Where's the circle? And how does it relate to where e^(-x^2) comes from? Help fund future projects: https://www.patreon.com/3blue1brown Special thanks to these supporters: https://www.3blue1brown.com/lessons/gaussian-integral#thanks An equally valuable form of support is to simply share the videos. The artwork in this video is by Kurt Bruns, aided by Midjourney Here are several other good posts about the classic Poisson proof vcubingx: https://www.youtube.com/watch?v=9CgOthUUdw4 BriTheMathGuy: https://www.youtube.com/watch?v=S79KPrIm_Gc Dr. Alter's math library: https://idan-alter.github.io/2023/02/20/Gaussian-Integral.html And if you'd like to see many other variations on approaching this integral, take a look at this expository paper from Keith Conrad: https://kconrad.math.uconn.edu/blurbs/analysis/gaussianintegral.pdf Correction: 13:46 - The denominator should read 2πσ^2 Timestamps: 0:00 - The statistician's friend 3:44 - The classic proof 12:47 - The Herschel-Maxwell derivation 21:55 - Reflecting back on the proof 23:50 - A bonus problem Thanks to these viewers for their contributions to translations French: Jeremy Senn, Laurent Joubert German: dreivmeister, lprecord, mathiashaebich Hebrew: David Bar-On, Omer Tuchfeld Hindi: prateekbansal97 Spanish: Joel Tovar Turkish: kilobravo3 ------------------ These animations are largely made using a custom python library, manim. See the FAQ comments here: https://www.3blue1brown.com/faq#manim https://github.com/3b1b/manim https://github.com/ManimCommunity/manim/ You can find code for specific videos and projects here: https://github.com/3b1b/videos/ Music by Vincent Rubinetti. https://www.vincentrubinetti.com/ Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com/ Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown