Long before the Nuttall-Pommerenke theorem was established, George Baker and his collaborators observed the phenomenon of spurious poles: several of the approximants could have poles which in no way were related to those of the underlying function. However, those poles affected convergence only in a small neighbourhood, and there were usually very few of these “bad” approximants. Thus, one might compute [ n/n ] , n = 1 , 2 , 3 , . . . 50, and find a definite convergence trend in 45 of the approximants, with five of the 50 approximants displaying pathological behaviour. The curious thing (contrary to expectation) is that the five bad approximants could be distributed anywhere in the 50, and need not be the first few. Nevertheless, after omitting the “bad” approximants, one obtained a clear convergence trend. This seemed to be a characteristic of the Pad´e method, and Baker et al. formulated a now famous conjecture [4]. There are now many forms of the conjecture; we shall concentrate on the following form: