I'd like to fit a branch of the Lambert function :

-ProductLog[-1, x]

in the interval $[-0.35,0]$ with a function of the form $-a\log(-bx)+c+d x$ or $-a\log(-x)+c+d x$ or something like that. I saw that the functions

-Log[-x] + 1.5 + 3 x

were not so bad but I'd like to fit better...

The command $Fit$ is not good for that kind of fitting, and Fitting one function to another doesn't look appropriate either.

How could I proceed plz ?



int[a_?NumericQ, c_?NumericQ, d_?NumericQ] :=NIntegrate[( \[Minus]a Log[\[Minus]x] + c + d x -ProductLog[-1, x])^2, {x, -.35, 0}]
opt = NMinimize[int[a, c, d], {a, c, d}]
(*{0.000222356, {a -> -1.13033, c -> -1.27779, d -> -2.88088}}*)

check result

Plot[{ \[Minus]a Log[\[Minus]x] + c + d x,ProductLog[-1, x]} /. opt[[2]] // Evaluate, {x, -.35, 0}]

enter image description here


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