# Scorer functions

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where …
##### 2: 9.1 Special Notation
 $k$ nonnegative integer, except in §9.9(iii). …
The main functions treated in this chapter are the Airy functions $\operatorname{Ai}\left(z\right)$ and $\operatorname{Bi}\left(z\right)$, and the Scorer functions $\operatorname{Gi}(z)$ and $\operatorname{Hi}(z)$ (also known as inhomogeneous Airy functions). Other notations that have been used are as follows: $\operatorname{Ai}\left(-x\right)$ and $\operatorname{Bi}\left(-x\right)$ for $\operatorname{Ai}\left(x\right)$ and $\operatorname{Bi}\left(x\right)$ (Jeffreys (1928), later changed to $\operatorname{Ai}\left(x\right)$ and $\operatorname{Bi}\left(x\right)$); $U(x)=\sqrt{\pi}\operatorname{Bi}\left(x\right)$, $V(x)=\sqrt{\pi}\operatorname{Ai}\left(x\right)$ (Fock (1945)); $A(x)=3^{-\ifrac{1}{3}}\pi\operatorname{Ai}\left(-3^{-\ifrac{1}{3}}x\right)$ (Szegő (1967, §1.81)); $e_{0}(x)=\pi\operatorname{Hi}(-x)$, $\widetilde{e}_{0}(x)=-\pi\operatorname{Gi}(-x)$ (Tumarkin (1959)).
##### 3: 9.17 Methods of Computation
The former reference includes a parallelized version of the method. … The methods for $\operatorname{Ai}'\left(z\right)$ are similar. For quadrature methods for Scorer functions see Gil et al. (2001), Lee (1980), and Gordon (1970, Appendix A); but see also Gautschi (1983). … See also Fabijonas et al. (2004). For the computation of the zeros of the Scorer functions and their derivatives see Gil et al. (2003c).
##### 4: 9.16 Physical Applications
Reference to many of these applications as well as to the theory of elasticity and to the heat equation are given in Vallée and Soares (2010): a book devoted specifically to the Airy and Scorer functions and their applications in physics. … In the case of the rainbow, the scattering amplitude is expressed in terms of $\operatorname{Ai}\left(x\right)$, the analysis being similar to that given originally by Airy (1838) for the corresponding problem in optics. An application of the Scorer functions is to the problem of the uniform loading of infinite plates (Rothman (1954b, a)).
##### 5: 9.19 Approximations
###### §9.19(iv) ScorerFunctions
• MacLeod (1994) supplies Chebyshev-series expansions to cover $\operatorname{Gi}\left(x\right)$ for $0\leq x<\infty$ and $\operatorname{Hi}\left(x\right)$ for $-\infty. The Chebyshev coefficients are given to 20D.

• ##### 6: 9.18 Tables
###### §9.18(vi) ScorerFunctions
• Scorer (1950) tabulates $\operatorname{Gi}\left(x\right)$ and $\operatorname{Hi}\left(-x\right)$ for $x=0(.1)10$; 7D.

• Rothman (1954a) tabulates $\int_{0}^{x}\operatorname{Gi}\left(t\right)\,\mathrm{d}t$, $\operatorname{Gi}'\left(x\right)$, $\int_{0}^{x}\operatorname{Hi}\left(-t\right)\,\mathrm{d}t$, $-\operatorname{Hi}'\left(-x\right)$ for $x=0(.1)10$; 7D.

• National Bureau of Standards (1958) tabulates $A_{0}(x)\equiv\pi\operatorname{Hi}\left(-x\right)$ and $-A_{0}^{\prime}(x)\equiv\pi\operatorname{Hi}'\left(-x\right)$ for $x=0(.01)1(.02)5(.05)11$ and $1/x=0.01(.01)0.1$; $\int_{0}^{x}A_{0}(t)\,\mathrm{d}t$ for $x=0.5,1(1)11$. Precision is 8D.

• Gil et al. (2003c) tabulates the only positive zero of $\operatorname{Gi}'\left(z\right)$, the first 10 negative real zeros of $\operatorname{Gi}\left(z\right)$ and $\operatorname{Gi}'\left(z\right)$, and the first 10 complex zeros of $\operatorname{Gi}\left(z\right)$, $\operatorname{Gi}'\left(z\right)$, $\operatorname{Hi}\left(z\right)$, and $\operatorname{Hi}'\left(z\right)$. Precision is 11 or 12S.

##### 8: 9.10 Integrals
9.10.1 $\int_{z}^{\infty}\operatorname{Ai}\left(t\right)\,\mathrm{d}t=\pi\left(% \operatorname{Ai}\left(z\right)\operatorname{Gi}'\left(z\right)-\operatorname{% Ai}'\left(z\right)\operatorname{Gi}\left(z\right)\right),$
9.10.2 $\int_{-\infty}^{z}\operatorname{Ai}\left(t\right)\,\mathrm{d}t=\pi\left(% \operatorname{Ai}\left(z\right)\operatorname{Hi}'\left(z\right)-\operatorname{% Ai}'\left(z\right)\operatorname{Hi}\left(z\right)\right),$
9.10.3 $\int_{-\infty}^{z}\operatorname{Bi}\left(t\right)\,\mathrm{d}t=\int_{0}^{z}% \operatorname{Bi}\left(t\right)\,\mathrm{d}t=\pi\left(\operatorname{Bi}'\left(% z\right)\operatorname{Gi}\left(z\right)-\operatorname{Bi}\left(z\right)% \operatorname{Gi}'\left(z\right)\right)\\ =\pi\left(\operatorname{Bi}\left(z\right)\operatorname{Hi}'\left(z\right)-% \operatorname{Bi}'\left(z\right)\operatorname{Hi}\left(z\right)\right).$
For the functions $\operatorname{Gi}$ and $\operatorname{Hi}$ see §9.12. …
##### 9: 11.11 Asymptotic Expansions of Anger–Weber Functions
For the Scorer function $\operatorname{Hi}$ see §9.12(i). …
##### 10: Bibliography G
• A. Gil, J. Segura, and N. M. Temme (2002b) Algorithm 822: GIZ, HIZ: two Fortran 77 routines for the computation of complex Scorer functions. ACM Trans. Math. Software 28 (4), pp. 436–447.
• A. Gil, J. Segura, and N. M. Temme (2003c) On the zeros of the Scorer functions. J. Approx. Theory 120 (2), pp. 253–266.