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:(adsbygoogle = window.adsbygoogle || []).push({});

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

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# The generalized Lambert W-function

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