Real valued functions for the bfkl eigenvalue

We consider known expressions for the eigenvalue of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation in N = 4 super Yang-Mills theory as a real valued function of two variables. We define new real valued functions of two complex conjugate variables that have a definite complexity analogous to the weight of the nested harmonic sums. We argue that those functions span a general space of functions for the BFKL eigenvalue at any order of the perturbation theory.