Necessary criterion for expressing f(a + b)

[sultan]

Homework Statement

You are given a function f(x). Determine the necessary criterion for expressing f(a + b) as a finite combination of a, b, f(a), f(b), and possibly a finite set of known mathematical constants, for any a and b from the domain of definition.

For example, if f(x) = x squared, then f(a + b) = (a + b) squared = a^2+ b^2 + 2ab = f(a) + f(b) + 2ab.
If f(x) = e^x, then f(a + b) = f(a) * f(b). For f(x) = 1/x: f(a+b)=1/ab * f(f(a)+f(b))

Clarification: by "combination" i mean any composition of the form G_f(a, b). For example, sin[(f(a/2)^2 + f(b)^2 + 15].

Relevant Equations

f(a+b)=G(f, a, b)

It's easy to see that any polynomial is function of that class. Also it seems that composition of exponent and polynomial is good as well. G_f(a, b) should be equal G_f(b, a) as the f(a+b) is symmetrical.
Does anyone have information about this or at least related to it?

 

[fresh_42]

Given any function $f$ is way too general to draw any conclusions. In case $f$ is differentiable, we have things like Rolle's theorem or the Mean Value Theorem.

 

[realJohn]

The key is understanding what kind of functions let you write f(a + b) in terms of f(a), f(b) and constants.

 

[WWGD]

I assume your expression of f(a+b) is , unless f is extremely simple, intended to be a good approximation ( Edit: In the Limit), to f(a+b) , rather than equal to it?

 

[mathwonk]

A classic example: e^(a+b) = e^a.e^b, and as consequences of this (since e^iz = cos(z) + isin(z)), the sin and cosine formulas. These "addition formulas" interested a classmate of mine at Brandeis in the late 60's, and he wrote a PhD thesis on what functions satisfy such addition formulas. I don't know the exact result but as I recall, it was essentially only the classically known ones, (including elliptic functions?), but his conditions may have been more restrictive than yours.

The devil, AI, says this:

"AI Overview
In 1970, the Brandeis University PhD thesis entitled "Functions Satisfying Addition Formulas" was authored by Charles William Miller. While a full abstract was not found, the title suggests it belongs to the field of functional equations and the theory of special functions. "

but I do not remember that name as being the correct one.

You may find interesting the book Topics in complex analysis, vol. 1, by Carl Siegel, on elliptic integrals, and perhaps continuing to "abelian integrals", a subject initiated obviously by Abel.

Thinking back briefly, there seems to be a clear connection with periodic functions, so on the complex plane, either exponential, trig or elliptic functions, periodic wrt Euclidean translations, and then more generally functions on the disc periodic wrt, lets see now, a discrete group of non euclidean motions. See Siegel's vol.2 for this topic.

The point is that functions periodic on the plane for a lattice of translations, define functions on the torus, a group, (trig functions, periodic on the real line define functions on the circle, also a group, a key point in Fourier analysis).

Functions periodic on the disc for an appropriate non Euclidean group define functions on an arbitrary Riemann surface of genus ≥ 2, (whose universal covering space is the disc). Abelian integrals are multi valued functions on a Riemann surface, defined by integrating a differential form which is everywhere holomorphic on the Riemann surface.


here you go: check this out, the classic result seems due to Weierstrass:
https://www.johndcook.com/blog/2023/07/14/addition-laws/