Thanks for watching everyone! I'm overwhelmed by the response to this video - 100k views is more than I dared to hope for!

I've got a couple quick clarifications:

5:26 - This cannot hold for every x - only for values where the domain of the function allows the formula to make sense. It turns out that this excludes non-positive integers. Some people rightly pointed out that the recursive formula seems to imply that 0! = 0 * (-1)! = 0., but this assumes that (-1)! exists and is finite. In fact it was that exact formula that led to the conclusion that there must be an asymptote at -1. (6:33)

9:08 - We might guess that we can make the function behave better by taking its reciprocal, which would make it flatten out and rapidly approach 0. This is actually one of the first things I tried, but unfortunately it doesn't work. It would work the function approached any value except for 0, but since the factorials are all about multiplication, and since 0 * anything = 0, we don't get any new information.

0:04 - So I wasn't actually in middle school. In my memory I was in the 8th grade, but I checked the Wayback Machine, and the version of the site I remember didn't exist until my first year of high school.

21:27 - The proof that I have the easiest time understanding is "Proof 2" on this ProofWiki page: https://proofwiki.org/wiki/Integral_Form_of_Gamma_Function_equivalent_to_Euler_Form

Another note - This also works for complex numbers! You can just plug a complex number in for x, and it will converge. I made sure I never mentioned real numbers and instead said "any number" or "non-integer", so that I didn't accidentally exclude complex numbers.

i have no clue what is going on

γ not proven as irrational is crazy!!!

You could make a series out of this where you explain how extensions of different discrete functions are derived! You could call it “Points that connect.”

0:14 "Plugging in different functions in a graphing calculator is a weird pastime"

You know I'm something of a mathematician myself.

13:24 N = 49 if you were curious

I call uppercase sigma bigma

This was one of the most well put together math videos I have ever seen. Please do not stop making content because you truly have incredible potential as a math explainer

That was a fantastic Vsauce "or is it" with the music

2:30 This so true! Lectures in university are usually about proving as many theorems, lemmas and formulas as possible during certain period despite the fact that it completely misses the point of sharing a proof with students. The fact itself that you’d shown a certain proof to a student doesn’t matter, what matters is student understanding why formula or theorem is the way it is and gaining additional intuition about the topic.

The vsauce music caught me off guard

I was shocked to see that you only have two videos. The production of this and the explanation were both fantastic. Keep it up, I'll be there to watch anything else you put out!

Truly appreciate the cultured reference to Vsauce at 10:36

18:00 i dont even understand anything anymore im here for the animation ASMR

“I can show that Mascheroni is actually an imaginary number masquerading as an irrational, I have a proof of this theorem, but there is not enough space in this margin"

Dude, your channel is out of this world! I already considered this video one of the best math-related ones I've seen in a long while, several mitutes befor its end. However, when I saw the definition of gamma appear so naturally from the derivative of x!, I literally started screaming "It's gamma! GAMMA!" before the limit even appeared. This video reminded me of how much I - who dropped of a STEM major in favor of a Humanites one - still love math, and why. Thank you so, so, SO much! 😍

10:38 Hey, Lines That Connect here!

Nice easter egg at 20:37 with the taxi zipping across the screen, representing the fact that 1729 is a taxicab number!

mhm mhm i see (understood absolutely nothing but is entertained)

This was very well done! I actually used the gamma function in my own SoME2 submission and wished I could have included a derivation of it, at least as a side resource. But now I can just point to this video!