# 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 $$x < 0$$.

gsl_sf_lambert_W0(x)

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

gsl_sf_lambert_Wm1(x)

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