But what is a Laplace Transform?
Visualizing the most important tool for differential equations.
For a while, ever since I made a video about Fourier Transforms, one of the most requested topics on the channel has been its close cousin, the Laplace Transform.
I’ve been working hard on a mini-series about this topic, to be inserted into the differential equation series, and the main part is now out.
This chapter visualizes what this transform is, how it’s defined, and how it exposes the exponential pieces lurking inside a function.
Creating these visuals was a real joy. In particular, one of the steps to understanding what it’s really doing is to understand what it means to integrate a complex-valued function, and building up a machine to do that piece by piece and watching what it does is, to me at least, extremely satisfying.
The previous chapter, for those who missed it, talked about how to interpret complex exponentials, from a physical point of view, and why those functions are, in a certain sense, the “atoms of calculus”.
Next up, we’ll delve into the relationship between a derivative of a function and its Laplace Transform, which makes clear why it’s such a useful tool for differential equations. After that, we’ll step back and talk about how you could have reinvented the Laplace Transform for yourself, which walks us down a path exposing its relationship to Fourier transforms, as well as the formula for the inverse Laplace Transform. This will show a completely different way to understand how it breaks down functions as combinations of exponentials.
Stay tuned.
The most disappointing thing about getting my EE degree was when we got to Laplace transforms, we were presented with the integral without any derivation. We immediately focused on using it with constants, exponentials, sin, cos, unit step, polynomials... Then learned to use partial fractions to get the ratio of polynomials into a form to do the reverse transform. And it was useful because it's easy to express a circuit with resistors, inductors, and capacitors directly into the s plane. But just being handed that initial integral like it's unknowable magic makes one lose one's footing in a way.
This series is going a long way to getting that footing back.
Do Legendre transform next. Even postdocs in statmech I've talked to don't have an intuition for how exactly it encodes information between conjugate variables. This is an excellent paper I would love to animate that gives clean intuition: https://www2.ph.ed.ac.uk/~mevans/sp/LT070902.pdf