Scorer's function

File:Mplwp Scorers Gi Hi.svg

Graph of <math>\mathrm{Gi}(x)</math> and <math>\mathrm{Hi}(x)</math>

In mathematics, the Scorer's functions are special functions studied by Template:Harvtxt 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 functions:

<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>

• 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

• 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

• 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

• 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

• Template:Dlmf
• Template:Citationf(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}}

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