# Tagged Questions

For questions related to the Lambert-W or product log function. This is the inverse function of $f(z) = ze^z$.

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### Is this metric space normable?

Given the function $\rho:\mathbb{R}^n\to\mathbb{R}_+$ defined by $$\rho(x)=\log\left(\frac{\sum\left|x_i\right| e^{\left|x_i\right|}}{W\left(\sum\left|x_i\right| e^{\left|x_i\right|}\right)}\right)$$ ...
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### Is it possible to solve for $x$ using the lambert W function in the expression ${\ln\left(x\right)}=(t-x)^2$?

${\ln\left(x\right)}=(t-x)^2$ $\pm\sqrt{\ln\left(x\right)}+x=t$ $\mathrm{e}^{\sqrt{\ln\left(x\right)}+x}=e^t$ And that is as close as I can get it to the form $x\mathrm{e}^x$. What do I do next? ...
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### Any way to characterize this family of polynomials?

I have a family of polynomials generated by the recurrence relation $P_{n+1}(w) = (1+w)P_n ^{\ \prime}(w) -(3n-1 +nw)P_n(w) \\ P_1(w) =1$ The family is related to the Lambert $W$-function by its ...
1answer
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### Is there a relation between the branches of the Lambert function?

Is it possible to express $W_{-1}(z)$ exactly by a closed-form expression, allowing the principal branch function $W_0$ ? Update: I found this related post: http://mathoverflow.net/a/196321.
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### Asymptotic evaluation of a quantity

Can we say that the following quantity (a recursion of logarithms): $W_{-1}(x)=\ln \cfrac{-x}{-\ln \cfrac{-x}{-\ln \cfrac{-x}{...}}}$ is $\Theta(\ln x)$? i.e., asimptotically upper and lower bounded?...
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### $\log(x)$ as iteration-series: how can this be made correct?

I was tinkering with the question whether the logarithm $\log(x)$ can be expressed by some more useful series than by the Mercator series (in terms of (1+x)) for a certain question. One idea ...
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### What is the solution for $y(t)=e^{-\frac{t}{\tau y(t)}}$?

A simple quadratic flow model leads to the following apparently simple equation $$y(t)=e^{-\frac{t}{\tau y(t)}}$$ where the flow, $y$ is a function of time, $t$ and $\tau$ is a constant. But is ...
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### Function inversion (analytical)

Can $t(x)$ be found from: $$A \, t + B\ln\frac{1-t}{t}=x \; ?$$ Here, $A>0, \; B < 0$ and $0 \lt t \lt 1$. The $t(x)$ should be given in analytical form (even if you use, say, Lambert's W - ...
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### Sign of a quantity involving the Lambert function

Is the following quantity $$\frac{W(\ln x)}{x}\left(1-\frac{1}{1+W(\ln x)}\right)$$ positive for $x \geq 1$ ? Thank you very much.
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### How do you solve x^2 = log^2(x)

I read a page that said that the limit as $x$ approaches infinity of (polynomial function)/(logarithmic function) = infinity and that the limit as $x$ approaches infinity of (logarithmic function)/...
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