The Riemann surface of the \(r\)-Lambert function (Q2043744)

The \(r\)-Lambert function of the title of the paper is a generalization of the classical Lambert \(W\) function which is the (multi-valued) inverse of the complex function \(z \exp (z)\) (with various applications in mathematics, physics and science). Considering such a generalization, the \(r\)-Lambert function is defined as the inverse of the function \(z \exp (z) + rz\). ``In the present paper, we construct the Riemann surface of this function. It turns out that this surface has some peculiar properties, therefore it might be useful for demonstration purposes, for those who would like to see non-standard examples of Riemann surfaces coming from complex function theory.'' ``In order to construct the Riemann surface of the \(r\)-Lambert function, we need information about the branches, branch cuts, and several mapping properties of our function.'' This is based on a previous paper by the present author [J. Math. Anal. Appl. 455, No. 1, 538--553 (2017; Zbl 1371.33036)], cf. also a paper by the present author \textit{Á. Baricz} [Trans. Am. Math. Soc. 369, No. 11, 7917--7934 (2017; Zbl 1375.33034)].