6.1 Generalisation of the Airy Integral

Airy functions are solutions to the Airy differential equation y ?? ? xy = 0. A generalisation of these functions may be made in considering the solutions of the second order differential equation of the kind: y ?? + C x ^ky = 0.

Since the Airy function is defined by the integral (2.19)

we can generalise this integral as follows. We put, according to Watson (1966),

where , n ? 2, and F is the hypergeometric function defined by [Abramowitz & Stegun (1965)]

Thus we obtain

So the generalisation of the Airy integral is given by

As particular cases, we have (formula (2.19)) and (formula (2.125)). ^[1]

Like in the case of Airy functions (cf. 2.2.4), we can express the integrals (6.3), (6.4) and (6.5), thanks to the Bessel functions I, J and K, according to the parity of n.

We shall not detail the calculations (see for instance Watson, 1966), but if n is even, Ci _n( ?) and Si _n( ?) are solutions of the differential equation

with Wi = Ci _n( ?), Si _n( ?). The function Ei _n( ?) is a solution to the equation

These three functions may be expressed in the following form, with ? > 0,

and

For example, in the case n = 4, we obtain

where P