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Zbl 0824.33002
Reid, W.H.
Integral representations for products of Airy functions. (English)
[J] Z. Angew. Math. Phys. 46, No.2, 159-170 (1995). ISSN 0044-2275; ISSN 1420-9039/e

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.
[L.Gatteschi (Torino)]

MSC 2000:: *33C10 Cylinder functions, etc.

Keywords: Airy functions

Cited in: Zbl 0879.33002 Zbl 0874.33004 Zbl 0874.33003

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