what I just realized: if loc is fixed, we cannot use the first equation for scale in book "Statistical distributions". this is because the first equation is derived by using the MLE condition for loc (i.e., loc has been replaced  by an expression involving scale only, the inverse of get_loc_from_scale). but if loc is fixed, the derivate w.r.t. loc cannot be assumed to be zero, so we cannot do that. maybe it is worth adding a comment?