<Home>Qualifying Exams><Analysis>Fall 2020 - Problem 7

Fall 2020 - Problem 7

harmonic functions

Let $\Delta_j = \set{z \mid \abs{z - a_j} \leq r_j}$ , $1 \leq j \leq n$ , be a collection of disjoint closed disks, with radii $r_j \geq 0$ , all contained in the open unit disk $\D$ of the complex plane. Let $\Omega = \D \setminus \p{\bigcup_j \Delta_j}$ and let $\func{u}{\Omega}{\R}$ be harmonic. prove that there exist real numbers $c_1, \ldots, c_n$ such that

u\p{z} - \sum_{j=1}^n c_j\log\,\abs{z - a_j}

is the real part of a (single-valued) analytic function on $\Omega$ . Show also that the choice of $c_1, \ldots, c_n$ is unique.

Solution.

Let $u$ be a harmonic function on $\Omega$ , and consider its conjugate differential

{}^*\diff{u} = -\pder{u}{y} \,\diff{x} + \pder{u}{x} \,\diff{y}.

For each $j$ , let $c_j = \int_{\partial\Delta_j} {}^*\diff{u}$ and consider

v\p{z} = \sum_{j=1}^n \frac{c_j}{2\pi} \log\,\abs{z - a_j}.

Observe that ${}^*\diff{v} = \sum_{j=1}^n \frac{c_j}{2\pi} \,\diff\theta_j$ so that

\int_{\partial\Delta_k} {}^*\diff{v} = \sum_{j=1}^n c_j\delta_{jk} = c_k \implies \int_{\partial\Delta_k} {}^*\diff\p{u - v} = 0

by construction. Let $f = u_x - iu_y$ so that $f \,\diff{z} = \diff{u} + i{}^*\diff{u}$ . Hence, by Cauchy's theorem and taking imaginary parts, we see that for any closed $\gamma \subseteq \Omega$ ,

\int_\gamma {}^*\diff\p{u - v} = \sum_{j=1}^n W\p{\gamma; a_j}\int_{\partial\Delta_k} {}^*\diff\p{u - v} = 0,

where $W\p{\gamma; a_j}$ is the winding number of $\gamma$ around $a_j$ . Thus, $u - v$ has a harmonic conjugate, i.e., $u - v$ is the real part of a holomorphic function on $\Omega$ . To show uniqueness, observe that

\sum_{j=1}^n c_j \log\,\abs{z - a_j}

is the real part of a holomorphic function on $\Omega$ if and only if $c_j = 0$ for all $j$ .