mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Scorer's functions are

special function Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by ...

s 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 function In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai(''x'') and the related function Bi(''x''), are Linear in ...

s: :

\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

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

\mathrm(x)+\mathrm(x)\equiv \mathrm(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

References

* * Special functions {{mathanalysis-stub