斯科惹函数

斯科惹函数(Scorers functions)是下列方程的两个解

${\displaystyle y''(x)-x\ y(x)={\frac {1}{\pi }}}$

${\displaystyle \mathrm {Gi} (x)={\frac {1}{\pi }}\int _{0}^{\infty }\sin \left({\frac {t^{3}}{3}}+xt\right)\,dt,}$

${\displaystyle \mathrm {Hi} (x)={\frac {1}{\pi }}\int _{0}^{\infty }\exp \left(-{\frac {t^{3}}{3}}+xt\right)\,dt.}$

Scorers Gi function

Scorers Hi Function

也可以通过艾里函数定义：

${\displaystyle {\begin{aligned}\mathrm {Gi} (x)&{}=\mathrm {Bi} (x)\int _{x}^{\infty }\mathrm {Ai} (t)\,dt+\mathrm {Ai} (x)\int _{0}^{x}\mathrm {Bi} (t)\,dt,\\\mathrm {Hi} (x)&{}=\mathrm {Bi} (x)\int _{-\infty }^{x}\mathrm {Ai} (t)\,dt-\mathrm {Ai} (x)\int _{-\infty }^{x}\mathrm {Bi} (t)\,dt.\end{aligned}}}$

幂级数展开

${\displaystyle Gi(z)=\sum _{k=0}^{\infty }cos({\frac {(2k-1)*\pi }{3}})\Gamma ({\frac {k+1}{3}})*{\frac {(3^{1/3}*z)^{k}}{k!}}}$

${\displaystyle Hi(z)={\frac {3^{-2/3}}{\pi }}\sum _{k=0}^{\infty }\Gamma ({\frac {(2k+1)*\pi }{3}}{\bigr )}{\frac {(3^{1/3}*z)^{k}}{k!}}}$

参考文献

• Olver, F. W. J., Scorer functions, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (编), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248
• Scorer, R. S., Numerical evaluation of integrals of the form ${\displaystyle I=\int _{x_{1}}^{x_{2}}f(x)e^{i\phi (x)}dx}$ and the tabulation of the function ${\displaystyle {\rm {Gi}}(z)={\frac {1}{\pi }}\int _{0}^{\infty }{\rm {sin}}\left(uz+{\frac {1}{3}}u^{3}\right)du}$, The Quarterly Journal of Mechanics and Applied Mathematics, 1950, 3: 107–112, ISSN 0033-5614, MR 0037604, doi:10.1093/qjmam/3.1.107