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The generalized Lambert W-function

  1. Feb 10, 2005 #1

    saltydog

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    Can anyone here help me derive the generalized Lambert function? I'm working on a solution for an ODE from the homework group which involves this function. This is what I have so far:

    The W-function is defined as the inverse of the following:

    [tex]
    f(x)=xe^x=y
    [/tex]

    then:

    [tex]f^{-1}(y)=x=W(y) [/tex]

    with W being the Lambert W-function for [itex]y>-e^{-1}[/itex]

    I need help showing the following:

    If:

    [tex] g(x)=x^2e^x=y [/tex]

    then:

    [tex] g^{-1}(y)=2W(\frac{\sqrt y}{2}) [/tex]

    and in general if:

    [tex] h(x)=x^ne^x=y[/tex]

    then:

    [tex] h^{-1}(y)=nW(\frac{y^\frac{1}{n}}{n})
    [/tex]

    Thanks,
    Salty
     
  2. jcsd
  3. Feb 10, 2005 #2

    StatusX

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    you need to get one side into the form:

    [tex]f(x) e^{f(x)}[/tex]

    then take the W of both sides, which will leave f(x) here.
     
  4. Feb 10, 2005 #3

    dextercioby

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    Where did you come up with that factor "n" in front of the W function...?

    Daniel.
     
  5. Feb 10, 2005 #4

    saltydog

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    Mathematica reports the results as I stated but I'd like to understand how it's arriving at it. I've checked it with real numbers (I don't have a support contract and they don't like me bothering them).
     
  6. Feb 10, 2005 #5

    saltydog

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    Thanks StatusX. I don't see that but will try and work with that logic in the morning.

    Salty
     
  7. Feb 10, 2005 #6

    dextercioby

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    It's straightforward.
    [tex] x^{n}e^{x}=y \Rightarrow xe^{\frac{x}{n}}=y^{\frac{1}{n}}\Rightarrow \frac{x}{n}e^{\frac{x}{n}}=\frac{y^{\frac{1}{n}}}{n} [/tex]

    Apply the W (Lambert function) on the last equality and u'll get
    [tex] \frac{x}{n}=W(\frac{y^{\frac{1}{n}}}{n}) \Rightarrow x=n W(\frac{y^{\frac{1}{n}}}{n}) [/tex]

    which is the inverse function of the one you started with.

    Daniel.
     
  8. Feb 11, 2005 #7

    saltydog

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    Yep, would not have figured that on my own. I mean, it took me a while to even see what you were doing. Thanks Daniel.
    I'll follow-up with a report (and plot) in the homework section for this problem. I know they're long-gone to other things but I tell you what, they missin' out (and they wouldn't like me as their teacher because I'd make them do this extra stuf for at least some of the problems).

    Salty
     
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