楕円有理関数
低次の楕円有理関数は次のようになる:
R 1 ( ξ , x ) = x {\displaystyle R_{1}(\xi ,x)=x\,}
R 2 ( ξ , x ) = ( t + 1 ) x 2 − 1 ( t − 1 ) x 2 + 1 {\displaystyle R_{2}(\xi ,x)={\frac {(t+1)x^{2}-1}{(t-1)x^{2}+1}}} ここで t ≡ 1 − 1 ξ 2 {\displaystyle t\equiv {\sqrt {1-{\frac {1}{\xi ^{2}}}}}}
R 3 ( ξ , x ) = x ( 1 − x p 2 ) ( x 2 − x z 2 ) ( 1 − x z 2 ) ( x 2 − x p 2 ) {\displaystyle R_{3}(\xi ,x)=x\,{\frac {(1-x_{p}^{2})(x^{2}-x_{z}^{2})}{(1-x_{z}^{2})(x^{2}-x_{p}^{2})}}} G ≡ 4 ξ 2 + ( 4 ξ 2 ( ξ 2 − 1 ) ) 2 / 3 {\displaystyle G\equiv {\sqrt {4\xi ^{2}+(4\xi ^{2}(\xi ^{2}\!-\!1))^{2/3}}}} ここで x p 2 ≡ 2 ξ 2 G 8 ξ 2 ( ξ 2 + 1 ) + 12 G ξ 2 − G 3 − G 3 {\displaystyle x_{p}^{2}\equiv {\frac {2\xi ^{2}{\sqrt {G}}}{{\sqrt {8\xi ^{2}(\xi ^{2}\!+\!1)+12G\xi ^{2}-G^{3}}}-{\sqrt {G^{3}}}}}} また x z 2 = ξ 2 / x p 2 {\displaystyle x_{z}^{2}=\xi ^{2}/x_{p}^{2}}