I was talking with an acquaintance and he proposed an interesting function:
Consider a number $x$. Take the fraction $\frac{x}{x - 1}$, and raise it to the power $\frac{3x}{3x - 1}$. Iterate this process until the value converges.
Mathematically, this function looks like:
$$ f(x) = \left(\frac{3^0 x}{(3^0x)-1}\right) ^{\left(\frac{3^1x}{(3^1x)-1}\right) ^{\left(\frac{3^2x}{(3^2x)-1}\right) ^{\left(\frac{3^3x}{(3^3x)-1}\right) ^{...}}}} $$
This is a power tower where the exponents converge to 1, so $f(x)$ converges if it exists (DNE for $x = 1$). For example:
$$ f(2) = \frac{2}{1} ^{\frac{6}{5} ^{\frac{18}{17} ^{\frac{54}{53} ^{...}}}} \approx 2.47363... $$
The function for $x \in (-100, 100)$ looks like this; I think it's rather interesting how it differs from its "parent" function, $g(x) = \frac{x}{x - 1}$, (in particular in the range domain $x \in (-1, 0)$, shown here).
Interestingly, while $f(x)$ is not defined at $x = 1$, the function is complex-valued in the range domain $x \in (0, 1)$, although unfortunately I lack the wherewithal to provide a nice-looking contour plot of the complex outputs.
I'd like to ask whether some of the more advanced mathematicians on this site can comment on either of both of:
The integral $\int_{1^+}^{\infty} \left[f(x) - g(x)\right] dx$. It seems tantalizingly similar to the Euler-Mascheroni constant in theme.
The behavior of $f(x)$ in the
rangedomain $x \in (0, 1)$. I'm not especially well-versed in complex analysis, so I would love any commentary.
Thanks!