斯科惹函数

斯科惹函数(Scorers functions)是下列方程的两个解

:y*(x) - x\ y(x) = \frac{1}{\pi}

:\mathrm{Gi}(x) = \frac{1}{\pi} \int_0^\infty \sin\left(\frac{t^3}{3} + xt\right)\, dt,
:\mathrm{Hi}(x) = \frac{1}{\pi} \int_0^\infty \exp\left(-\frac{t^3}{3} + xt\right)\, dt.

也可以通过艾里函数定义：

:\begin{align}
\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{align}

幂级数展开
Gi(z)=\sum_{k=0}^{\infty}cos(\frac{(2k-1)\pi}{3})\Gamma(\frac{k+1}{3})\frac{(3^{1/3}*z)^k}{k!}

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
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*{{Citation | last1=Scorer | first1=R. S. | title=Numerical evaluation of integrals of the form I=\int^{x_2}_{x_{1}}f(x)e^{i\phi(x)}dx and the tabulation of the function {\rm Gi} (z)=\frac{1}{\pi}\int^\infty_0{\rm sin}\left(uz+\frac 13 u^3\right)du | doi=10.1093/qjmam/3.1.107 | mr=0037604 |id=| year=1950 | journal=The Quarterly Journal of Mechanics and Applied Mathematics | issn=0033-5614 | volume=3 | pages=107–112}}