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Scorer Function

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ScorerFunction

The Scorer functions Gi(z) and Hi(z) are entire functions that provide particular solutions of the inhomogeneous Airy differential equation. They satisfy

Gi^('')(z)-zGi(z)=-1/pi
(1)
Hi^('')(z)-zHi(z)=1/pi.
(2)

For real x, the integral representations are

Gi(x)=1/piint_0^inftysin((t^3)/3+xt)dt
(3)
Hi(x)=1/piint_0^inftye^(-(t^3)/3+xt)dt.
(4)

The second integral converges for every complex argument. The first is an oscillatory improper integral on the real axis, with the entire function obtained by analytic continuation (Scorer 1950, NIST DLMF, §9.12).

The relation

Gi(z)+Hi(z)=Bi(z),
(5)

where Bi(z) is an Airy function, connects the two particular solutions. The Scorer functions are implemented in the Wolfram Language as ScorerGi[z] and ScorerHi[z].

AiryGiReIm
AiryGiContours
AiryHiReIm
AiryHiContours

The Airy functions give the alternative representations

Gi(z)=1/3Bi(z)+int_0^z[Ai(z)Bi(t)-Ai(t)Bi(z)]dt
(6)
=1/3Bi(z)-(z^2_1F_2(1;4/3,5/3;1/9z^3))/(2pi)
(7)
Hi(z)=2/3Bi(z)+int_0^z[Ai(t)Bi(z)-Ai(z)Bi(t)]dt
(8)
=2/3Bi(z)+(_1F_2(1;4/3,5/3;1/9z^3)z^2)/(2pi),
(9)

where _pF_q(a_1,...,a_p;b_1,...,b_q;z) is a generalized hypergeometric function.

See also

Airy Differential Equation, Airy Functions

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References

NIST Digital Library of Mathematical Functions. "Scorer Functions." §9.12. https://dlmf.nist.gov/9.12.Scorer, R. S. "Numerical Evaluation of Integrals of the Form I=int_(x_1)^(x_2)f(x)e^(iphi(x))dx and the Tabulation of the Function Gi(z)=1/piint_0^inftysin(uz+1/3u^3)du." Quart. J. Mech. Appl. Math. 3, 107-112, 1950. https://doi.org/10.1093/qjmam/3.1.107.

Cite this as:

Weisstein, Eric W. "Scorer Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ScorerFunction.html

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