Functional equations

1 Dan Piponi’s functional logarithms

Nice hack, Dan Piponi — Logarithms and exponentials of functions:

A popular question in mathematics is this: given a function $f$, what is its “square root” $g$ in the sense that $g(g(x))=f(x)$. […] I want to approach the problem indirectly. When working with real numbers we can find square roots, say, by using $\sqrt{x}=\exp (\frac{1}{2}\log x)$. I want to use an analogue of this for functions. So my goal is to make sense of the idea of the logarithm and exponential of a formal power series as composable functions.

2 Tom Leinster’s course

Tom Leinster taught a punchy course on functional equations (course notes here):

Today was a warm-up, focusing on Cauchy’s functional equation: which functions $f:\mathbb{R}\to \mathbb{R}$ satisfy

$$f(x+y)=f(x)+f(y)\mathrm{\forall }x,y\in \mathbb{R}?$$

He goes on to talk about Shannon entropy from a functional equation perspective, which is a refreshing derivation.

3 References