On functions K and E generated by a sequence of moments

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

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