Mathematica Eterna
Open Access

Abstract

The asymptotics of the generalised Bessel function

R. B. Paris

We demonstrate how the asymptotics for large |z| of the generalised Bessel function 0Ψ1(z) = X∞ n=0 z n Γ(an + b)n! , where a > −1 and b is any number (real or complex), may be obtained by exploiting the well-established asymptotic theory of the generalised Wright function pΨq(z). A summary of this theory is given and an algorithm for determining the coefficients in the associated exponential expansions is discussed in an appendix. We pay particular attention to the case a = − 1 2 , where the expansion for z → ±∞ consists of an exponentially small contribution that undergoes a Stokes phenomenon. We also examine the different nature of the asymptotic expansions as a function of arg z when −1 < a < 0, taking into account the Stokes phenomenon that occurs on the rays arg z = 0 and arg z = ±π(1 + a) for the associated function 1Ψ0(z). These regions are more precise than those given by Wright in his 1940 paper. Numerical computations are carried out to verify several of the expansions developed in the paper.