How to Take the Factorial of Any Number
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it as an intuitive motivation for why the gamma function is the right extension of the factorial, and as the source of the limit-product derivation they then ask follow-up questions about.
Recommended by 1 people across 1 different months, 2023 to 2025.
What it is
A 26-minute animated walkthrough by Lines That Connect that derives the gamma function as a natural extension of the factorial to fractional, irrational and complex numbers. It builds from the recursive formula and logarithms to the limit product form, then ends with a surprising connection. Animations are made with Manim.
What people said
Recommends Polygon · 2023-12 · Mathematics Stack Exchange · on “How can construction of the gamma function be motivated?”
This video might be helpful: [this video] though it's kind of long. In short, we might want a function which satisfies the factorial's recursive formula, where the log of the function becomes "linear" (second derivative approaches 0), as x approaches infinity. There's only one such function.
Cites anddrypv7 · 2023-04 · Mathematics Stack Exchange · on “Can i exchange limit and derivation for this sequence?”
i was watching a video [this video] on extending the factorial and got confused about one step in particular. In 22:45 he claims the following: $\frac{d}{dx} \left( \lim_{N \to \infty} \sum_{k=1}^{N} \left( \ln{\frac{k}{x+k}} \right) +x \cdot \ln N \right)=\lim_{N \to \infty} \left( \sum_{k=1}^{N} \left(\frac{d}{dx}\left( \ln{\frac{k}{x+k}} \right)\right) +\frac{d}{dx} \left( x \cdot \ln N \right) \right)$ To be able to swap derivative and the limit, the sequence has to be uniformly convergent. From a post i posted earlier, it appears not to be the case. Am i missing something? My thoughts: Is this step actually legal? Is there any paper that derives the factorial as seen in video? Thank you for
Cites carbonSoda · 2024-09 · Mathematics Stack Exchange · on “Equivalence product form of the Gamma function”
From the LinesThatConnect video [this video] we could derive the factorial function $$ \lim _{N \rightarrow \infty} N^x \prod_{k=1}^N \frac{k}{k+x} $$ and it’s indeed equivalent to $\Gamma(x+1)$ according to wolframAlpha. But I can’t find any way to prove this. Could anyone provide some proof outline or insights?
Cites Cheerful Parsnip · 2025-07 · Mathematics Stack Exchange · on “Can someone explain to me how a formula for the sine function is derived?”
That's a cool idea to use that $\sin x\approx x$ near $0$ together with a recursion to extend the approximation to the entire regime, end even characterize sine. It's similar in philosophy to how this video extends factorial to real numbers: [this video]