The intuition behind Fourier and Laplace transforms I was never taught in school

Finally some dark theme animations that don't destroy your eyes at 3 AM

Wow, Laplace transform is like Fourier but it looks for exponentials as well as sinusoids? Holy crap, now it makes sense why the transfer function of a control system is the laplace transform of the system dynamics, you're looking at the exponential part for stability, and the sinusoidal part can tell you about performance. Fascinating!

As an electrical engineering student who wants to understand these concepts, this video is gold!

I find this even more intuitive than 3b1b's, and that is saying something. Considering how good his explanations are, there is no over estimating how good this is! Thank you!

HOLY SHIT this is the best explanation i ever got, i've been looking for so long for videos that would give the intuition behind those notions and i couldn't find one that resonated with my logic but this one hit hard, thank you so much!!!!

That's one of the best explanations of Fourier Transform I have ever seen !!!

as a physicist, i found this to be the best, most intuitive no bullshit explanation of the fourier transform. the orthogonality and completeness relations make perfect sense now. fucking awesome job dawg!

Been working with integral transforms since ~7years and this is the first video that actually gave me a graphical understanding of the transform itself. Awesome videos, mate!

This is really what I was looking for - a visualization that is explained slowly enough to catch its underlying thought while teaching a concept like FT. My professor always reminds us of the importance of understanding the underlying working principle, but fails to explain it in a way that would allow it. So really thank you for your effort in creating these animations.

3Major1Prep?

You literally blew my mind. I have studied control systems and we've continuously used the concept of poles but after listening only do I understand the intuition behind using Laplace transforms. This is an absolute genius and a work of art! Thank you so much for this video @zach star.

14:01 can we just take a minute to appreciate how intuitively powerful this animation is?

This is one of those great pieces of content where I can come back to it after months at a time and get something new out of it each time. Great work, Zach.

As Leonard Euler, I confess that it’s too damn complicated.

Ok... I've been using LaPlace for control systems for years and never truly understood what was happening behind the scenes, I came here to understand how the hell the Fourier Series work and I'm completely mind blowed, congratulations, I'll have a hard time sleeping tonight with all the concepts and ideas taking shape in my little brain.

The connection between using impulses to find frequencies of the signal and showing a continuous fourier transform as a magnitude plot at varying frequencies, THEN showing how laplace is a generalisation. I'm amazed :) looking forward to my signals class now

I've always thought the simplest way to understand Fourier series is from a linear algebra perspective: First, define continuous function space, where the "inner product" of two functions is the integral of their product. Then the infinite set of Sin(nwt) and Cos(nwt) functions form an orthoganal basis that spans this space,. So any continuous function can be expressed as a linear combination of these "eigen functions". I haven't thought about this in years, but I think I still have the terminology right. Also, the Laplace transform is a special integration technique for solving the convolution integral. I believe that is where it came from. You should explain Laplace by introducing the concept of linear systems and superposition, then show how this leads to the convolution integral, then show how Laplace came up with a brilliant shortcut for solving these without having to do integrals. Just saying that's how I learned it centuries ago.

I've always looked to understand the fourier transform and series..... all I can say is I landed on GOLD today. Thank you very much for the video. Very intuitive and I always love intution first before I dive into the calculations.

This is incredibly helpful. Currently reviewing my understanding on Fourier Transforms and this really helps me visualize and intuitively understand it

Wooow, I used the FFT (DFT) for years now and know the math. But honestly, watching 3B1B and the Veritasium version of it rather got me confused about my previous assumption. They do an ok job but they try to squash too much detail into a short YT video. I think what you did right here is the perfect explanation of how the FFT works, on a level that one can actually really fit in a youtube video. Well done!