  We study the asymptotic behaviour of the entire function \[ E(z) = \sum_{n\ge
0} \frac{z^n}{\gamma (n+1)} \] and the analytic function \[ K(z) = \frac1{2\pi
{\rm i}}\, \int_{c-{\rm i}\infty}^{c+{\rm i}\infty} z^{-s}\gamma (s)\, {\rm
d}s\,, \] which naturally appear in various classical problems of analysis.
