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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 $w'''- 4zw'- 2w=0$, which is satisfied by $$w(z)= c\sb 1 \text{Ai} (z)+ c\sb 2 \text{Ai} (z) \text{Bi} (z)+ c\sb 3 \text{Bi} (z).$$ Then he looks for solutions in the form of Laplace contour integrals. This approach leads to a number of interesting representations for $\text{Ai}\sp 2 (z)$, $\text{Ai} (z) \text{Bi} (z)$ and $\text{Bi}\sp 2 (z)$. 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.

33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
Full Text: DOI
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