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Moment integrals of powers of Airy functions. (English) Zbl 0784.33003

An analytic approach to compute integrals: ${J}_{n}\left(\alpha \right)={\int }_{0}^{\infty }{z}^{n}{\left[Ai\left(z\right)\right]}^{\alpha }dz$, ${J}_{n}^{\text{'}}\left(\alpha \right)={\int }_{0}^{\infty }{\left[A{i}^{\text{'}}\left(z\right)\right]}^{\alpha }dz$ is presented, where $A{i}^{\text{'}}\left[z\right]$ is the derivative of the Airy function $Ai\left(z\right)$ and $\alpha$ is any real number. Its mathematical basis lies on the introduction of an auxiliary function:

${i}_{k,\beta }\left(\alpha \right)={\int }_{0}^{\infty }{z}^{k}{\left[Ai\left(z\right)\right]}^{\beta }{\left[A{i}^{\text{'}}\left(z\right)\right]}^{\alpha -\beta }dz,\phantom{\rule{1.em}{0ex}}\text{with}\phantom{\rule{4.pt}{0ex}}\beta \le \alpha ·$

and reduction to a linear partial difference equation with two variables and then derivation of recurrence relations for ${J}_{n}$ and ${J}_{n}^{\text{'}}$.

##### MSC:
 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$
##### References:
 [1] J. R. Albright and E. P. Gavthas,Integrals involving Airy Functions, J. Phys. A: Math. Gen.20, 2663-2665 (1985). [2] J. Kelber, Ch. Schnittler and H. Ubensee,Integral Formulae for Airy Functions, Wissenschaftliche Zeitschrift der Technischen Hochschule Ilmenau31, 147-156 (1985). [3] A. Apelblat,Table of Definite and Indefinite Integrals, Elsevier, Amsterdam 1983. [4] A. I. Nikishov and V. I. Ritus,Group Theoretical Methods in Physics v. I, II, VNU Science Press, Utrecht 1986. [5] L. W. Pearson,A Scheme for Automatic Computation of Fock-type Integrals, Institute of Electrical and Electronic Engineers, Transactions on Antennas and Propagation35, 1111-1118 (1987). · Zbl 0946.78517 · doi:10.1109/TAP.1987.1143985 [6] C. M. Bender, K. A. Milton, S. S. Pinsky and L. M. Simmons,A New Perturbative Approach to Nonlinear Problems, J. Math. Phys.30, 1447-1455 (1989). · Zbl 0684.34008 · doi:10.1063/1.528326 [7] B. J. Laurenzi,An Analytic Solution to the Thomas-Fermi Equation, J. Math. Phys.31, 2535-2537 (1990). · Zbl 0743.34021 · doi:10.1063/1.528998 [8] M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions, Natl. Bur. of Standards Applied Math. Series55, Chap. 10 (1964). [9] R. E. Mickens,Difference Equations, Van Nostrand Reinhold, New York 1987. [10] ibid., ref. 8., Chap. 15. [11] ibid., ref. 8., Chap. 25. [12] ibid., ref. 8., Chap. 6.