ルジャンドルのカイ関数
χ 2 ( x ) + χ 2 ( 1 / x ) = π 2 4 − i π 2 | ln  x | ( x > 0 ) . {\displaystyle \chi _{2}(x)+\chi _{2}(1/x)={\frac {\pi ^{2}}{4}}-{\frac {i\pi }{2}}|\ln x|\qquad (x>0).}
d d x χ 2 ( x ) = a r c t a n h x x . {\displaystyle {\frac {d}{dx}}\chi _{2}(x)={\frac {{\rm {arctanh\,}}x}{x}}.}