ガウス関数の原始関数の一覧
∫ − ∞ ∞ x 2 φ ( x ) n d x = 1 n 3 ( 2 π ) n − 1 {\displaystyle \int _{-\infty }^{\infty }x^{2}\phi (x)^{n}\,dx={\frac {1}{\sqrt {n^{3}(2\pi )^{n-1}}}}}
∫ − ∞ 0 φ ( a x ) Φ ( b x ) d x = 1 2 π | a | ( π 2 − arctan  ( b | a | ) ) {\displaystyle \int _{-\infty }^{0}\phi (ax)\Phi (bx)dx={\frac {1}{2\pi |a|}}\left({\frac {\pi }{2}}-\arctan \left({\frac {b}{|a|}}\right)\right)}
∫ 0 ∞ φ ( a x ) Φ ( b x ) d x = 1 2 π | a | ( π 2 + arctan  ( b | a | ) ) {\displaystyle \int _{0}^{\infty }\phi (ax)\Phi (bx)\,dx={\frac {1}{2\pi |a|}}\left({\frac {\pi }{2}}+\arctan \left({\frac {b}{|a|}}\right)\right)}