Lambert W Functions

Lambert’s \(W\) functions, \(W(x)\), are defined to be solutions of the equation \(W(x) \exp(W(x)) = x\). This function has multiple branches for \(x < 0\); however, it has only two real-valued branches. We define \(W_0(x)\) to be the principal branch, where \(W > -1\) for \(x < 0\), and \(W_{-1}(x)\) to be the other real branch, where \(W < -1\) for \(x < 0\).


This routine computes the principal branch of the Lambert \(W\) function, \(W_0(x)\).


This routine computes the secondary real-valued branch of the Lambert \(W\) function, \(W_{-1}(x)\).