Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

Section 12.4 Answers

  • Page ID
    30088
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    1. \(u(r,\theta )=\alpha_{0}\frac{\ln r/\rho }{\ln \rho_{0}/\rho}+ \sum_{n=1}^{\infty}\frac{r^{n}\rho^{-n}-\rho^{n}r^{-n}}{\rho_{0}^{n}\rho^{-n}-\rho^{n}\rho_{0}^{-n}}(\alpha_{n}\cos n\theta +\beta_{n}\sin n\theta )\quad \alpha_{0}=\frac{1}{2\pi }\int_{-\pi }^{\pi }f(\theta )d\theta ,\) and \(\alpha_{n}=\frac{1}{\pi }\int_{-\pi }^{\pi }f(\theta )\cos n\theta d\theta,\quad \beta_{n}=\frac{1}{\pi }\int_{-\pi }^{\pi}f(\theta )\sin n\theta d\theta,\quad n=1,2,3,\ldots\)

    2. \(u(r,\theta )= \sum_{n=1}^{\infty}\alpha_{n}\frac{\rho_{0}^{-n\pi /\gamma} r^{n\pi /\gamma}-\rho_{0}^{n\pi /\gamma}r^{-n\pi /\gamma}}{\rho_{0}^{-n\pi /\gamma}\rho^{n\pi /\gamma}-\rho_{0}^{n\pi /\gamma}\rho^{-n\pi /\gamma}}\sin\frac{n\pi\theta}{\gamma}\quad\alpha_{n}=\frac{1}{\gamma}\int_{0}^{\gamma}f(\theta )\sin\frac{n\pi\theta}{\gamma}d\theta ,\quad n=1,2,3,\ldots \)

    3. \(u(r,\theta )=\rho\alpha_{0}\ln\frac{r}{\rho_{0}}+\frac{\rho\gamma}{\pi} \sum_{n=1}^{\infty}\frac{\alpha_{n}}{n}\frac{\rho_{0}^{-n\pi /\gamma} r^{n\pi /\gamma}-\rho_{0}^{n\pi /\gamma}r^{-n\pi /\gamma}}{\rho_{0}^{-n\pi /\gamma}\rho^{n\pi /\gamma}-\rho_{0}^{n\pi /\gamma}\rho^{-n\pi /\gamma}}\cos\frac{n\pi\theta}{\gamma},\quad\alpha_{0}=\frac{1}{\gamma}\int_{0}^{\gamma}f(\theta )d\theta ,\quad \alpha_{n}=\frac{2}{\gamma }\int_{0}^{\gamma}f(\theta )\cos\frac{n\pi\theta }{\gamma}d\theta ,\quad n=1,2,3,\ldots\)

    4. \(u(r,\theta )=\sum_{n=1}^{\infty}\alpha_{n}\frac{r^{(2n-1)\pi /2\gamma}}{\rho^{(2n-1)\pi /2\gamma}}\cos\frac{(2n-1)\pi\theta}{2\gamma}\quad\alpha_{n}=\frac{2}{\gamma}\int_{0}^{\gamma}f(\theta )\cos\frac{(2n-1)\pi\theta}{2\gamma}d\theta ,n=1,2,3,\ldots\)

    5. \(u(4,\theta )=\frac{2\gamma\rho_{0}}{\pi}\sum_{n=1}^{\infty}\frac{\alpha_{n}}{2n-1}\frac{\rho ^{-(2n-1)\pi /2\gamma}r^{(2n-1)\pi /2\gamma}+\rho^{(2n-1)\pi /2\gamma}r^{-(2n-1)\pi /2\gamma} }{\rho^{-(2n-1)\pi /2\gamma}\rho_{0}^{(2n-1)\pi /2\gamma}-\rho^{(2n-1)\pi /2\gamma}\rho_{0}^{-(2n-1)\pi /2\gamma} }\sin\frac{(2n-1)\pi\theta}{2\gamma}, \quad\alpha_{n}=\frac{2}{\gamma}g(\theta )\sin\frac{(2n-1)\pi\theta }{2\gamma}d\theta ,\quad n=1,2,3,\ldots  \)

    6. \(u(r,\theta )=\alpha_{0}+\sum_{n=1}^{\infty}\alpha_{n}\frac{r^{n\pi /\gamma}}{\rho^{n\pi /\gamma}}\cos\frac{n\pi\theta }{\gamma }\quad\alpha_{0}=\frac{1}{\gamma}\int_{0}^{\gamma}f(\theta )d\theta ,\quad \alpha_{n}=\frac{2}{\gamma }\int_{0}^{\gamma }f(\theta )\cos\frac{n\pi\theta }{\gamma} d\theta ,\quad n=1,2,3,\ldots\)

    7. \(v_{n}(r,\theta )=\frac{r^{n}}{n\rho^{n-1}}(\alpha_{n}\cos n\theta +\sin n\theta )\quad u(r,\theta )=c+\sum_{n=1}^{\infty}\frac{r^{n}}{n\rho^{n-1}}(\alpha _{n}\cos n\theta +\beta_{n}\sin n\theta )\quad \alpha_{n}=\frac{1}{\pi}\int_{-\pi }^{\pi}f(\theta )\cos n\theta d\theta ,\quad \beta_{n}=\frac{1}{\pi }\int_{-\pi }^{\pi }f(\theta )\sin n\theta d\theta ,\quad n=1,2,3,\ldots \)