# inhomogeneous Airy functions

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##### 1: 9.1 Special Notation
 $k$ nonnegative integer, except in §9.9(iii). …
The main functions treated in this chapter are the Airy functions $\mathrm{Ai}\left(z\right)$ and $\mathrm{Bi}\left(z\right)$, and the Scorer functions $\mathrm{Gi}(z)$ and $\mathrm{Hi}(z)$ (also known as inhomogeneous Airy functions). …
##### 2: 9.12 Scorer Functions
9.12.6 $\mathrm{Gi}\left(0\right)=\tfrac{1}{2}\mathrm{Hi}\left(0\right)=\tfrac{1}{3}% \mathrm{Bi}\left(0\right)={1\Big{/}\!\left(3^{7/6}\Gamma\left(\tfrac{2}{3}% \right)\right)=0.20497\;55424\ldots,}$
9.12.7 $\mathrm{Gi}'\left(0\right)=\tfrac{1}{2}\mathrm{Hi}'\left(0\right)=\tfrac{1}{3}% \mathrm{Bi}'\left(0\right)=1\Big{/}\left(3^{5/6}\Gamma\left(\tfrac{1}{3}\right% )\right)=0.14942\;94524\ldots.$
9.12.13 $\mathrm{Gi}\left(z\right)=e^{\mp\pi i/3}\mathrm{Hi}\left(ze^{\pm 2\pi i/3}% \right)\pm i\mathrm{Ai}\left(z\right),$
##### 3: 9.10 Integrals
9.10.1 $\int_{z}^{\infty}\mathrm{Ai}\left(t\right)\mathrm{d}t=\pi\left(\mathrm{Ai}% \left(z\right)\mathrm{Gi}'\left(z\right)-\mathrm{Ai}'\left(z\right)\mathrm{Gi}% \left(z\right)\right),$
9.10.3 $\int_{-\infty}^{z}\mathrm{Bi}\left(t\right)\mathrm{d}t=\int_{0}^{z}\mathrm{Bi}% \left(t\right)\mathrm{d}t=\pi\left(\mathrm{Bi}'\left(z\right)\mathrm{Gi}\left(% z\right)-\mathrm{Bi}\left(z\right)\mathrm{Gi}'\left(z\right)\right)\\ =\pi\left(\mathrm{Bi}\left(z\right)\mathrm{Hi}'\left(z\right)-\mathrm{Bi}'% \left(z\right)\mathrm{Hi}\left(z\right)\right).$
##### 4: Bibliography E
• H. Exton (1983) The asymptotic behaviour of the inhomogeneous Airy function ${\rm Hi}(z)$ . Math. Chronicle 12, pp. 99–104.
##### 6: 9.13 Generalized Airy Functions
###### §9.13(i) Generalizations from the Differential Equation
Swanson and Headley (1967) define independent solutions $A_{n}\left(z\right)$ and $B_{n}\left(z\right)$ of (9.13.1) by …
##### 7: Bibliography L
• Soo-Y. Lee (1980) The inhomogeneous Airy functions, ${\rm Gi}(z)$ and ${\rm Hi}(z)$ . J. Chem. Phys. 72 (1), pp. 332–336.
• ##### 8: Bibliography G
• A. Gil, J. Segura, and N. M. Temme (2001) On nonoscillating integrals for computing inhomogeneous Airy functions. Math. Comp. 70 (235), pp. 1183–1194.
• ##### 9: Bibliography M
• A. J. MacLeod (1994) Computation of inhomogeneous Airy functions. J. Comput. Appl. Math. 53 (1), pp. 109–116.
• ##### 10: Bibliography B
• H. F. Baker (1995) Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions. Cambridge University Press, Cambridge.
• P. Baldwin (1985) Zeros of generalized Airy functions. Mathematika 32 (1), pp. 104–117.
• P. Baldwin (1991) Coefficient functions for an inhomogeneous turning-point problem. Mathematika 38 (2), pp. 217–238.
• J. S. Ball (2000) Automatic computation of zeros of Bessel functions and other special functions. SIAM J. Sci. Comput. 21 (4), pp. 1458–1464.
• W. Barrett (1981) Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.