In this video, we build on my last two videos by exploring connections between the gamma function (the extended factorials), the digamma function (the extended harmonic numbers), and trigonometry. We derive Euler's Sine Product Formula, which we then use to prove the gamma and digamma functions' reflection formulas. Finally, we derive a related formula for calculating cotangent.

Watch my previous two videos here:
Extending the Harmonic Numbers to the Reals:    • Extending the Harmonic Numbers to the...  
How to Take the Factorial of Any Number:    • How to Take the Factorial of Any Number  

An Elementary Proof of the Sine Product Formula:
https://www.researchgate.net/publicat...

The animations in this video were made with Manim: https://www.manim.community/

Music credits:
Fluidscape by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 license. https://creativecommons.org/licenses/...
Night Music by Kevin Macleod
Space Chillout by penguinmusic
river - Calm and Relaxing Piano Music by HarumachiMusic
Surrealism (Ambient Mix) by Andrewkn
... And a couple of my own songs:
  / the-fog  
  / thanks-for-watching  


Chapters:
00:00 Intro
0:43 Background and Notation
3:24 The Digamma-Cotangent Connection
5:09 The Gamma-Sine Connection
6:04 The Sine Product Formula
9:59 Proving the Gamma-Sine Connection
12:22 The value of (1/2)!
13:07 Proving the Digamma-Cotangent Connection
14:21 The True Logarithmic Derivative
15:52 An Infinite Sum for Cotangent
17:46 Final Thoughts