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Lambert W Function

  1. Oct 8, 2006 #1
    How can I use the W function to solve for t?

    a = bt + e^(ct),

    where a,b,c,e are known constants. e is Euler's number. t is the unknown?

    I have only seen examples where a = 0. Thanks.
  2. jcsd
  3. Oct 8, 2006 #2


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    Science Advisor

    Do a little thinking and a little algebra to get it into the right form.
    Lambert's W function is the inverse function to f(x)= xex so you have to change variables to get your equation into that form. From
    a= bt+ ect, a- bt= ect. Okay, let u= a- bt. Then t= -(u-a)/b so
    [tex]e^{ct}= e^{\frac{-cu}{b}}e^{\frac{a}{b}}= u[/tex]
    [tex]e^{\frac{a}{b}}= ue^{\frac{cu}{b}}[/tex]
    Now let [itex]x= \frac{cu}{b}[/itex] so [itex]u= \frac{b}{c}x[/itex] and
    [tex]e^{\frac{a}{b}}= \frac{b}{c}xe^x[/tex]
    [tex] \frac{c}{b}e^{\frac{a}{b}}= xe^x[/tex]
    so that
    [tex]x= W(\frac{c}{b}e^{\rac{a}{b}})[/tex]
  4. Oct 8, 2006 #3
    Thanks a lot. That's cool!
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