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:
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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/
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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/
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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