Looking for Guidance: Behavior of Gamma(like) Function…

2018-03-27 19:11:31

On my own time, I've been trying to learn as much as I can about the upper levels of mathematics. I recently came across the Gamma function:

$$\Gamma(n) = (n-1)! = \int_{0}^{\infty}(t^{x-1}e^{-x})dt = \int_{0}^{1}(-\ln(t))^{x-1}dt$$

Therefore, obviously $x! = \int_{0}^{1}(-\ln(t))^{x-1}dt$ (this can also be verified by graphing both functions). This reminded me of something I thought about a long time ago: $f(x)=(x!)^{\frac{1}{x}}$. Now that I understand more about mathematics, I (like many others both in general and on stackoverflow) was able to prove that $(x!)^{\frac{1}{x}}$ diverges to $\infty$. However, when calculating $\lim_{x\to0}(x!)^{\frac{1}{x}}$, my intial guess that $\lim_{x\to0}(x!)^{\frac{1}{x}}=\gamma=0.57721...$ was proven wrong. I found that $\lim_{x\to0}(x!)^{\frac{1}{x}}\approx0.5615...$. This leads me to my first question. Does this number have any significance? Might it have any importance other than being the arbitrary number that's the ans