Scorer's function

File:Mplwp Scorers Gi Hi.svg of $\backslash mathrm\{Gi\}(x)$ and $\backslash mathrm\{Hi\}(x)$]]

In mathematics, the Scorer's functions are special functions studied by {{harvtxt|Scorer|1950}} and denoted Gi(x) and Hi(x).

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

:$y\text{'}\text{'}(x)\; -\; x\backslash \; y(x)\; =\; \backslash frac\{1\}\{\backslash pi\}$

and are given by

:$\backslash mathrm\{Gi\}(x)\; =\; \backslash frac\{1\}\{\backslash pi\}\; \backslash int\_0^\backslash infty\; \backslash sin\backslash left(\backslash frac\{t^3\}\{3\}\; +\; xt\backslash right)\backslash ,\; dt,$

:$\backslash mathrm\{Hi\}(x)\; =\; \backslash frac\{1\}\{\backslash pi\}\; \backslash int\_0^\backslash infty\; \backslash exp\backslash left(-\backslash frac\{t^3\}\{3\}\; +\; xt\backslash right)\backslash ,\; dt.$

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

:$\backslash begin\{align\}$

\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}

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

:$\backslash mathrm\{Gi\}(x)+\backslash mathrm\{Hi\}(x)\backslash equiv\; \backslash mathrm\{Bi\}(x)$

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