  We demonstrate how the asymptotics for large $|z|$ of the generalised Bessel
function \[{}_0\Psi_1(z)=\sum_{n=0}^\infty\frac{z^n}{\Gamma(an+b) n!},\] 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
${}_p\Psi_q(z)$. 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 $z\to\pm\infty$ 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 $\arg\,z=\pm\pi(1+a)$ for the
associated function ${}_1\Psi_0(z)$. These regions are more precise than those
given by Wright in his 1940 paper. Numerical computations are carried to verify
several of the expansions developed in the paper.
