Analytic structure and explicit solution of an important implicit equation

The equationz=2G(z)-expG(z)+1 (and similar ones obtained from it by substitutions) appears in connection with a variety of problems ranging from pure mathematics (combinatorics; some first order, nonlinear differential equations) over statistical thermodynamics to renormalization theory. It is therefore of interest to solve this equation forG(z) explicitly. It turns out, after study of the complex structure of thez andG planes, that an explicit integral representation ofG(z) can be given, which may be directly used for numerical calculations of high precision.