8.2 調和関数の定積分
$k_{xm} \ne \hat{k}_{xm'}$ のとき，
\begin{eqnarray}
\hat{X}_{mm'}^{ \{ \substack{\sin \\ \cos} }
&=& \int_{x_{min}}^{x_{max}} \begin{matrix} \sin \\ \cos \end{matrix} \big\{ k_{xm} (x +x_1) \big\}
\cdot \begin{matrix} \sin \\ \cos \end{matrix} \big\{ \hat{k}_{xm'} (x +x_2) \big\} dx
\nonumber \\
&=& \left[ \mp \frac{\sin \big\{ k_{xm} (x +x_1) + \hat{k}_{xm'} (x +x_2) \big\} }{2(k_{xm}+\hat{k}_{xm'})} \right.
\nonumber \\
&&\left. + \frac{\sin \big\{ k_{xm} (x +x_1) - \hat{k}_{xm'} (x +x_2) \big\} }{2(k_{xm}-\hat{k}_{xm'})} \right]_{x_{min}}^{x_{max}}
\end{eqnarray}
また，$k_{xm} = \hat{k}_{xm'} \ne 0$ のとき，
\begin{eqnarray}
\hat{X}_{mm'}^{\{ \substack{\sin \\ \cos}}
&=& \int_{x_{min}}^{x_{max}} \begin{matrix} \sin \\ \cos \end{matrix} \big\{ k_{xm} (x +x_1) \big\}
\cdot \begin{matrix} \sin \\ \cos \end{matrix} \big\{ k_{xm} (x +x_2) \big\} dx
\nonumber \\
&=& \left[ \mp \frac{\sin \big\{ k_{xm} (2x + x_1 + x_2) \big\} }{4 k_{xm}}
+ \frac{x \cos \big\{ k_{xm} (x_1-x_2) \big\} }{2} \right]_{x_{min}}^{x_{max}}
\end{eqnarray}
$k_{xm} = \hat{k}_{xm'} = 0$ のとき，上側は被積分項がゼロ，下側は，
\begin{gather}
\big[ x \big]_{x_{min}}^{x_{max}} =x_{max}-x_{min}
\end{gather}
同様にして，$k_{yn} \ne \hat{k}_{yn'}$ のとき，
\begin{eqnarray}
\hat{Y}_{nn'}^{\{ \substack{\sin \\ \cos}}
&=& \int_{y_{min}}^{y_{max}} \begin{matrix} \sin \\ \cos \end{matrix} \big\{ k_{yn} (y +y_1) \big\}
\cdot \begin{matrix} \sin \\ \cos \end{matrix} \big\{ \hat{k}_{yn'} (y +y_2) \big\} dy
\nonumber \\
&=& \left[ \mp \frac{\sin \big\{ k_{yn} (y +y_1) + \hat{k}_{yn'} (y +y_2) \big\} }{2(k_{yn}+\hat{k}_{yn'})} \right.
\nonumber \\
&&\left. + \frac{\sin \big\{ k_{yn} (y +y_1) - \hat{k}_{yn'} (y +y_2) \big\} }{2(k_{yn}-\hat{k}_{yn'})} \right]_{y_{min}}^{y_{max}}
\end{eqnarray}
また，$k_{yn} = \hat{k}_{yn'} \ne 0$ のとき，
\begin{eqnarray}
\hat{Y}_{nn'}^{\{ \substack{\sin \\ \cos}}
&=& \int_{y_{min}}^{y_{max}} \begin{matrix} \sin \\ \cos \end{matrix} \big\{ k_{yn} (y +y_1) \big\}
\cdot \begin{matrix} \sin \\ \cos \end{matrix} \big\{ k_{yn} (y +y_2) \big\} dy
\nonumber \\
&=& \left[ \mp \frac{\sin \big\{ k_{yn} (2y + y_1 + y_2) \big\} }{4 k_{yn}}
+ \frac{y \cos \big\{ k_{yn} (y_1-y_2) \big\} }{2} \right]_{y_{min}}^{y_{max}}
\end{eqnarray}
$k_{yn} = \hat{k}_{yn'} = 0$ のとき，上側は被積分項がゼロ，下側は，
\begin{gather}
\big[ y \big]_{y_{min}}^{y_{max}} =y_{max}-y_{min}
\end{gather}