Scorer's Function

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

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

:$y''(x) - x\ y(x) = \frac$

and are given by

:$\mathrm(x) = \frac \int_0^\infty \sin\left(\frac + xt\right)\, dt,$
:$\mathrm(x) = \frac \int_0^\infty \exp\left(-\frac + xt\right)\, dt.$

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

:$\begin \mathrm(x) &= \mathrm(x) \int_x^\infty \mathrm(t) \, dt + \mathrm(x) \int_0^x \mathrm(t) \, dt, \\ \mathrm(x) &= \mathrm(x) \int_^x \mathrm(t) \, dt - \mathrm(x) \int_^x \mathrm(t) \, dt. \end$

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

References

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Special functions

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