How to Take the Factorial of Any Number

In this video, I walk through the derivation of an extension of the factorial function that works for any number: fractional, irrational, and even complex! This turns out to be a very important function, known as the gamma function, which has many surprising connections, one of which I explore in the last chapter of the video. The animations in this video were made with Manim, an open-source Python library for making math animations, originally created by 3Blue1Brown. https://www.manim.community/ My Previous video: Extending the Harmonic Numbers to the Reals: https://www.youtube.com/watch?v=9p_U_o1pMKo Chapters: 0:00 Introduction 1:38 A few Disclaimers 3:58 The Recursive Formula 6:50 The Super Recursive Formula 8:45 A minor setback 10:28 Logarithms 15:21 Deriving the Solution 19:26 Our Constraints 20:25 History and Conventions 22:16 The Miracle 25:44 The End I went for some more relaxing background music this time. Hopefully it doesn't put you to sleep! Creative Commons music used in this video: ------------------------------------------ Enchanted Journey by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 license. https://creativecommons.org/licenses/by/4.0/ Source: http://incompetech.com/music/royalty-free/index.html?isrc=USUAN1100799 Artist: http://incompetech.com/ ------------------------------------------ Fluidscape by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 license. https://creativecommons.org/licenses/by/4.0/ Source: http://incompetech.com/music/royalty-free/index.html?isrc=USUAN1100393 Artist: http://incompetech.com/ ------------------------------------------ Kevin MacLeod is a life-saver! Other music used in this video: - River by HarumachiMusic - Night Music by Kevin MacLeod - Moon Men by Jake Chudnow And a couple of my own songs: - https://soundcloud.com/lines-that-connect/thanks-for-watching - https://soundcloud.com/lines-that-connect/the-fog #SoME2 #Mathematics #Education