On functions K and E generated by a sequence of moments

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.