Proving a Problem has a Closed Form Solution

Question

I have been working on how deduce the radius of a circle based only on knowing the length of a chord within the circle and the area of the segment the chord creates. This restricts the radius to only one possibility but I can't seem to find a closed form solution for finding the radius using the given information. I am not interested in the answer to the question but I am interested in how one would go about proving whether or not this problem and others like it have a closed form solution. What field should I be looking in to or papers should I be reading in order to work on proving whether or not this problem and others like it have a closed form solution or not?

Answer 1

You have to find a mathematical description of your problem. This can be e.g. an equation, an equation system or a function. Each simple geometric shape can be described by an associated equation. For circles, it is the circle equation. If you have the equation in explict form (means in closed form), you can ask if this kind of equations can be solved explicitly (means in closed form).

You are asking for a general method for solving a given equation in closed form. There is no comprehensive method known. Here are some references for the Elementary functions and for the Liouvillian functions.

The problem of existence of elementary inverses of elementary functions is solved in Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90 and in Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math. 101 (1979) (4) 743-759.

It is easy to prove a theorem that is in a certain sense opposite to Ritt's theorem: If $f$ is a function with $f=f_1\circ\ldots\circ f_n$, where $n\in\mathbb{N}_{\ge 1}$ and $\forall i\in\{1,...n\}\colon f_i\colon D_i\subseteq\mathbb{C}^{k_i}\to\mathbb{C}^{k_i}$, for each partial inverse $\phi$ of $f$, $\phi=\phi_n\circ\ldots\circ \phi_1$ holds, where $\forall i\in\{1,...n\}\colon \phi_i$ is a partial inverse of $f_i$.

The question for closed-form solutions of equations that are values of elementary functions of rational numbers (the Elementary numbers) is asked in Chow, T. Y.: What is a Closed-Form Number? Amer. Math. Monthly 106 (1999) (5) 440-448.

The problem of solving a given ordinary equation of certain kinds in a differential field (e.g. in the Elementary functions or in the Liouvillian functions) is solved in Rosenlicht, M.: On the explicit solvability of certain transcendental equations. Publications mathématiques de l'IHÉS 36 (1969) 15-22.

Another method is decribed in Khovanskii, A.: Topological Galois Theory. Solvability and Unsolvability of Equations in Finite Terms. Springer 2014 and articles of A. Khovanskii and Y. Burda. It is applied in the following article.
Belov-Kanel, A.; Malistov, A.; Zaytsev, R.: Solvability of equations in elementary functions. Journal of Knot Theory and Its Ramifications 29 (2020) (2) 204-205

I suppose all of these methods could be extended to other classes of closed-form functions.

Look also for Lambert W, Generalized Lambert W, Hyper Lambert W and Leal functions.

Answer 2

It is hard to make a general statement to this effect. In your case, without writing down equations, I can already tell that your equations have a combination of $\sin{\theta}$ and $\theta$, which rarely produces closed form solutions. Then again, some equations can surprise: today, someone posted an awful equation to which some very bright person deduced a closed-form solution.

One result that may have some relevance is Liouville's Theorem which states what kind of functions have closed-form antiderivatives. Other than that, I know of no general statement on what may be solved with a closed form.