エアリー関数
Ai(x), Bi(x) およびそれらの導関数の x = 0 における値は
Ai  ( 0 ) = 1 3 2 / 3 Γ ( 2 / 3 ) , Ai ′  ( 0 ) = − 1 3 1 / 3 Γ ( 1 / 3 ) , Bi  ( 0 ) = 1 3 1 / 6 Γ ( 2 / 3 ) , Bi ′  ( 0 ) = 3 1 / 6 Γ ( 1 / 3 ) {\displaystyle {\begin{aligned}\operatorname {Ai} (0)&={\frac {1}{3^{2/3}\Gamma (2/3)}},&\operatorname {Ai} '(0)&=-{\frac {1}{3^{1/3}\Gamma (1/3)}},\\\operatorname {Bi} (0)&={\frac {1}{3^{1/6}\Gamma (2/3)}},&\operatorname {Bi} '(0)&={\frac {3^{1/6}}{\Gamma (1/3)}}\end{aligned}}}
Gi  ( x ) = Bi  ( x ) ∫ x ∞ Ai  ( t ) d t + Ai  ( x ) ∫ 0 x Bi  ( t ) d t , Hi  ( x ) = Bi  ( x ) ∫ − ∞ x Ai  ( t ) d t − Ai  ( x ) ∫ − ∞ x Bi  ( t ) d t {\displaystyle {\begin{aligned}\operatorname {Gi} (x)&=\operatorname {Bi} (x)\int _{x}^{\infty }\operatorname {Ai} (t){\mathit {dt}}+\operatorname {Ai} (x)\int _{0}^{x}\operatorname {Bi} (t){\mathit {dt}},\\[8pt]\operatorname {Hi} (x)&=\operatorname {Bi} (x)\int _{-\infty }^{x}\operatorname {Ai} (t){\mathit {dt}}-\operatorname {Ai} (x)\int _{-\infty }^{x}\operatorname {Bi} (t){\mathit {dt}}\end{aligned}}}
と書くこともできる。