Evelyn Lamb and Kevin Knudson have started a new podcast called My Favorite Theorem. In each episode, they have a mathematician as a guest, and that guest talks about their favorite theorem. They also have to pair that theorem with something, often a food. This inspired me to think about which theorem is my favorite (not difficult) and what food I would pair with it (more difficult).
My Favorite Theorem: Liouville’s Theorem
My favorite theorem is one I love so much that I actually made a video about it for one of my grad school applications. Liouville’s Theorem is a result in complex analysis that states that if a function is bounded and entire, then it is constant.
To break that down a bit, we start with a function f that maps the set of complex numbers to itself. We first assume that it is bounded. This means that for all complex numbers z, the magnitude of f(z) is at most some finite real number M. We next assume that f is entire, which means that for every complex number z, f(z) is holomorphic. You can think of a holomorphic function as the complex-valued equivalent of a real-valued differentiable function. It’s worth noting that being holomorphic is actually a really strong condition. In the real case, a function being differentiable doesn’t mean it’s twice differentiable, and a twice differential function might not be thrice differentiable, and so on. But a holomorphic function is infinitely differentiable. That property ends up being a big part of what makes complex analysis so beautiful.
Okay, so we have this function that is bounded and differentiable at every point in the complex plane. Then Liouville’s Theorem tells us that actually, there isn’t a lot of choice about what our function can be. It has to be constant.
The first time I saw this, I was flabbergasted. It sounds like such a strong statement, and it was one of the few results in complex analysis up to that point that had surprised me. And then my professor wrote out the proof, and it instantly became my favorite theorem because the proof takes only two or three lines.
What makes the proof so short is that it uses a couple of really high-powered results. The first result the proof uses is that all holomorphic functions are analytic, which means that locally, the function can be given by a convergent power series. The fact that in Liouville’s Theorem we consider only entire functions means that “locally” is actually the entire plane. We write the function as a Taylor series around 0 with coefficients ak. To find those coefficients, we use our second high-powered result, Cauchy’s Integral Formula. Cauchy’s Integral Formula tells us that the nth derivative of a holomorphic function at point w is related to the contour integral along a path around w of f(z)/(z-w)n+1 dz.
From there, we can bound the size of ak by finding upper bounds on this integral. To do that, we first use that f is bounded, which bounds the f(z) portion. Then we note that we’re integrating around a circle, and the denominator of the integrand is a power of the radius. We can bound the integral around the circle by multiplying our bound on the integrand by 2πR, where R is the radius. For every k greater than or equal to 1, this means we still have a positive power of R in the denominator of our upper bound. But our function is entire, so we can make that radius arbitrarily large, making the coefficient ak = 0. The zeroth coefficient, a0, is the only one that is not necessarily 0, but it’s the constant term, so our function is then constant.
So there you go! Once you know Cauchy’s Integral Formula and that holomorphic functions are analytic, Liouville’s Theorem falls right out, even though it sounds really strong and impressive.
It took me a while to come up with a food to go with Liouville’s Theorem! My first thought was durian ice cream sandwiches. Durian is really strong, but durian ice cream isn’t so much. That seemed to fit the idea of Liouville’s Theorem sounding strong but having a simple proof, but I didn’t like the implication that something was diluted. Then I thought about simple foods with fancy names and made with high-powered ingredients. One of my ideas for that was a croque monsieur, but that’s (a) not a particularly fancy name, and (b) not a fancy name at all if French is your first language. (Bread, cheese, and ham are pretty high-powered ingredients, though.)
And then I landed on it: the buckeye.
Okay, so “buckeye” isn’t the fanciest name, but it’s a little opaque, and buckeyes look fancy. But they’re actually pretty easy to make. A buckeye is a essentially peanut butter fudged dipped in chocolate, but there’s a section (ideally a circle) of peanut butter that’s not chocolate-coated. They require five ingredients and under an hour to make, and chocolate and peanut butter are pretty much the highest-power ingredients in my books.
So there you go. The buckeye is the food world’s equivalent of Liouville’s Theorem. Go forth to study complex analysis and enjoy chocolate and peanut butter in all their variations!