There is obviously a set of situations where one lack an algorithm to compute the exact solution of an equation via symbolic manipulation only, for example `x = sin(x)`. 


One has to resort to numerical analysis and iteration methods (for example) to estimate a value resolving the above.


My question is: is there an accepted mathematical proof or set of evidence demonstrating that it is impossible to resolve this equation via symbolic manipulation or is there a possibility that one comes with a solution through some clever trick in the future?


If such a proof exist, which mathematical concepts do I need to Google for further investigation?



Proof there is no algorithm to compute the intersection of a line and sinusoidal wave?
Your question is addressed in my paper [What is a closed-form number?](http://alum.mit.edu/www/tchow/closedform.pdf)


The first step is to decide which "symbols" or functions you accept as furnishing a "symbolic solution." In my paper I focus on perhaps the most restrictive set, namely exp and log and the arithmetic operations.


The second issue you have to address is whether you're asking for a symbolic expression for a *function* or for a symbolic expression for a *number* (namely, the root of some equation that you're trying to solve). This distinction seems not to have been emphasized in the literature prior to my paper. There is quite a bit of literature about closed-form *functions*, but knowing, for example, that the smallest positive root of the equation $kx = \sin x$, thought of as a *function of $k$*, has no closed-form expression, doesn't answer the question of whether there is some ad hoc expression for *specific values of $k$*.


In my paper I show that [Schanuel's conjecture](https://en.wikipedia.org/wiki/Schanuel%27s_conjecture) implies that there is no closed-form expression for the roots of equations such as $x + e^x = 0$. I think that this is pretty much the best answer we can currently give to this type of question.


