Comments on “Asymptotic expansion of a Bessel function integral using hypergeometric functions” by L.J. Landau and N.J. Luswili

In a recent paper Landau and Luswili (J. Comput. Appl. Math. 132 (2001) 387) used generalized hypergeometric functions to obtain a complete asymptotic expansion for the integral ∫ 0 π / 2 J μ ( λ sin θ ) J ν ( λ sin θ ) d θ , where J μ is the μ th-order Bessel function of the first kind and λ is a large parameter tending to infinity. The purpose of this note is to point out that the same complete asymptotic expansion for this integral (as well as another one for a Hankel-type integral) has previously been obtained by Stoyanov et al. (J. Comput. Appl. Math. 50 (1994) 533) by using the same method. In addition, an alternative, simpler representation of the algebraic series contribution to the asymptotic expansion is provided. A few errors are also corrected and additional relevant references indicated.