Integral representations for products of Airy functions
type: Item
instance of: scholarly article
title: Integral representations for products of Airy functions
zbMATH Open document ID: 0824.33002
DOI: 10.1007/BF00944750
author: W. H. Reid
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1995-11-13T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 33C10
zbMATH DE Number: 759665
zbMATH Keywords: Airy functions
reviewed by: Luigi Gatteschi
MaRDI profile type: Publication
cites work: Q4059363
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cites work: An Exact Solution of the Orr-Sommerfeld Equation for Plane Couette Flow
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description: scientific article; zbMATH DE number 759665