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Scorer's function - Wikipedia

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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 ) − x   y ( x ) = 1 π {\displaystyle y''(x)-x\ y(x)={\frac {1}{\pi }}}

and are given by

G i ( x ) = 1 π ∫ 0 ∞ sin ⁡ ( t 3 3 + x t ) d t , {\displaystyle \mathrm {Gi} (x)={\frac {1}{\pi }}\int _{0}^{\infty }\sin \left({\frac {t^{3}}{3}}+xt\right)\,dt,}

H i ( x ) = 1 π ∫ 0 ∞ exp ⁡ ( − t 3 3 + x t ) d t . {\displaystyle \mathrm {Hi} (x)={\frac {1}{\pi }}\int _{0}^{\infty }\exp \left(-{\frac {t^{3}}{3}}+xt\right)\,dt.}

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

G i ( x ) = B i ( x ) ∫ x ∞ A i ( t ) d t + A i ( x ) ∫ 0 x B i ( t ) d t , H i ( x ) = B i ( x ) ∫ − ∞ x A i ( t ) d t − A i ( x ) ∫ − ∞ x B i ( t ) d t . {\displaystyle {\begin{aligned}\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{aligned}}}

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

G i ( x ) + H i ( x ) ≡ B i ( x ) {\displaystyle \mathrm {Gi} (x)+\mathrm {Hi} (x)\equiv \mathrm {Bi} (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 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 Scorer function Hi(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D

• 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 = ∫ x 1 x 2 f ( x ) e i ϕ ( x ) d x {\displaystyle I=\int _{x_{1}}^{x_{2}}f(x)e^{i\phi (x)}dx} and the tabulation of the function G i ( z ) = 1 π ∫ 0 ∞ s i n ( u z + 1 3 u 3 ) d u {\displaystyle {\rm {Gi}}(z)={\frac {1}{\pi }}\int _{0}^{\infty }{\rm {sin}}\left(uz+{\frac {1}{3}}u^{3}\right)du} ", The Quarterly Journal of Mechanics and Applied Mathematics, 3: 107–112, doi:10.1093/qjmam/3.1.107, ISSN 0033-5614, MR 0037604

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