Find the range of a univariate function over a specific interval of the variable

I would like to find the range of the function below, but only for values of $a$ strictly greater than zero.

v(a_)=-a*Log(1-(a - a E^(-1 - 1/a + ProductLog(E^(1 + 1/a))))/a)

In other words: what is the range of $v(a)$, for/conditional on $a>0$ (reals).


  • I have tried this: (ie without restricting)
    FunctionRange(v(a), a, y), but the computation time is too long, I can’t manage to get an output.

  • I also have tried to define the $v(a)$ function over the domain that interests me using this:

    v(a_ /; 0 < a) := -a*Log(1-(a - a E^(-1 - 1/a + ProductLog(E^(1 + 1/a))))/a) (solution found here), and then FunctionRange(v(a), a, y), but the computation time seems to bee stil too long. Probably due to the fact that my restricted definition doesn’t seem to have worked: even in that case, FunctionDomain(v(a),a)returns $a≠0$ (instead of $a>0$ as I would have expected since it is what I defined in the first place).

EDIT

  • This doesn’t seem to work either: Assuming(a>0,FunctionRange(v(a), a, y)).