i^i, visualized, and explained. Full playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDP5CVelJJ1bNDouqrAhVPev Home page: https://www.3blue1brown.com Brought to you by you: https://3b1b.co/thanks Mistakes: At 1:06:20, when changing r to equal 0.69*i, I said "this is what we might think of as (2i)^x", but that is not correct. It's what we'd think of as [Exp(ln(2)*i)]^x for whatever complex number Exp(ln(2)*i) is. Video by Matt Parker: https://youtu.be/9tlHQOKMHGA Video by Red Pen Black Pen https://youtu.be/ABk1HK2AR2E ------------------ The live question setup with stats on-screen is powered by Itempool. https://itempool.com/ Curious about other animations? https://www.3blue1brown.com/faq#manim Music by Vincent Rubinetti. 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 If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people. ------------------ 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_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown ... https://www.youtube.com/watch?v=pq9LcwC7CoY

(Sine waves / boundary condition) + linearity + Fourier = Solution Home page: https://www.3blue1brown.com Brought to you by you: http://3b1b.co/de3thanks More about the heat equation, with a derivation in terms of slope corresponding to heat flow from MIT OCW: https://ocw.mit.edu/courses/mathematics/18-303-linear-partial-differential-equations-fall-2006/lecture-notes/heateqni.pdf If you want to learn more about Fourier series, here are a few great videos/posts which I think you'll enjoy: Mathologer's: https://youtu.be/qS4H6PEcCCA The Coding Train: https://youtu.be/Mm2eYfj0SgA ... https://www.youtube.com/watch?v=ToIXSwZ1pJU