3.1.1 In terms of Airy functions

From the formulae (2.127) and (2.128), we deduce the expressions of the primitives of Ai( x) and Bi( x) [Abramowitz & Stegun (1965)]

3.1.2 Ascending series

The ascending series of the primitives of Airy functions are [Abramowitz & Stegun (1965)].

the series F( x) and G( x) being defined by integration term-by-term of the series f and g (cf. 2.1.4)

where the constants c ₁ and c ₂ are defined in 2.1.4.2 c ₁ = Ai(0) and c ₂ = Ai ?(0).

3.1.3 Asymptotic series

For x >> 1 (and ), the first terms of the asymptotic series of the primitives of the homogeneous Airy functions are [Abramowitz & Stegun (1965)]

These series are obtained by integrating term-by-term the series defined in 2.1.4.

3.1.4 Primitive of Scorer functions

Gordon (1970) also gives some primitives implying the inhomogeneous function Gi( x). The primitive ? Gi[ ?( x + ?)] d x seems unable to be expressed simply in terms of Airy functions. Nevertheless, we can calculate

3.1.5 Repeated primitives

For all the primitives given below, the integration constant has been omitted. If y is any linear combination of Airy function and y ? its derivative, we note y ₁ its primitive. Then we have

From which we find