查看“︁斯科惹函数”︁的源代码

[[File:Scorers Gi function.png|thumb|Scorers  Gi  function]]
[[File:Scorers Hi function.png|thumb|Scorers Hi  Function]]
'''斯科惹函数'''(Scorers functions)是下列方程的两个解

:<math>y''(x) - x\ y(x) = \frac{1}{\pi}</math> 

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

也可以通过[[艾里函数]]定义：

:<math>\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}
</math>

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

<math>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!}</math>

==参考文献==
<references/>
* {{dlmf|title= Scorer functions |id=9.12|first=F. W. J.|last= Olver}}
*{{Citation | last1=Scorer | first1=R. S. | title=Numerical evaluation of integrals of the form <math>I=\int^{x_2}_{x_{1}}f(x)e^{i\phi(x)}dx</math> and the tabulation of the function <math>{\rm Gi} (z)=\frac{1}{\pi}\int^\infty_0{\rm sin}\left(uz+\frac 13 u^3\right)du</math> | 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}}

[[Category:特殊函数]]