\begin{eqnarray} U(c,ϕ)& =& {{A}_{0}\over 2} +{ \mathop{∑ }}_{n=1}^{∞}{c}^{n}\left ({A}_{ n}\mathop{ cos}\nolimits nϕ + {B}_{n}\mathop{ sin}\nolimits nϕ\right ) %& \\ & =& \left \{\array{ 100\quad &\text{if $0 < ϕ < π$}\cr 0 \quad &\text{if $π < ϕ < 2π$} } \right .\quad .%&(8.29)\\ \end{eqnarray}

\begin{eqnarray} {A}_{0}& =& {1\over π}{\mathop{\mathop{\mathop{∫ }\nolimits }}\nolimits }_{0}^{π}100\kern 1.66702pt dϕ = 100, %& \\ {c}^{n}{A}_{ n}& =& {1\over π}{\mathop{\mathop{\mathop{∫ }\nolimits }}\nolimits }_{0}^{π}100\mathop{cos}\nolimits nϕ\kern 1.66702pt dϕ = {100\over nπ} \mathop{sin}\nolimits (nϕ){|}_{0}^{π} = 0, %& \\ {c}^{n}{B}_{ n}& =& {1\over π}{\mathop{\mathop{\mathop{∫ }\nolimits }}\nolimits }_{0}^{π}100\mathop{sin}\nolimits nϕ\kern 1.66702pt dϕ = −{100\over nπ} \mathop{cos}\nolimits (nϕ){|}_{0}^{π} %& \\ & =& \left \{\array{ 200∕(nπ)\quad &\text{if $n$ is odd}\cr 0 \quad &\text{if $n$ is even} } \right .\quad .%&(8.30)\\ \end{eqnarray}