4.1.1 Causal relations

With the help of the integral representation of Ai( x), Gi( x), Ai ²( x) and Ai( x) Bi( x) respectively given by the formulae (2.20), (2.125), (2.148) and (2.149), we can write [Scorer (1950); Aspnes (1967)]

These functions being analytic in the complex plane, the real and imaginary parts may be written as a Hilbert transform. We then have

and conversely

where is the Cauchy principal value. From the relation (4.2), we can also deduce

and conversely

With the help of the relation (4.1), we can write

So that when ? ? 0, this last relation becomes

that is to say

and

It can be seen that these last two expressions are the beginning of the expansions given by Lee (1980)

and

4.1.2 Green function of the Airy equation

In this section, we are going to build the Green function satisfying the differential equation [Moyer (1973 ); Burnett & Belsley (1983)]

The integral expression of this function given by Lukes & Somaratna (1969), is

with

We can write, thanks to the integral representation (2.20) of the function Ai( x)

for 0 ? arg( z ? z ?) ? ?/2, and

for 0 ? arg( z ? ? z) ? ?/2. Comparing Eqs. (4.13) (4.15), we obtain

Thanks to the relation

we can then write G( x, x ?) as...