ルジャンドルのカイ関数
∫ 0 π / 2 arcsin  ( r sin  θ ) d θ = χ 2 ( r ) {\displaystyle \int _{0}^{\pi /2}\arcsin(r\sin \theta )d\theta =\chi _{2}\left(r\right)}
∫ 0 π / 2 arctan  ( r sin  θ ) d θ = − 1 2 ∫ 0 π r θ cos  θ 1 + r 2 sin 2  θ d θ = 2 χ 2 ( 1 + r 2 − 1 r ) {\displaystyle \int _{0}^{\pi /2}\arctan(r\sin \theta )d\theta =-{\frac {1}{2}}\int _{0}^{\pi }{\frac {r\theta \cos \theta }{1+r^{2}\sin ^{2}\theta }}d\theta =2\chi _{2}\left({\frac {{\sqrt {1+r^{2}}}-1}{r}}\right)}