A.1 The Homogeneous Functions

Except the numerical tables of Miller (1946) and Abramowitz & Stegun (1965), the calculation of the Airy functions can be carried out by means of the algorithms of Gordon (1969, 1970) and Lee (1980). Indeed, if we use a Gauss method of quadrature, the Airy function Ai( x) should be replaced by the following sums (here we highlight the formulae (2.32) and (2.34)), where , x > 0

and the Bi( x) function by (formulae (2.33) and (2.35))

w _i are the weight factors corresponding to the integration points x _i [Gordon (1970)]. In Table A.1, we give the partition of for the calculation of the homogeneous Airy functions. We give in Table A.2 the values of w _i and x _i for integration in ten points. In all the cases, the neighbourhood of the origin will be calculated with the help of the ascending series, given by the formulae (2.37) and (2.38).

Table A.1: Partition of for the calculation of the homogeneous Airy functions

x	Ai( x)	Bi( x)
x < ?3.7	integration A.2	integration A.4
?3.7 < x < 2.35	series 2.37	series 2.38
2.35 < x < 8.5	integration A.1	series 2.38
8.5 < x	integration A.1	integration A.3