Integral representations for products of Airy functions
This paper is concerned with a method for obtaining integral representations for the products of Airy functions. The author considers first the differential equation \(w'''- 4zw'- 2w=0\), which is satisfied by \[ w(z)= c_ 1 \text{Ai} (z)+ c_ 2 \text{Ai} (z) \text{Bi} (z)+ c_ 3 \text{Bi} (z). \] Then he looks for solutions in the form of Laplace contour integrals. This approach leads to a number of interesting representations for \(\text{Ai}^ 2 (z)\), \(\text{Ai} (z) \text{Bi} (z)\) and \(\text{Bi}^ 2 (z)\). Further results include some analogues of Airy's integrals for \(\text{Ai} (x)\), the analogue for Airy functions of Nicholson's integral for Bessel functions, and a simple derivation of some Mellin transforms.
scientific article; zbMATH DE number 759665