Any chance your figures are an homage to Geometry and the Imagination from 60 years before? They remind me of its beautiful line weights & dashing, and its intricate detail. It's interesting to consider the impact technology has had in the 100 years since. Thx for sharing.

Appreciate the recommendation!

Sure thing! timothychow.net/forcing.pdf michaelnotebook.com/df/index…

Yes! I've been wondering if discovery fiction (in the sense of @michael_nielsen) and open-exposition problems (as introduced by Timothy Chow) might start to receive greater attention. It's also interesting to consider how AI might assist in these more subjective areas.

📢Stunning math & AI update "I still did not see how to prove this inequality, but I decided to try my luck giving it to ChatGPT Pro, which... gave me a duality-based proof..."—Terence Tao (3/23/26) How Fields medalist Tao used AI in his latest work: terrytao.wordpress.com/2026/…

There's a precise connection between the discrete generating functions from de Moivre (they solve recurrence relations, i.e. difference equations) and the continuous Laplace transform (it solves differential equations). Very cool story, but that's for another day!

So there you have it. You now have the short version (in a single tweet), and a longer version haha. I hope you don't mind the extra detail. That's not the full story, of course. What about other uses of the Laplace transform, like solving differential equations?

The only convenient base is e. And the integral is most useful if it actually converges... so customarily, we use e^(-s) as our base (if s>0, this is a number that's less than one, making the integral converge for reasonable f(t)).

The Laplace transform is essentially a continuous generating function! Recall the OGF that transforms a sequence a(n) into the sum of a(n) x^n. The Laplace transform is the integral (continuous sum) of f(t) x^t. But base x isn't convenient...

The expectation is an integral, and the exponential e^(tX) gives it that Laplace vibe. And this isn't too surprising, actually... THE BIG FINALE. THE DENOUEMENT (had to check the spelling on that one haha...) DRUM ROLL PLEASE...

By design, if we determine a closed form of E[e^(tX)], we can get our moments simply by computing derivatives. Nice! Calculus for the win. How might we get a closed form? That brings us to the variation of the Laplace transform that you mentioned...

Now we recognize the series inside our single expectation E[...]. It's just e^(Xt), or as it's usually written, e^(tX). And voila! We have the _moment generating function_ E[e^(tX)] of a random variable X, a different way to specify its probability distribution.

But we want to know the generating function that encodes all this, so that we can calculate any of these coefficients at will, with a closed formula. So, we might try to gather up our terms, using linearity of expectation: = E[1 + Xt + (Xt)^2 / 2! + ...]

That's some motivation, anyway. And that means we want the EGF (recall the Taylor series example). Here's that: 1 + E[X]*t + E[X^2]/2! * t^2 + ... We just "hang" the moments onto the exponential series 1 + t + t^2/2! etc. (make them the coefficients of those terms).

We know the moments are E[X], E[X^2], ... We'd like to know the generating function that encodes these. In practical terms, we'd like to have a simple way of computing them. Calculus is designed to make hard calculations easy, so derivatives fit the bill...

Continuing in probability theory, we soon find that the _moments_ of a random variable are useful, as soon as we begin to consider the mean and variance. And this brings us to the _moment generating function_. Let's derive it again. This time, with the core concepts in place.

If we're doing calculus, we may use the derivatives of an analytic function as our sequence (here, the EGF is natural). If we're doing probability, we may use the mass of a discrete random variable as the sequence. The OGF is the _probability generating function_.

What uses we put these to depends on us. If we're doing enumerative combinatorics, we may use a counting sequence, like the combinations C(n, 0), C(n, 1), C(n, 2), ... that I mentioned. The generating function for these is (1+t)^n via the binomial theorem, as I mentioned.

Anyway, to recap, a generating function is intuitively a closed-form expression that encodes a sequence. We've seen OGFs and EGFs. There are others. In, OGFs and EGFs, the terms of the sequence become the coefficients in the power series with terms t^n and t^n/n!, respectively.