact subsets of $A_{r, R}(0)$.
   6. Suppose now that $f$ is holomorphic on an open set containing $\overline{A_{r, R}(z_0)}$. Show that there are coefficients $(a_n)_{n\in\Z}$ so that \[f(z) = \sum_{n=-\infty}^{\infty} a_n(z-z_0)^n\] uniformly on compact subsets of $A_{r, R}(z_0)$.
   7. For $|z| \neq 1$, compute \[\int_{C_1} \frac{1}{w(w-z)}\,dw.\]

2. Obstructions to primitives

   Suppose $\Omega_1 \subseteq \C$ is open and simply connected, $s_1, \ldots, s_n \in \Omega_1$ are distinct, $S = \set{s_1, \ldots, s_n}$, and $\Omega = \Omega_1 \setminus S$.
   1. Let $f : \Omega \to \C$ be holomorphic. Show that $f$ admits a primitive if and only if for every $s \in S$, $\res_s(f) = 0$.

   2. Show that there are holomorphic functions $\varphi_1, \ldots, \varphi_n : \Omega \to \C$ so that for any $f : \Omega \to \C$ holomorphic, there are coefficients $\alpha_1, \ldots, \alpha_n \in \C$ so that \[f + \alpha_1\varphi_1 + \alpha_2\varphi_2 + \ldots + \alpha_n\varphi_n\] admits a primitive.

      (What you have proven is that if $\mathcal{H}_\Omega$ is the vector space of functions holomorphic on $\Omega$ and $\mathcal{H}^0_{\Omega}$ is the vector space of functions on $\Omega$ which admit primitives, then \[\dim_\C\paren{\mathcal H_\Omega / \mathcal H^0_\Omega} = n\text{.)}\]