Scorer's function

Graph of $\mathrm{G}\mathrm{i}(x)$ and $\mathrm{H}\mathrm{i}(x)$

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

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

$y^{″}(x)-xy(x)=\frac{1}{\pi }$

and are given by

$\mathrm{G}\mathrm{i}(x)=\frac{1}{\pi }\underset{0}{\overset{\mathrm{\infty }}{\int }}\sin (\frac{t^3}{3}+xt)dt,$

$\mathrm{H}\mathrm{i}(x)=\frac{1}{\pi }\underset{0}{\overset{\mathrm{\infty }}{\int }}\exp (-\frac{t^3}{3}+xt)dt.$

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

$\begin{array}{cc}\mathrm{G}\mathrm{i}(x)& =\mathrm{B}\mathrm{i}(x)\underset{x}{\overset{\mathrm{\infty }}{\int }}\mathrm{A}\mathrm{i}(t)dt+\mathrm{A}\mathrm{i}(x)\underset{0}{\overset{x}{\int }}\mathrm{B}\mathrm{i}(t)dt,\\ \mathrm{H}\mathrm{i}(x)& =\mathrm{B}\mathrm{i}(x)\underset{-\mathrm{\infty }}{\overset{x}{\int }}\mathrm{A}\mathrm{i}(t)dt-\mathrm{A}\mathrm{i}(x)\underset{-\mathrm{\infty }}{\overset{x}{\int }}\mathrm{B}\mathrm{i}(t)dt.\end{array}$

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

$\mathrm{G}\mathrm{i}(x)+\mathrm{H}\mathrm{i}(x)\equiv \mathrm{B}\mathrm{i}(x)$

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  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
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  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
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  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
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  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

References

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