When trying to solve the equation $y^y = \frac{\ln^{y(1+c)}n}{n}$ , I've found the result $$y=\frac{-\ln n}{W(-\ln^{-c}n)}$$ where $c$ is a positive constant and $W$ is the Lambert function.
The result should be positive, and the result I obtain seems to be positive, but I don't like the form.
Is it possible to obtain a better form for this result ?
Thank you very much.
Well, you could rewrite the last line as
$$y=e^{W(-\ln^{-c}n)}$$
$$=e^{W\left(\frac{-1}{\ln^cn}\right)}$$
For clarity, I'm going to write the stuff inside the $W$,
$$\frac{-1}{\ln^cn}=\frac1{\left(e^{\frac\pi ci}\ln(n)\right)^c}$$
$$y=e^{W\left(\frac1{\left(e^{\frac\pi ci}\ln(n)\right)^c}\right)}$$
Got rid of the negatives?