Editing Scorer's function

[[File:Mplwp Scorers Gi Hi.svg|300px|thumb|[[Graph of a function|Graph]] of <math>\mathrm{Gi}(x)</math> and <math>\mathrm{Hi}(x)</math>]]
In [[mathematics]], the '''Scorer's functions''' are [[special function]]s studied by {{harvtxt|Scorer|1950}} and denoted Gi(''x'') and Hi(''x''). 

Hi(''x'') and -Gi(''x'') solve the equation 

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

and are given by

:<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>

The Scorer's functions can also be defined in terms of [[Airy function]]s:

:<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>

It can also be seen, just from the integral forms, that the following relationship holds:

:<math>\mathrm{Gi}(x)+\mathrm{Hi}(x)\equiv \mathrm{Bi}(x)</math>
<gallery>
File:Plot of the Scorer function Gi(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|Plot of the Scorer function Gi(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
File:Plot of the derivative of the Scorer function Hi'(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|Plot of the derivative of the Scorer function Hi'(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
File:Plot of the derivative of the Scorer function Gi'(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|Plot of the derivative of the Scorer function Gi'(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
File:Plot of the Scorer function Hi(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|Plot of the Scorer function Hi(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
</gallery>

==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:Special functions]]


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