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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 1 Ai(z)+c 2 Ai(z)Bi(z)+c 3 Bi(z)·

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


MSC:
33C10Bessel and Airy functions, cylinder functions, 0 F 1
References:
[1]F. W. J. Olver,Asymptotics and Special Functions, Academic Press, New York and London 1974.
[2]B. J. Laurenzi,Moment integrals of powers of Airy functions, Z. angew. Math. Phys.44, 891-908 (1993). · Zbl 0784.33003 · doi:10.1007/BF00942815
[3]M. V. Berry,Uniform asymptotic smoothing of Stokes’s discontinuities, Proc. Roy. Soc. London Ser.A422, 7-21 (1989). · Zbl 0683.33004 · doi:10.1098/rspa.1989.0018
[4]F. W. J. Olver,On Stokes’ phenomenon and converging factors, inAsymptotic and Computational Analysis (R. Wong, ed.), Marcel Dekker, New York and Basel 1990, pp. 329-355.
[5]E. T. Copson,Asymptotic Expansions, Cambridge Univ. Press, London and New York 1965.
[6]W. H. Reid,An exact solution of the Orr-Sommerfeld equation for plane Couette flow, Studies in Appl. Math.61, 83-91 (1979).
[7]G. N. Watson,A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge Univ. Press, London and New York 1944.
[8]A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi,Higher Transcendental Functions, vol. 1, McGraw-Hill Book Co., Inc., New York 1953.