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:
Γ(n)=(n−1)!=∫∞0(tx−1e−x)dt=∫10(−ln(t))x−1dt
Therefore, obviously x!=∫10(−ln(t))x−1dt (this can also be verified by graphing both functions). This reminded me of something I thought about a long time ago: f(x)=(x!)1x. Now that I understand more about mathematics, I (like many others both in general and on stackoverflow) was able to prove that (x!)1x diverges to ∞. However, when calculating lim, 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 answer to this question? [ANSWERED BY R_Berger]
Moving on to my second question. When I graphed this to verify my solution, I was surprised to see that the graph was practically a straight line (I expected more than a negligible curve of some type). Taking the derivative of this function would obviously be shown as \frac{d}{dx}\left(\int_0^1\left(-\ln\left(t\right)\right)^{x}dt\right)^{\frac{1}{x}}. I was unable to find the above derivative either by hand or by using a calculator. So, is there a way to take the above derivative or any other variant of x! (or \Gamma(x-1))? [NO POINTERS GIVEN YET]
Lastly, most importantly, and my primary reason for asking this question... My ultimate goal is finding \lim_{x\to\infty}\frac{d}{dx}\left(\int_0^1\left(-\ln\left(t\right)\right)^{x}dt\right)^{\frac{1}{x}}. If you have any pointers or small hints on anything I could look into, that would very much be appreciated. [NO POINTERS GIVEN YET]
If you take the time to read and answer this, I thank you very much in advance.
No comments:
Post a Comment