Real values of a function involving the Lambert $W(x)$ function

Question

I have the following function: $$y=-\dfrac{W\left(-\ln(k)\right)}{\ln(k)}$$ where $W(x)$ is the Lambert $W$ function defined as the solution of the equation: $$x=W(x)e^{W(x)}$$ If $k\in\mathbb{R}$ and $k\gt 0$, for what values of $k$, $y$ is real? I found heuristically that the value of $k$ should be in the range $1.44\lt k\lt 1.45$. How can I found it analitically, or how can I get the best approximation for the maximum value of $k$ for which the solution is real? Thanks.

Answer 1

The inverse function of $y$ is given by $\left(\frac{1}{y}\right)^{-\frac{1}{y}}$, hence it is enough to locate the maximum of $g(x)=x^{-x}$ over $\mathbb{R}^+$ to get that the domain of $y$ is given by $$ \left(0,\left(\frac{1}{e}\right)^{-\frac{1}{e}}\right).$$

Answer 2

Notice, for the 'Product Log Function' (or the Lambert $\text{W}$-function) is defined as follows:

$$f(z)=ze^z\to z=f^{-1}(ze^z)=\text{W}(ze^z)$$

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Now, we know that for $k\in\mathbb{R}^+$ (real, and bigger than zero) $\ln(k)$ is well defined.

Now, your question is about:

$$y(k)=k^{k^{k^{k^{\dots}}}}=-\frac{\text{W}\left(-\ln(k)\right)}{\ln(k)}$$

Eisenstein's (1844) considered this series of the infinite power tower. $y(k)$ converges iff $e^{-e}\le k\le e^{\frac{1}{e}}$; OEIS A073230 and A073229), as shown by Euler (1783) and Eisenstein (1844) (Le Lionnais 1983; Wells 1986, p. 35).

So, the domain of $y(k)$:

$$\left[0,e^{\frac{1}{e}}\right]$$

Answer 3

The Lambert W function has two real branches. For $W_0$, your domain is $(0,\exp(1/e))$ as noted by Jack. For $W_{-1}$, your domain is $(1,\exp(1/e))$.

$y=-\dfrac{W_0\left(-\ln(k)\right)}{\ln(k)}$ (red) and $y=-\dfrac{W_{-1}\left(-\ln(k)\right)}{\ln(k)}$ (blue)