斯科惹函數 (Scorers functions)是下列方程的兩個解
Scorers Gi function Scorers Hi Function
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{\displaystyle y''(x)-x\ y(x)={\frac {1}{\pi }}}
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{\displaystyle \mathrm {Gi} (x)={\frac {1}{\pi }}\int _{0}^{\infty }\sin \left({\frac {t^{3}}{3}}+xt\right)\,dt,}
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{\displaystyle \mathrm {Hi} (x)={\frac {1}{\pi }}\int _{0}^{\infty }\exp \left(-{\frac {t^{3}}{3}}+xt\right)\,dt.}
也可以通過艾里函數 定義：
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{\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}}}
冪級數展開
編輯
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{\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!}}}
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{\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!}}}
參考文獻
編輯
Olver, F. W. J., Scorer functions , Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (編), NIST Handbook of Mathematical Functions , Cambridge University Press, 2010, ISBN 978-0521192255MR 2723248 Scorer, R. S., Numerical evaluation of integrals of the form
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{\displaystyle I=\int _{x_{1}}^{x_{2}}f(x)e^{i\phi (x)}dx}
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{\displaystyle {\rm {Gi}}(z)={\frac {1}{\pi }}\int _{0}^{\infty }{\rm {sin}}\left(uz+{\frac {1}{3}}u^{3}\right)du}
3 : 107–112, ISSN 0033-5614 MR 0037604 doi:10.1093/qjmam/3.1.107 
