ルジャンドルの関係式

${\displaystyle K(k)E\left({\sqrt {1-k^{2}}}\right)+E(k)K\left({\sqrt {1-k^{2}}}\right)-K(k)K\left({\sqrt {1-k^{2}}}\right)={\frac {\pi }{2}}}$

証明

{\displaystyle {\begin{aligned}k{\frac {d}{dk}}E(k)&=k{\frac {d}{dk}}{\frac {\pi }{2}}\left(1-\sum _{n=1}^{\infty }{\left({\frac {(2n-1)!!}{(2n)!!}}\right)^{2}{\frac {k^{2n}}{2n-1}}}\right)\\&={\frac {\pi }{2}}\sum _{n=1}^{\infty }\left({\frac {(2n-1)!!}{(2n)!!}}\right)^{2}\left(-{\frac {1}{2n-1}}-1\right)k^{2n}\\&=E(k)-K(k)\end{aligned}}}
{\displaystyle {\begin{aligned}k{\frac {d}{dk}}K(k)&=k{\frac {d}{dk}}{\frac {\pi }{2}}\left(1+\sum _{n=1}^{\infty }{\left({\frac {(2n-1)!!}{(2n)!!}}\right)^{2}k^{2n}}\right)\\&={\frac {\pi }{2}}\sum _{n=1}^{\infty }{\left({\frac {(2n-1)!!}{(2n)!!}}\right)^{2}\left({\frac {(2n)^{2}}{(2n-1)^{2}}}(2n-2+1)-{\frac {1}{2n-1}}-1\right)k^{2n}}\\&=k^{2}\left(k{\frac {d}{dk}}K(k)+K(k)\right)+E(k)-K(k)\end{aligned}}}
${\displaystyle k(1-k^{2}){\frac {d}{dk}}K(k)=E(k)-(1-k^{2})K(k)}$

から、微分方程式

{\displaystyle {\begin{aligned}{\frac {d}{dk}}\left(k(1-k^{2}){\frac {d}{dk}}K(k)\right)&={\frac {E(k)-K(k)}{k}}-(1-k^{2}){\frac {E(k)-(1-k^{2})K(k)}{k(1-k^{2})}}+2kK(k)\\&={\frac {E(k)-K(k)-E(k)+(1-k^{2})K(k)}{k}}+2kK(k)\\&=kK(k)\\\end{aligned}}}

が得られるが、ここで${\displaystyle k'={\sqrt {1-k^{2}}}}$とすれば

{\displaystyle {\begin{aligned}{\frac {d}{dk}}\left(k(1-k^{2}){\frac {d}{dk}}K(k')\right)&={\frac {dk'}{dk}}{\frac {d}{dk'}}\left(k(1-k^{2}){\frac {dk'}{dk}}{\frac {d}{dk'}}K(k')\right)\\&={\frac {k}{\sqrt {1-k^{2}}}}{\frac {d}{dk'}}\left(k^{2}{\sqrt {1-k^{2}}}{\frac {d}{dk'}}K(k')\right)\\&={\frac {k}{k'}}{\frac {d}{dk}}\left((1-k'^{2})k'{\frac {d}{dk'}}K(k')\right)\\\end{aligned}}}

であるから${\displaystyle K'(k)=K(k')}$も同じ微分方程式の解になる。${\displaystyle Y(k)={\sqrt {k(1-k^{2})}}K(k)}$とすれば

{\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}}}

となり、${\displaystyle Y'(k)={\sqrt {k(1-k^{2})}}K'(k)}$も同様である。故に

${\displaystyle {\frac {{\frac {d^{2}}{dk^{2}}}Y(k)}{Y(k)}}=-{\frac {(1+k^{2})^{2}}{4k^{2}(1-k^{2})^{2}}}={\frac {{\frac {d^{2}}{dk^{2}}}Y'(k)}{Y'(k)}}}$

であるから

${\displaystyle Y(k){\frac {d^{2}}{dk^{2}}}Y'(k)-Y'(k){\frac {d^{2}}{dk^{2}}}Y(k)=0}$
${\displaystyle {\sqrt {k(1-k^{2})}}K(k){\frac {d^{2}}{dk^{2}}}\left({\sqrt {k(1-k^{2})}}K'(k)\right)-{\sqrt {k(1-k^{2})}}K'(k){\frac {d^{2}}{dk^{2}}}\left({\sqrt {k(1-k^{2})}}K(k)\right)=0}$

が成立する。積分して整理すると

${\displaystyle k(1-k^{2})\left(K(k){\frac {d}{dk}}K'(k)-K'(k){\frac {d}{dk}}K(k)\right)=C}$

となり、これに

${\displaystyle {\frac {d}{dk}}K(k)={\frac {E(k)-(1-k^{2})K(k)}{k(1-k^{2})}}}$
${\displaystyle {\frac {d}{dk}}K'(k)={\frac {d}{dk}}K\left({\sqrt {1-k^{2}}}\right)={\frac {-k}{\sqrt {1-k^{2}}}}{\frac {E\left({\sqrt {1-k^{2}}}\right)-k^{2}K\left({\sqrt {1-k^{2}}}\right)}{{\sqrt {1-k^{2}}}\;k^{2}}}=-{\frac {E'(k)-k^{2}K'(k)}{k(1-k^{2})}}}$

を代入すると

${\displaystyle K(k)E'(k)+E(k)K'(k)-K(k)K'(k)=-C}$

が得られる。不完全楕円積分の極限を用いて

{\displaystyle {\begin{aligned}-C&=K(k)E'(k)+E(k)K'(k)-K(k)K'(k)\\&=\lim _{x\to 1}F(x,0)E(x,1)+E(x,0)F(x,1)-F(x,0)F(x,1)\\&=\lim _{x\to 1}\sin ^{-1}x\left(x+\tanh ^{-1}x-\tanh ^{-1}x\right)\\&={\frac {\pi }{2}}\\\end{aligned}}}

が得られる。