Let $$M(z)=1+\sum_{n=1}^{\infty} \frac{z^{n}}{n^{n}}$$
Q1: Is there a standard name for this function (besides "comparison function")? I'm interested in it as an analogue to the exponential function because $M(1)$ and $M(-1)$ can very easily be expressed in terms of the sophomore's dream integrals: $M(1)=1+\int_{0}^{1} \frac{dx}{x^{x}}$ and $M(-1)=1-\int_{0}^{1} x^{x} dx$.
Q2: Are there standard names for the obvious analogues for the regular and hyperbolic, sine and cosine with $M(z)$ in the place of $\exp(z)$?
