Functional equations

Putting the funk in functions

Miscellaneous tricks with functions. The purer side of the functional wrangling which gets you, e.g. variational approximation.

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 \(\sqrt{x}=\exp\left(\frac{1}{2}\log x\right)\). 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.

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

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

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


Aubrun, Guillaume, and Ion Nechita. 2011. The Multiplicative Property Characterizes \(\ell_p\) and \(L_p\) Norms.” Confluentes Mathematici 03 (04): 637–47.
Granas, Andrzej, and James Dugundji. 2003. Fixed Point Theory. Springer Monographs in Mathematics. New York, NY: Springer New York.

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