I'm particularly interested in transcendental equations but searching in internet gives me only results about the classical linear-exponential equation (which is solved with Lambert's W) and its generalizations or some trigonometric equations like $\cos x=x$ or $\tan ax=bx$ (which are solved with Riemann's method). So I was wondering what families or kinds of transcendental equations do we know how to solve in closed form ? I know that with Newton's method or other approximation methods we can almost solve every equation but I'm searching for closed form ones.
Are there any books that explains general method on how to solve some equations other than the few I mentioned ?
Or some articles whose analyze some kinds of them ?
Or can directly someone list me all the families of equation we can solve in closed form ?
Thx for every contribute :)
"Closed form" means expressions of allowed functions (Wikipedia: Closed-form expression). If an equation is solvable in closed form depends therefore on the functions you allow.
A general method for solving a given equation $H(x)=0$ is to apply the compositional inverse $H^{-1}$ of $H$: $x=H^{-1}(0)$. In general, $H$ and $H^{-1}$ are correspondences. But often it is possible to split the problem into subproblems where $H$ and $H^{-1}$ are functions.
For applying this method, $H$ and $H^{-1}$ have to be known. That means they have to be in closed form.
For the elementary functions, there is a structure theorem of J. F. Ritt (mentioned below) that can help to decide if a given kind of equations of elementary functions can be solved by transforming a given equation by applying only elementary functions.
The elementary functions are according to Liouville and Ritt those functions of one variable which are obtained in a finite number of steps by performing algebraic operations and taking exponentials and logarithms (Wikipedia: Elementary function).
The incomprehensibly unfortunately hardly noticed theorem of Joseph Fels Ritt in [Ritt 1925] Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90 answers which kinds of Elementary functions can have an inverse which is an Elementary function. You can also take the method of Rosenlicht, M.: On the explicit solvability of certain transcendental equations. Publications mathématiques de l'IHÉS 36 (1969) 15-22.
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 $f_i$.
Lin and Chow prove that certain kinds of equations aren't solvable by "Elementary numbers.
Ferng-Ching Lin: Schanuel's Con-jecture Implies Ritt's Conjectures. Chin. J. Math. 11 (1983) (1) 41-50
Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448
If $f$ can be decomposed into compositions of algebraic functions and other known Standard functions than $\exp$ and $\ln$, an analog theorem to the theorem of Ritt of [Ritt 1925] could be applied. I hope to prove such a generalization of Ritt's theorem for this class of functions.