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Integral representations for products of Airy functions. (English) Zbl 0824.33002

This paper is concerned with a method for obtaining integral representations for the products of Airy functions. The author considers first the differential equation ${w}^{\text{'}\text{'}\text{'}}-4z{w}^{\text{'}}-2w=0$, which is satisfied by

$w\left(z\right)={c}_{1}\text{Ai}\left(z\right)+{c}_{2}\text{Ai}\left(z\right)\text{Bi}\left(z\right)+{c}_{3}\text{Bi}\left(z\right)·$

Then he looks for solutions in the form of Laplace contour integrals. This approach leads to a number of interesting representations for ${\text{Ai}}^{2}\left(z\right)$, $\text{Ai}\left(z\right)\text{Bi}\left(z\right)$ and ${\text{Bi}}^{2}\left(z\right)$. Further results include some analogues of Airy’s integrals for $\text{Ai}\left(x\right)$, the analogue for Airy functions of Nicholson’s integral for Bessel functions, and a simple derivation of some Mellin transforms.

##### MSC:
 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$
Airy functions
##### References:
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