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% \mondai{$B:n@.<T(B}{$B:n@.G/7nF|(B}{$BEE;R%a!<%k(B}{$BLdBjHV9f(B}{$BLdBjJ8(B}{$B2rEz(B}
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\mondai
{g140826 $B?eLn(B $BMNJe(B}
{01/12/25}
{g140826@mail.ecc.u-tokyo.ac.jp}
{8-6}
{$B5e%Y%C%;%k4X?t(B$j_\ell (x),n_\ell (x)$$B$N(B$\ell=0,1,2$$B$N4X?t7A$r(B(8.57)$B$h$j5a$a$h!#(B}
{
(8.57)$B$h$j!"(B
\begin{equation}
j_\ell (x) = \left( { - x} \right)^\ell\left( {\frac{1}{x}\frac{d}{dx}} 
\right)^\ell\frac{\sin x}{x}
\end{equation}
\begin{equation}
n_\ell (x) = - \left( { - x} \right)^\ell\left( {\frac{1}{x}\frac{d}{dx}} 
\right)^\ell\frac{\cos x}{x}
\end{equation}
$B$G$"$k$+$i!"$3$l$i$K(B$\ell=0,1,2$$B$rBeF~$9$l$P$h$$!#(B
\begin{eqnarray}
j_0 (x) = 1 \cdot 1 \cdot \frac{\sin x}{x} = \frac{\sin x}{x}
\end{eqnarray}
\begin{eqnarray}
j_1 (x) &=& \left( { - x} \right)^1\left( {\frac{1}{x}\frac{d}{dx}} \right)^1\frac{\sin x}{x}\nonumber\\
& = &( - x) \cdot \frac{1}{x} \cdot \frac{x\cos x - \sin x}{x^2}\nonumber\\
& = &\frac{\sin x - x\cos x}{x^2}
\end{eqnarray}
\begin{eqnarray}
j_2 (x) &=& \left( { - x} \right)^2\left( {\frac{1}{x}\frac{d}{dx}} \right)^2\frac{\sin x}{x}\nonumber\\
& = &x^2\left( {\frac{1}{x}\frac{d}{dx}} \right)\frac{x\cos x - \sin x}{x^3} \nonumber\\ 
& = &- x^2 \cdot \frac{1}{x} \cdot \frac{x^3\left({\cos x - x\sin x - \cos x}
 \right) - 3x^2\left( {x\cos x - \sin x} \right)}{x^6} \nonumber\\ 
& = &\frac{ - x^4\sin x - 3x^3\cos x + 3x^2\sin x}{x^5}\nonumber\\
& = &- \frac{\left( {x^2 - 3} \right)\sin x + 3x\cos x}{x^3}
\end{eqnarray}
\begin{eqnarray}
n_0 (x) = - 1 \cdot 1 \cdot \frac{\cos x}{x} =  - \frac{\cos x}{x}
\end{eqnarray}
\begin{eqnarray}
n_1 (x) &=& - \left( { - x} \right)^1\left( {\frac{1}{x}\frac{d}{dx}} \right)^1\frac{\cos x}{x}\nonumber\\
& = &x \cdot \frac{1}{x}\frac{ - x\sin x - \cos x}{x^2}\nonumber\\
& = &- \frac{\cos x + x\sin x}{x^2}
\end{eqnarray}
\begin{eqnarray}
n_2 (x) &=& - \left( { - x} \right)^2\left( {\frac{1}{x}\frac{d}{dx}} \right)^2\frac{\cos x}{x}\nonumber\\
& = &- x^2\left( {\frac{1}{x}\frac{d}{dx}} 
\right)\frac{ - \left( {x\sin x + \cos x} \right)}{x^3} \nonumber\\ 
& = &x^2 \cdot \frac{1}{x} \cdot \frac{x^3\left({ - \sin x + \sin x + x\cos x} 
\right) - 3x^2\left( {\cos x + x\sin x} \right)}{x^6} \nonumber\\ 
& = &\frac{x^4\cos x - 3x^2\cos x - 3x^3\sin x}{x^5}\nonumber\\
& = &\frac{\left( {x^2 - 3} \right)\cos x - 3x\sin x}{x^3}
\end{eqnarray}
}
