ルジャンドルの関係式
d 2 d k 2 Y ( k ) = k ( 1 − k 2 ) d 2 d k 2 K ( k ) + ( 1 − 3 k 2 ) k ( 1 − k 2 ) d d k K ( k ) + 3 k 4 − 6 k 2 − 1 4 k ( 1 − k 2 ) k ( 1 − k 2 ) K ( k ) = 1 k ( 1 − k 2 ) ( k ( 1 − k 2 ) d 2 d k 2 K ( k ) + ( 1 − 3 k 2 ) d d k K ( k ) + 3 k 4 − 6 k 2 − 1 4 k ( 1 − k 2 ) K ( k ) ) = 1 k ( 1 − k 2 ) ( d d k ( k ( 1 − k 2 ) d d k K ( k ) ) + 3 k 4 − 6 k 2 − 1 4 k ( 1 − k 2 ) K ( k ) ) = 1 k ( 1 − k 2 ) ( k K ( k ) + 3 k 4 − 6 k 2 − 1 4 k ( 1 − k 2 ) K ( k ) ) = − ( 1 + k 2 ) 2 4 k 2 ( 1 − k 2 ) 2 Y ( k ) {\displaystyle {\begin{aligned}{\frac {d^{2}}{dk^{2}}}Y(k)&={\sqrt {k(1-k^{2})}}{\frac {d^{2}}{dk^{2}}}K(k)+{\frac {(1-3k^{2})}{\sqrt {k(1-k^{2})}}}{\frac {d}{dk}}K(k)+{\frac {3k^{4}-6k^{2}-1}{4{\sqrt {k(1-k^{2})}}\;k(1-k^{2})}}K(k)\\&={\frac {1}{\sqrt {k(1-k^{2})}}}\left(k(1-k^{2}){\frac {d^{2}}{dk^{2}}}K(k)+(1-3k^{2}){\frac {d}{dk}}K(k)+{\frac {3k^{4}-6k^{2}-1}{4k(1-k^{2})}}K(k)\right)\\&={\frac {1}{\sqrt {k(1-k^{2})}}}\left({\frac {d}{dk}}\left(k(1-k^{2}){\frac {d}{dk}}K(k)\right)+{\frac {3k^{4}-6k^{2}-1}{4k(1-k^{2})}}K(k)\right)\\&={\frac {1}{\sqrt {k(1-k^{2})}}}\left(kK(k)+{\frac {3k^{4}-6k^{2}-1}{4k(1-k^{2})}}K(k)\right)\\&=-{\frac {(1+k^{2})^{2}}{4k^{2}(1-k^{2})^{2}}}Y(k)\\\end{aligned}}}