ルジャンドルのカイ関数
∫ 0 π / 2 arctan  ( p sin  θ ) arctan  ( q sin  θ ) d θ = π χ 2 ( 1 + p 2 − 1 p ⋅ 1 + q 2 − 1 q ) {\displaystyle \int _{0}^{\pi /2}\arctan(p\sin \theta )\arctan(q\sin \theta )d\theta =\pi \chi _{2}\left({\frac {{\sqrt {1+p^{2}}}-1}{p}}\cdot {\frac {{\sqrt {1+q^{2}}}-1}{q}}\right)}
∫ 0 α ∫ 0 β d x d y 1 − x 2 y 2 = χ 2 ( α β ) i f | α β | ≤ 1 {\displaystyle \int _{0}^{\alpha }\int _{0}^{\beta }{\frac {dxdy}{1-x^{2}y^{2}}}=\chi _{2}(\alpha \beta )\qquad {\rm {if}}~~|\alpha \beta |\leq 1}