- #1
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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
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