Euler's Formula & Laplace Transform Basics
Explore the physics behind Euler's formula, harmonic oscillators, and complex exponents in ODEs. Next: Laplace Transform tutorial. π

3Blue1Brown
696.6K views β’ Oct 5, 2025

About this video
The simple harmonic oscillator and the fundamental role of complex exponents for ODEs.
Next chapter on the Laplace Transform video: https://youtu.be/j0wJBEZdwLs
Instead of sponsored ad reads, these lessons are funded directly by viewers: https://3b1b.co/support
An equally valuable form of support is to simply share the videos.
Home page: https://www.3blue1brown.com
Timestamps:
0:00 - Intro
1:51 - Euler's formula explained dynamically
9:27 - The harmonic oscillator
21:08 - General linear equations
22:47 - Motivating the Laplace Transform
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These animations are largely made using a custom Python library, manim. See the FAQ comments here:
https://3b1b.co/faq#manim
Music by Vincent Rubinetti.
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. If you're reading the bottom of a video description, I'm guessing you're more interested than the average viewer in lessons here. It would mean a lot to me if you chose to stay up to date on new ones, either by subscribing here on YouTube or otherwise following on whichever platform below you check most regularly.
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Facebook: https://www.facebook.com/3blue1brown
Patreon: https://patreon.com/3blue1brown
Website: https://www.3blue1brown.com
Next chapter on the Laplace Transform video: https://youtu.be/j0wJBEZdwLs
Instead of sponsored ad reads, these lessons are funded directly by viewers: https://3b1b.co/support
An equally valuable form of support is to simply share the videos.
Home page: https://www.3blue1brown.com
Timestamps:
0:00 - Intro
1:51 - Euler's formula explained dynamically
9:27 - The harmonic oscillator
21:08 - General linear equations
22:47 - Motivating the Laplace Transform
------------------
These animations are largely made using a custom Python library, manim. See the FAQ comments here:
https://3b1b.co/faq#manim
Music by Vincent Rubinetti.
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. If you're reading the bottom of a video description, I'm guessing you're more interested than the average viewer in lessons here. It would mean a lot to me if you chose to stay up to date on new ones, either by subscribing here on YouTube or otherwise following on whichever platform below you check most regularly.
Mailing list: https://3blue1brown.substack.com
Twitter: https://twitter.com/3blue1brown
Bluesky: https://bsky.app/profile/3blue1brown.com
Instagram: https://www.instagram.com/3blue1brown
Reddit: https://www.reddit.com/r/3blue1brown
Facebook: https://www.facebook.com/3blue1brown
Patreon: https://patreon.com/3blue1brown
Website: https://www.3blue1brown.com
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Video Information
Views
696.6K
Likes
30.6K
Duration
27:49
Published
Oct 5, 2025
User Reviews
4.8
(139)