The Lambert W Function

has a countably infinite number of solutions, which are denoted by Wk(z) for integers k. Each choice of k specifies a branch of the Lambert W function. By convention, only the branches k = 0 (called the principal branch) and k = −1 are real-valued for any z; the range of every other branch excludes the real axis, although the range of W1(z) includes (−∞,−1/e] in its closure. Only W0(z) contains positive values in its range (see figure 1). When z = −1/e (the only nonzero branch point), there is a double root w = −1 of the basic equation wew = z. The conventional choice of branches assigns W0(−1/e) = W−1(−1/e) = −1