楕円積分
F ( x , 0 ) = ∫ 0 x 1 1 − t 2 d t = ∫ 0 sin − 1  x 1 1 − sin 2  θ ( sin  θ ) ′ d θ = sin − 1  x F ( x , 1 ) = ∫ 0 x 1 1 − t 2 d t = ∫ 0 tanh − 1  x 1 1 − tanh 2  θ ( tanh  θ ) ′ d θ = tanh − 1  x E ( x , 0 ) = ∫ 0 x 1 1 − t 2 d t = F ( x , 0 ) = sin − 1  x E ( x , 1 ) = ∫ 0 x d t = x {\displaystyle {\begin{aligned}F(x,0)&=\int _{0}^{x}{{\frac {1}{\sqrt {1-t^{2}}}}dt}=\int _{0}^{\sin ^{-1}x}{\frac {1}{\sqrt {1-\sin ^{2}\theta }}}(\sin \theta )'d\theta =\sin ^{-1}x\\F(x,1)&=\int _{0}^{x}{{\frac {1}{1-t^{2}}}dt}=\int _{0}^{\tanh ^{-1}x}{\frac {1}{1-\tanh ^{2}\theta }}(\tanh \theta )'d\theta =\tanh ^{-1}x\\E(x,0)&=\int _{0}^{x}{{\frac {1}{\sqrt {1-t^{2}}}}dt}=F(x,0)=\sin ^{-1}x\\E(x,1)&=\int _{0}^{x}{dt}=x\end{aligned}}}