Scorer's function

File:Mplwp Scorers Gi Hi.svg 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

$$\begin{gathered} y^{″}(x)-x \\ y(x)=\frac{1}{\pi } \end{gathered}$$

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:

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

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

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

• 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
  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
• 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
  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
• 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
  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
• 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
  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

• Olver, F. W. J. (2010), "Scorer functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
• Scorer, R. S. (1950), "Numerical evaluation of integrals of the form $I=\underset{x_1}{\overset{x_2}{\int }}f(x)e^{i\varphi (x)}dx$ and the tabulation of the function $\mathrm{G}\mathrm{i}(z)=\frac{1}{\pi }\underset{0}{\overset{\mathrm{\infty }}{\int }}\mathrm{s}\mathrm{i}\mathrm{n}(uz+\frac{1}{3}{u}^3)du$", The Quarterly Journal of Mechanics and Applied Mathematics, 3: 107–112, doi:10.1093/qjmam/3.1.107, ISSN 0033-5614, MR 0037604