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LambertW function homework

  1. Oct 24, 2014 #1
    If, ##f(x)=x^x##, then, f-1(x)=?
  2. jcsd
  3. Oct 24, 2014 #2
    I don't think there's a closed form expression for that.
  4. Oct 24, 2014 #3


    Staff: Mentor

  5. Oct 27, 2014 #4
  6. Oct 27, 2014 #5


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    Science Advisor

    Nicely done! I misread your answer at first and thought you had it wrong but saw that you are correct after working it out for myself. The Lambert W function is inverse to [itex]f(x)= xe^x[/itex] but taking the logarithm of both sides of [itex]x^x= y[/itex] gives [itex]xln(x)= ln(y)[/itex] not [itex]xe^x= y[/itex].

    Instead, once you have [itex]xln(x)= ln(y)[/itex], let [itex]u= ln(x)[/itex]. Of course, then, [itex]x= e^{ln(x)}= e^u[/itex] so the equation becomes
    [itex]ue^u= ln(y)[/itex], [itex]u= ln(x)= W(ln(y))[/itex] so that, as you say, [itex]x= exp(W(ln(y))[/itex].
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